https://conwaylife.com/w/api.php?action=feedcontributions&feedformat=atom&user=DavidConant 2026-11-11T03:05:30Z User contributions MediaWiki 1.45.3 https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=156973 2024-11-20T00:08:57Z

<p>DavidConant: /* Logic gates */</p> <hr /> <div>{{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if it is currently dead and exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With [[Hensel notation]]&lt;ref group=note&gt;[[Hensel notation]] is a [[rulestring]] notation for isotropic rules, supported in CA simulation software including [[Golly]] and [[LifeViewer]].&lt;/ref&gt; this is expressed as B2ae/S. The rule was included as a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Signal logic==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of Margolus oscillators (see [[#Other patterns|the following section]]), the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit stream generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes; and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 11, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo2$13bo39bo$12bo39bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = IBSG and a p8 banana gun based on it<br /> |style = width:300px;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form {{nowrap|8''n'' + 1}} are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is {{eq|{{times|3|8}}|24}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 5, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo2$21bo$20bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The #17 BSG<br /> |style = width:300px;<br /> }}<br /> <br /> It appears that the period of {{nowrap|''n''-th}} BSG follows Sloane's {{OEIS|A086839}} and {{nowrap|log₂(period + 1)}} almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> From another point of view, any BSG with width {{nowrap|8 ''n'' + 1}} generates the output of a linear-feedback shift register&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/Linear-feedback_shift_register Linear-feedback shift register] at Wikipedia&lt;/ref&gt; (LFSR). Only numbers from Sloane's {{OEIS|A123399}} generate maximum length LFSRs with periods of {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt; - 1}}.<br /> <br /> The highest number listed is 419 which has a period of {{nowrap|2&lt;sup&gt;419&lt;/sup&gt; - 1}}, about {{times|1.35|10&lt;sup&gt;126&lt;/sup&gt;}} before the output repeats.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,247<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,991<br /> |}<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt;-th}} place and the rest is sea of 0's. This is listed in Sloane's {{OEIS|A036987}}, the Fredholm-Rueppel sequence. An equivalent device akin to a [[caber tosser]] can be designed with only one unbounded replicator and two IBSG's, so that signals from the other stationary end can be fed into circuitry easily; the {{nowrap|''n''-th}} output starting from 0 arrives at the shown location at generation {{nowrap|2&lt;sup&gt;''n'' + 3&lt;/sup&gt; - 8}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 6, y = 5, rule = B2ae/S<br /> o4bo$o4bo2$3bo$2bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = XBSG, or Fredholm-Rueppel sequence generator<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 35, y = 34, rule = B2ae/S<br /> 21bo$20bo6$18b2o3$11bo$10bo7b2obo$10bo7b2o2bo2$15bo$16bo$18b2o6$20bo$<br /> 21bo2$11b2o5bo15bo$9bo3bo4bo15bo$8bo2$3bo3bo2bobo3bo3bo$3bo3bo8bo3bo$b<br /> o20bo$o11bo10bo$11bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The equivalent device with only one side of the [[bounding box]] growing infinitely<br /> |style = width:300px;<br /> }}<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to construct logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an [[intermittent stream|intermittent]] p8 input stream with an IBSG coming orthogonally at certain timing. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> {{EmbedViewer<br /> |rle = x = 98, y = 8, rule = B2ae/S<br /> 18bo47bo$17bo47bo2$o17bo4bo17bo24bo4bo24bo$bo16bo4bo16bo25bo4bo25bo2$<br /> 21bo47bo$20bo47bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = The BSGs used as signal sources are: {{eq|A|BSG2 width 17}}, {{eq|B|BSG3 width 25}}<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 77, y = 45, rule = B2ae/S<br /> 3bo$4bo16$6bo$3b2o2bo2$o$bo2$3b2o14$21bo31bo$20bo31bo2$4bo7bo8bo4bo8b<br /> o17bo4bo17bo$3bo7bo9bo4bo9bo16bo4bo16bo2$24bo31bo$23bo31bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = NOT A<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 179, y = 67, rule = B2ae/S<br /> 3bo$4bo8$6bo$3b2o2bo2$o59bo$bo59bo2$3b2o9$4bo$3bo4$21bo$20bo2$4bo16bo<br /> 4bo16bo$3bo17bo4bo17bo2$24bo$23bo23$51bo$50bo2$26bo24bo4bo24bo17bo4bo<br /> 17bo24bo4bo24bo$25bo25bo4bo25bo16bo4bo16bo25bo4bo25bo2$54bo47bo47bo$53b<br /> o47bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A AND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 205, y = 103, rule = B2ae/S<br /> 3bo$4bo23$53bo$6bo47bo$3b2o2bo2$o$bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o<br /> 2bo$3bo$30bo$31bo2$33b2o14$51bo$50bo2$34bo7bo8bo4bo8bo$33bo7bo9bo4bo9b<br /> o2$54bo$53bo13$71bo53bo47bo$70bo53bo47bo2$62bo8bo4bo8bo22bo16bo4bo16b<br /> o25bo4bo25bo$61bo9bo4bo9bo20bo17bo4bo17bo24bo4bo24bo2$74bo53bo47bo$73b<br /> o53bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NAND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 219, y = 107, rule = B2ae/S<br /> 3bo$4bo7$53bo$54bo$3b2o2$bo$o2$3b2o2bo$6bo8$4bo$3bo4$26bo$27bo2$4bo16b<br /> o4bo16bo$3bo17bo4bo17bo2$23bo$24bo23$56bo$57bo2$26bo24bo4bo24bo$25bo25b<br /> o4bo25bo2$53bo$54bo33$96bo34bo55bo$97bo32bo55bo2$82bo8bo4bo8bo8bo16bo<br /> 4bo16bo8bo24bo4bo24bo$81bo9bo4bo9bo6bo17bo4bo17bo6bo25bo4bo25bo2$93bo<br /> 40bo55bo$94bo38bo55bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A OR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 176, y = 83, rule = B2ae/S<br /> 3bo$4bo24$6bo$3b2o2bo2$o56bo$bo56bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o2b<br /> o$3bo$30bo$31bo2$33b2o14$51bo39bo52bo$50bo39bo52bo2$34bo7bo8bo4bo8bo8b<br /> o16bo4bo16bo5bo24bo4bo24bo$33bo7bo9bo4bo9bo6bo17bo4bo17bo3bo25bo4bo25b<br /> o2$54bo39bo52bo$53bo39bo52bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NOR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 250, y = 188, rule = B2ae/S<br /> 64bo$63bo2$35bo25bo4bo25bo$36bo24bo4bo24bo2$67bo$66bo6$33bo$34bo16$36b<br /> o$33b2o2bo2$30bo38bo$31bo36bo2$33b2o15$3bo$4bo$34bo$33bo6$6bo$3b2o2bo<br /> 2$o$bo2$3b2o9$4bo$3bo19$63bo$64bo24$66bo$63b2o2bo2$60bo38bo$61bo38bo2$<br /> 63b2o24$91bo$64bo25bo$63bo$74bo16bo4bo16bo$73bo17bo4bo17bo2$94bo$93bo<br /> 23$121bo39bo56bo$120bo39bo56bo2$112bo8bo4bo8bo8bo16bo4bo16bo9bo24bo4b<br /> o24bo$111bo9bo4bo9bo6bo17bo4bo17bo7bo25bo4bo25bo2$124bo39bo56bo$123bo<br /> 39bo56bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A XOR B<br /> |style = width:300px;<br /> }}<br /> <br /> ==Other patterns==<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 5, rule = B2ae/S<br /> bo$o$o$bo$2bo!<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> |position = right<br /> |style = width:300px;<br /> }}<br /> <br /> Despite its name, there are more &quot;naturalistic&quot; patterns of note beyond logic circuitry in Logic Rule. Consider a class of extended photons similar to the moon, with one &quot;wing&quot; fixed and the other of variable length. The first two cases are the banana (with length 0) and the moon (with length 1). At length 2 as shown to the right, the pattern is a [[predecessor]] of a period-2 spaceship. Depending on the length, these extended photons have different [[fate]]s and may become either a [[spaceship]] or a [[puffer]]. The first few cases are tabulated below, with links to the corresponding objects on [[Catagolue]] at their first occurrence.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view the fates of this class of patterns.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;wikitable sortable&quot;<br /> ! Length !! Fate<br /> |-<br /> | 0 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_16|patternname=A p2 spaceship (banana)}}<br /> |-<br /> | 1 || {{LinkCatagolue|style=raw|rule=b2aes|xq1_69|patternname=A p1 spaceship (moon)}}<br /> |-<br /> | 2 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_21ic|patternname=A p2 spaceship}}<br /> |-<br /> | 3 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_211ak8|patternname=A p2 spaceship}}<br /> |-<br /> | 4 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_8kahzw12442|patternname=A p4 spaceship}}<br /> |-<br /> | 5 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_0g8ka5ak8z34|patternname=A p4 spaceship}}<br /> |-<br /> | 6 || A p4 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2_12}} (duoplet, width 1) per period<br /> |-<br /> | 7 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_69g8g0g8gzx125ala52|patternname=A p4 spaceship}}<br /> |-<br /> | 8 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211ak8gzx4alalak8zy0g8521zx1|patternname=A p8 spaceship}}<br /> |-<br /> | 9 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_x25ak8gzxg8kalala4zx8la521z1221|patternname=A p8 spaceship}}<br /> |-<br /> | 10 || A p8 puffer, 2 xp2_12 per period<br /> |-<br /> | 11 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211248gzxlx1248gzy0hay2hazx58g0g8421|patternname=A p8 spaceship}}<br /> |-<br /> | 12 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp6_1248}} (width 1) per period<br /> |-<br /> | 13 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4_25ak8}} (width 2) per period<br /> |-<br /> | 14 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_0g8421z1}} (width 1) per period<br /> |-<br /> | 15 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_y1g842101248gzxahy7hazxg85y1g8421zxawg8421z488421|patternname=A p8 spaceship}}<br /> |-<br /> | 16 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy7hazy21248wg8421zy1g8ka521zx2521|patternname=A p16 spaceship}}<br /> |-<br /> | 17 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy712kzy21248y0g8421zy1g842h8421zxa18421|patternname=A p16 spaceship}}<br /> |-<br /> | 18 || A p16 puffer, 5 xp2_12 and 1 xp6_1248 per period<br /> |-<br /> | 19 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y4g842zy1g8421xg8421zxahxg8421zy1101|patternname=A p16 spaceship}}<br /> |-<br /> | 20 || A p16 puffer, 3 xp6_1248 per period<br /> |-<br /> | 21 || A p16 puffer, 2 xp4_25ak8 per period<br /> |-<br /> | 22 || A p16 puffer, 2 xp14_0g8421z1 per period<br /> |-<br /> | 23 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248zy1g8421y7g8421zxahy7g8421zy11248g0g8421|patternname=A p16 spaceship}}<br /> |-<br /> | 24 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_xg8421z421}} (width 1) per period<br /> |-<br /> | 25 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp12_xg8ka52z4a521}} (width 2) per period<br /> |-<br /> | 26 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248g}} (width 1) per period<br /> |-<br /> | 27 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp8_xg8kalak8z4alala521zx1}} (width 4) per period<br /> |-<br /> | 28 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_y3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 29 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y3g8ka52zwg8ka521z2521}} (width 2) per period<br /> |-<br /> | 30 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_1248gzy11248gzy61248}} (width 1) per period<br /> |-<br /> | 31 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahynhazy11248gydg8421zy61248gy3g8421zyb1240421|patternname=A p16 spaceship}}<br /> |-<br /> | 32 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn12kzy11248gyfg8421zy61248gy5g8421zyb12k0g8421zy8g8ka521zy3g8ka521zx8ka521|patternname=A p32 spaceship}}<br /> |-<br /> | 33 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_x58h248gzy21248h248gzy71248h248gzychaw1248gzy7g8421y51248gzy2g8421yf1248gzxk21yng85zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 34 || A p32 puffer, 14 xp2_12, 4 xp6_1248 and 1 xp4_25ak8 per period<br /> |-<br /> | 35 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gylg85zy61248gybg8421zyb12ky3g8421zy8g8421xg8421zy3g8421xg8421zx8421xg8421zx1240421|patternname=A p32 spaceship}}<br /> |-<br /> | 36 || A p32 puffer, 5 xp2_12, 4 xp6_1248, 1 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 37 || A p32 puffer, 5 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 38 || A p32 puffer, 4 xp2_12, 4 xp14_0g8421z1 and 1 xp14_xg8421z421 per period<br /> |-<br /> | 39 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_y1g842101248gzxahy71248gzy11248gy71248gzy61248gy71248gzyb12ky91248gzy8g8421yf1248gzy3g8421yng85zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p32 spaceship}}<br /> |-<br /> | 40 || A p32 puffer, 3 xp14_xg8421z421 and 1 xp2_12 per period<br /> |-<br /> | 41 || A p32 puffer, 3 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 42 || A p32 puffer, 2 xp2_12, 2 xp6_1248, 2 xp62_1248gzy11248g and 1 xp4_25ak8 per period<br /> |-<br /> | 43 || A p32 puffer, 2 xp8_xg8kalak8z4alala521zx1 per period<br /> |-<br /> | 44 || A p32 puffer, 2 xp126_y3g8421zwg8421z21 and 1 xp14_0g8kala4z12521 per period<br /> |-<br /> | 45 || A p32 puffer, 2 xp28_y3g8ka52zwg8ka521z2521 per period<br /> |-<br /> | 46 || A p32 puffer, 2 xp30_1248gzy11248gzy61248 per period<br /> |-<br /> | 47 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp12kzy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy41248gy7g8421zy91248g0g8421|patternname=A p32 spaceship}}<br /> |-<br /> | 48 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_y7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 49 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y7g8ka52zy2g8ka521z0g8ka521z121}} (width 2) per period<br /> |-<br /> | 50 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_y9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 51 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp24_8kalak8gzw125alalak8gzy3125alalak8gzy8125a521}} (width 4) per period<br /> |-<br /> | 52 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_1248gzy11248gzy61248gzyb1248g}} (width 1) per period<br /> |-<br /> | 53 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp124_ybg8ka52zy6g8ka521zy1g8ka521zg8ka521z01}} (width 2) per period<br /> |-<br /> | 54 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4094_ydg8421zy8g8421zy3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 55 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp16_wg842101248gzahy71248gzw1248gy71248gzy31248gy3g842zy81240421}} (width 8) per period<br /> |-<br /> | 56 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2046_1248gzy11248gzy61248gzyb1248gzyg1248}} (width 1) per period<br /> |-<br /> | 57 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp252_25ak8gzy0125ak8gzy5125ak8gzya125ak8gzyf125ak8}} (width 2) per period<br /> |-<br /> | 58 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_yhg8421zycg8421zy7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 59 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp56_8kalak8gzw125alalak8gzy3125alalak8gzy8125alalak8gzyd125alalak8zyi121}} (width 4) per period<br /> |-<br /> | 60 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp32766_yjg8421zyeg8421zy9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 61 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp60_yjg8ka52zyeg8ka521zy9g8ka521zy4g8ka521zxg8ka521z4a521}} (width 2) per period<br /> |-<br /> | 62 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248gzy61248gzyb1248gzyg1248gzyl1248g}} (width 1) per period<br /> |-<br /> | 63 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_ypg842101248gzykg8421y71248gzyfg8421yh1248gzyag8421yr1248gzy5g8421yzw1248gzy0g8421yzy81248gzx58gyzyeg85zy21248gyzy4g8421zy71248gyxg8421zychayqg8421zy7g8421yng8421zy2g8421yng8421zxk21yng8421zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 64 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzycg842zy41248gyzy2g8421zy91248gyvg8421zye1248gylg8421zyj1248gybg8421zyo1248gy1g8421zyrg8l8421zymg8ka521zyhg8ka521zycg8ka521zy7g8ka521zy2g8ka521zx4a521|patternname=A p64 spaceship}}<br /> |-<br /> | 65 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzyeg85zy41248gyzy4g8421zy91248gyxg8421zye1248gyng8421zyj1248gydg8421zyo1248gy3g8421zyrg850g8421zymg842h8421zyhg842h8421zycg842h8421zy7g842h8421zy2g842h8421zxk2h8421|patternname=A p64 spaceship}}<br /> |-<br /> | 66 || A p64 puffer, 41 xp2_12, 8 xp6_1248, 6 xp4_25ak8, 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 67 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_xk2101248gzy01248gx1248gzy51248gx1248gzya1248gx1248gzyf1248gx1248gzyk1248gx1248gzyp1248gx1248gzyqg842y41248gzylg8421yd1248gzygg8421yn1248gzybg8421yx1248gzy6g8421yzy41248gzy1g8421yzye12kzxahyzygg8421zy11248gyzy6g8421zy61248gyzg8421zyb12kyrg8421zy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p64 spaceship}}<br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Several trends can be found from the list for positive lengths.<br /> <br /> * Regardless of the type, a length-''n'' extended photon has a [[period]] of 2&lt;sup&gt;floor(log₂(''n''))&lt;/sup&gt;, i.e. Sloane's {{OEIS|A053644}}(''n'').<br /> * A length-''n'' extended photon will converge to a spaceship if ''n'' is in Sloane's {{OEIS|A099627}}. A vanishing T-square fractal&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/T-square_%28fractal%29 T-square fractal] at Wikipedia&lt;/ref&gt; can be seen at the tail for large enough ''n'' close to some power of 2.<br /> * A length-''n'' extended photon will converge to a [[clean]] puffer leaving one oscillator per period if ''n'' not in A099627 and {{nowrap|{{times|3|2&lt;sup&gt;''k''&lt;/sup&gt;}} ≤ ''n'' ≤ 2&lt;sup&gt;''k'' + 2&lt;/sup&gt;}} for some positive integer ''k''. Its trailing end is divided into two halves, one becomes a disappearing [[spark]] from a diagonal line of {{nowrap|2&lt;sup&gt;''k'' + 1&lt;/sup&gt; - 1}} cells while another evolves into the oscillator.<br /> ** The oscillators are square and oscillate according to a [[block cellular automaton]] on the [[Margolus neighbourhood]]. Previously the phenomenon was observed and studied in an outer-totalistic rule called [[2×2]], where the oscillators are made up of {{times|2|2}} [[block]]s, followed by several other non-totalistic rules later.&lt;ref name=&quot;post54201&quot; /&gt;&lt;ref name=&quot;post54268&quot; /&gt; Here the oscillators can be seen as composed of [[tub]]s, though it is not the smallest element because width-1 forms as labelled above exist. A width-1 diagonal line of 2''m'' cells has a period of Sloane's {{OEIS|A160657}}(''m''), and appears in the list above at length ''n'' = {{OEIS|A079946}}(''m''), or Sloane's {{OEIS|A049039}}(''m'')-th place among oscillators. Other diagonal widths are resulted from [[inflation]] by a factor of {{times|2&lt;sup&gt;''k''&lt;/sup&gt;|2&lt;sup&gt;''k''&lt;/sup&gt;}}, which also multiplies the period by 2&lt;sup&gt;''k''&lt;/sup&gt;.<br /> * An extended photon is [[dirty]] if its length does not satisfy any requirements above. In particular, a period-2&lt;sup&gt;''k''&lt;/sup&gt; extended photon produces the largest amount of [[ash]] if its length is {{nowrap|2&lt;sup&gt;''k''&lt;/sup&gt; + 2}}. Behind a puffer of this kind, straight diagonal lines replicate and decay in a complex manner leading to configurations with some self-similarity.<br /> <br /> [[Image:b2aes_complex_puffer.png|framed|center|The tail of a westbound length-16386 extended photon at generation 40960, showing both decaying and settled ash fields. Scale is 2^5:1.]]<br /> <br /> : However, no matter how dirty the ash is, it is guaranteed not to emit photons or replicators because B2e-governed evolution will only have new cells born at the same checkerboard parity, therefore prohibiting domino frontend from forming.<br /> * Regardless of the type, the reaction [[envelope]] of the tail is limited to one side of the extended photon. This implies that it can have two halves extended independently; not only their length can be different, but also they can operate at different periods with any [[phase]] difference.<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 3, rule = B2ae/S<br /> 2o$2bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Apart from the long extended photons above, the emergence of patterns may also be observed with smaller seeds, for instance the diagonally [[symmetry|symmetric]] [[tetraplet]] shown to the right. If random [[soup]]s are put behind the long diagonal line after a while to perturb the [[wavestretcher]], the outcome will be more complicated.&lt;ref name=&quot;post39221&quot; /&gt;<br /> <br /> ==Notes==<br /> &lt;references group=note&gt;<br /> &lt;/references&gt;<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54201&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54201<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 25, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54268&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54268<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 26, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post39221&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 39221<br /> |title = Re: Thread for basic questions<br /> |author = blah<br /> |date = January 10, 2017<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - note: the author claims to have all of the files for this web site, but it's not currently hosted anywhere. NOTE: The files for the original website are now permanently stored on a Google shared folder.<br /> * [https://drive.google.com/drive/folders/11x-Z3CzKPumuI5tqHWfWvY__7Yxq0urY Google Shared Folder] where all of the files for the original Logic Rule web page are stored.<br /> * {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> * {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where the author originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.<br /> <br /> [[Category:Isotropic non-totalistic rules]]<br /> [[Category:Rules without survival conditions]]<br /> [[Category:Rules without still lives]]</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=121052 2022-12-17T21:40:41Z

<p>DavidConant: /* External links */</p> <hr /> <div>{{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation {{iff}} it is currently dead and exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With [[Hensel notation]]&lt;ref group=note&gt;[[Hensel notation]] is a [[rulestring]] notation for isotropic rules, supported in CA simulation software including [[Golly]] and [[LifeViewer]].&lt;/ref&gt; this is expressed as B2ae/S. The rule was included as a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Signal logic==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of Margolus oscillators (see [[#Other patterns|the following section]]), the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit stream generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 11, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo2$13bo39bo$12bo39bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = IBSG and a p8 banana gun based on it<br /> |style = width:300px;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form {{nowrap|8''n'' + 1}} are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is {{eq|{{times|3|8}}|24}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 5, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo2$21bo$20bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The #17 BSG<br /> |style = width:300px;<br /> }}<br /> <br /> It appears that the period of {{nowrap|''n''-th}} BSG follows Sloane's {{OEIS|A086839}} and {{nowrap|log₂(period + 1)}} almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> From another point of view, any BSG with width {{nowrap|8 ''n'' + 1}} generates the output of a linear-feedback shift register&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/Linear-feedback_shift_register Linear-feedback shift register] at Wikipedia&lt;/ref&gt; (LFSR). Only numbers from Sloane's {{OEIS|A123399}} generate maximum length LFSRs with periods of {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt; - 1}}.<br /> <br /> The highest number listed is 419 which has a period of {{nowrap|2&lt;sup&gt;419&lt;/sup&gt; - 1}}, about {{times|1.35|10&lt;sup&gt;126&lt;/sup&gt;}} before the output repeats.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,247<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,991<br /> |}<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt;-th}} place and the rest is sea of 0's. This is listed in Sloane's {{OEIS|A036987}}, the Fredholm-Rueppel sequence. An equivalent device akin to a [[caber tosser]] can be designed with only one unbounded replicator and two IBSG's, so that signals from the other stationary end can be fed into circuitry easily; the {{nowrap|''n''-th}} output starting from 0 arrives at the shown location at generation {{nowrap|2&lt;sup&gt;''n'' + 3&lt;/sup&gt; - 8}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 6, y = 5, rule = B2ae/S<br /> o4bo$o4bo2$3bo$2bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = XBSG, or Fredholm-Rueppel sequence generator<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 35, y = 34, rule = B2ae/S<br /> 21bo$20bo6$18b2o3$11bo$10bo7b2obo$10bo7b2o2bo2$15bo$16bo$18b2o6$20bo$<br /> 21bo2$11b2o5bo15bo$9bo3bo4bo15bo$8bo2$3bo3bo2bobo3bo3bo$3bo3bo8bo3bo$b<br /> o20bo$o11bo10bo$11bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The equivalent device with only one side of the [[bounding box]] growing infinitely<br /> |style = width:300px;<br /> }}<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an [[intermittent stream|intermittent]] p8 input stream with an IBSG coming orthogonally at certain timing. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> {{EmbedViewer<br /> |rle = x = 98, y = 8, rule = B2ae/S<br /> 18bo47bo$17bo47bo2$o17bo4bo17bo24bo4bo24bo$bo16bo4bo16bo25bo4bo25bo2$<br /> 21bo47bo$20bo47bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = The BSGs used as signal sources are: {{eq|A|BSG2 width 17}}, {{eq|B|BSG3 width 25}}<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 77, y = 45, rule = B2ae/S<br /> 3bo$4bo16$6bo$3b2o2bo2$o$bo2$3b2o14$21bo31bo$20bo31bo2$4bo7bo8bo4bo8b<br /> o17bo4bo17bo$3bo7bo9bo4bo9bo16bo4bo16bo2$24bo31bo$23bo31bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = NOT A<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 179, y = 67, rule = B2ae/S<br /> 3bo$4bo8$6bo$3b2o2bo2$o59bo$bo59bo2$3b2o9$4bo$3bo4$21bo$20bo2$4bo16bo<br /> 4bo16bo$3bo17bo4bo17bo2$24bo$23bo23$51bo$50bo2$26bo24bo4bo24bo17bo4bo<br /> 17bo24bo4bo24bo$25bo25bo4bo25bo16bo4bo16bo25bo4bo25bo2$54bo47bo47bo$53b<br /> o47bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A AND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 205, y = 103, rule = B2ae/S<br /> 3bo$4bo23$53bo$6bo47bo$3b2o2bo2$o$bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o<br /> 2bo$3bo$30bo$31bo2$33b2o14$51bo$50bo2$34bo7bo8bo4bo8bo$33bo7bo9bo4bo9b<br /> o2$54bo$53bo13$71bo53bo47bo$70bo53bo47bo2$62bo8bo4bo8bo22bo16bo4bo16b<br /> o25bo4bo25bo$61bo9bo4bo9bo20bo17bo4bo17bo24bo4bo24bo2$74bo53bo47bo$73b<br /> o53bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NAND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 219, y = 107, rule = B2ae/S<br /> 3bo$4bo7$53bo$54bo$3b2o2$bo$o2$3b2o2bo$6bo8$4bo$3bo4$26bo$27bo2$4bo16b<br /> o4bo16bo$3bo17bo4bo17bo2$23bo$24bo23$56bo$57bo2$26bo24bo4bo24bo$25bo25b<br /> o4bo25bo2$53bo$54bo33$96bo34bo55bo$97bo32bo55bo2$82bo8bo4bo8bo8bo16bo<br /> 4bo16bo8bo24bo4bo24bo$81bo9bo4bo9bo6bo17bo4bo17bo6bo25bo4bo25bo2$93bo<br /> 40bo55bo$94bo38bo55bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A OR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 176, y = 83, rule = B2ae/S<br /> 3bo$4bo24$6bo$3b2o2bo2$o56bo$bo56bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o2b<br /> o$3bo$30bo$31bo2$33b2o14$51bo39bo52bo$50bo39bo52bo2$34bo7bo8bo4bo8bo8b<br /> o16bo4bo16bo5bo24bo4bo24bo$33bo7bo9bo4bo9bo6bo17bo4bo17bo3bo25bo4bo25b<br /> o2$54bo39bo52bo$53bo39bo52bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NOR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 250, y = 188, rule = B2ae/S<br /> 64bo$63bo2$35bo25bo4bo25bo$36bo24bo4bo24bo2$67bo$66bo6$33bo$34bo16$36b<br /> o$33b2o2bo2$30bo38bo$31bo36bo2$33b2o15$3bo$4bo$34bo$33bo6$6bo$3b2o2bo<br /> 2$o$bo2$3b2o9$4bo$3bo19$63bo$64bo24$66bo$63b2o2bo2$60bo38bo$61bo38bo2$<br /> 63b2o24$91bo$64bo25bo$63bo$74bo16bo4bo16bo$73bo17bo4bo17bo2$94bo$93bo<br /> 23$121bo39bo56bo$120bo39bo56bo2$112bo8bo4bo8bo8bo16bo4bo16bo9bo24bo4b<br /> o24bo$111bo9bo4bo9bo6bo17bo4bo17bo7bo25bo4bo25bo2$124bo39bo56bo$123bo<br /> 39bo56bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A XOR B<br /> |style = width:300px;<br /> }}<br /> <br /> ==Other patterns==<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 5, rule = B2ae/S<br /> bo$o$o$bo$2bo!<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> |position = right<br /> |style = width:300px;<br /> }}<br /> <br /> Despite its name, there are more &quot;naturalistic&quot; patterns of note beyond logic circuitry in Logic Rule. Consider a class of extended photons similar to the moon, with one &quot;wing&quot; fixed and the other of variable length. The first two cases are the banana (with length 0) and the moon (with length 1). At length 2 as shown to the right, the pattern is a [[predecessor]] of a period-2 spaceship. Depending on the length, these extended photons have different [[fate]]s and may become either a [[spaceship]] or a [[puffer]]. The first few cases are tabulated below, with links to the corresponding objects on [[catagolue]] at their first occurence.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view the fates of this class of patterns.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;wikitable sortable&quot;<br /> ! Length !! Fate<br /> |-<br /> | 0 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_16|patternname=A p2 spaceship (banana)}}<br /> |-<br /> | 1 || {{LinkCatagolue|style=raw|rule=b2aes|xq1_69|patternname=A p1 spaceship (moon)}}<br /> |-<br /> | 2 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_21ic|patternname=A p2 spaceship}}<br /> |-<br /> | 3 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_211ak8|patternname=A p2 spaceship}}<br /> |-<br /> | 4 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_8kahzw12442|patternname=A p4 spaceship}}<br /> |-<br /> | 5 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_0g8ka5ak8z34|patternname=A p4 spaceship}}<br /> |-<br /> | 6 || A p4 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2_12}} (duoplet, width 1) per period<br /> |-<br /> | 7 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_69g8g0g8gzx125ala52|patternname=A p4 spaceship}}<br /> |-<br /> | 8 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211ak8gzx4alalak8zy0g8521zx1|patternname=A p8 spaceship}}<br /> |-<br /> | 9 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_x25ak8gzxg8kalala4zx8la521z1221|patternname=A p8 spaceship}}<br /> |-<br /> | 10 || A p8 puffer, 2 xp2_12 per period<br /> |-<br /> | 11 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211248gzxlx1248gzy0hay2hazx58g0g8421|patternname=A p8 spaceship}}<br /> |-<br /> | 12 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp6_1248}} (width 1) per period<br /> |-<br /> | 13 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4_25ak8}} (width 2) per period<br /> |-<br /> | 14 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_0g8421z1}} (width 1) per period<br /> |-<br /> | 15 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_y1g842101248gzxahy7hazxg85y1g8421zxawg8421z488421|patternname=A p8 spaceship}}<br /> |-<br /> | 16 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy7hazy21248wg8421zy1g8ka521zx2521|patternname=A p16 spaceship}}<br /> |-<br /> | 17 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy712kzy21248y0g8421zy1g842h8421zxa18421|patternname=A p16 spaceship}}<br /> |-<br /> | 18 || A p16 puffer, 5 xp2_12 and 1 xp6_1248 per period<br /> |-<br /> | 19 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y4g842zy1g8421xg8421zxahxg8421zy1101|patternname=A p16 spaceship}}<br /> |-<br /> | 20 || A p16 puffer, 3 xp6_1248 per period<br /> |-<br /> | 21 || A p16 puffer, 2 xp4_25ak8 per period<br /> |-<br /> | 22 || A p16 puffer, 2 xp14_0g8421z1 per period<br /> |-<br /> | 23 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248zy1g8421y7g8421zxahy7g8421zy11248g0g8421|patternname=A p16 spaceship}}<br /> |-<br /> | 24 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_xg8421z421}} (width 1) per period<br /> |-<br /> | 25 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp12_xg8ka52z4a521}} (width 2) per period<br /> |-<br /> | 26 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248g}} (width 1) per period<br /> |-<br /> | 27 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp8_xg8kalak8z4alala521zx1}} (width 4) per period<br /> |-<br /> | 28 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_y3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 29 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y3g8ka52zwg8ka521z2521}} (width 2) per period<br /> |-<br /> | 30 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_1248gzy11248gzy61248}} (width 1) per period<br /> |-<br /> | 31 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahynhazy11248gydg8421zy61248gy3g8421zyb1240421|patternname=A p16 spaceship}}<br /> |-<br /> | 32 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn12kzy11248gyfg8421zy61248gy5g8421zyb12k0g8421zy8g8ka521zy3g8ka521zx8ka521|patternname=A p32 spaceship}}<br /> |-<br /> | 33 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_x58h248gzy21248h248gzy71248h248gzychaw1248gzy7g8421y51248gzy2g8421yf1248gzxk21yng85zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 34 || A p32 puffer, 14 xp2_12, 4 xp6_1248 and 1 xp4_25ak8 per period<br /> |-<br /> | 35 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gylg85zy61248gybg8421zyb12ky3g8421zy8g8421xg8421zy3g8421xg8421zx8421xg8421zx1240421|patternname=A p32 spaceship}}<br /> |-<br /> | 36 || A p32 puffer, 5 xp2_12, 4 xp6_1248, 1 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 37 || A p32 puffer, 5 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 38 || A p32 puffer, 4 xp2_12, 4 xp14_0g8421z1 and 1 xp14_xg8421z421 per period<br /> |-<br /> | 39 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_y1g842101248gzxahy71248gzy11248gy71248gzy61248gy71248gzyb12ky91248gzy8g8421yf1248gzy3g8421yng85zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p32 spaceship}}<br /> |-<br /> | 40 || A p32 puffer, 3 xp14_xg8421z421 and 1 xp2_12 per period<br /> |-<br /> | 41 || A p32 puffer, 3 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 42 || A p32 puffer, 2 xp2_12, 2 xp6_1248, 2 xp62_1248gzy11248g and 1 xp4_25ak8 per period<br /> |-<br /> | 43 || A p32 puffer, 2 xp8_xg8kalak8z4alala521zx1 per period<br /> |-<br /> | 44 || A p32 puffer, 2 xp126_y3g8421zwg8421z21 and 1 xp14_0g8kala4z12521 per period<br /> |-<br /> | 45 || A p32 puffer, 2 xp28_y3g8ka52zwg8ka521z2521 per period<br /> |-<br /> | 46 || A p32 puffer, 2 xp30_1248gzy11248gzy61248 per period<br /> |-<br /> | 47 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp12kzy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy41248gy7g8421zy91248g0g8421|patternname=A p32 spaceship}}<br /> |-<br /> | 48 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_y7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 49 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y7g8ka52zy2g8ka521z0g8ka521z121}} (width 2) per period<br /> |-<br /> | 50 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_y9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 51 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp24_8kalak8gzw125alalak8gzy3125alalak8gzy8125a521}} (width 4) per period<br /> |-<br /> | 52 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_1248gzy11248gzy61248gzyb1248g}} (width 1) per period<br /> |-<br /> | 53 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp124_ybg8ka52zy6g8ka521zy1g8ka521zg8ka521z01}} (width 2) per period<br /> |-<br /> | 54 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4094_ydg8421zy8g8421zy3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 55 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp16_wg842101248gzahy71248gzw1248gy71248gzy31248gy3g842zy81240421}} (width 8) per period<br /> |-<br /> | 56 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2046_1248gzy11248gzy61248gzyb1248gzyg1248}} (width 1) per period<br /> |-<br /> | 57 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp252_25ak8gzy0125ak8gzy5125ak8gzya125ak8gzyf125ak8}} (width 2) per period<br /> |-<br /> | 58 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_yhg8421zycg8421zy7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 59 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp56_8kalak8gzw125alalak8gzy3125alalak8gzy8125alalak8gzyd125alalak8zyi121}} (width 4) per period<br /> |-<br /> | 60 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp32766_yjg8421zyeg8421zy9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 61 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp60_yjg8ka52zyeg8ka521zy9g8ka521zy4g8ka521zxg8ka521z4a521}} (width 2) per period<br /> |-<br /> | 62 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248gzy61248gzyb1248gzyg1248gzyl1248g}} (width 1) per period<br /> |-<br /> | 63 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_ypg842101248gzykg8421y71248gzyfg8421yh1248gzyag8421yr1248gzy5g8421yzw1248gzy0g8421yzy81248gzx58gyzyeg85zy21248gyzy4g8421zy71248gyxg8421zychayqg8421zy7g8421yng8421zy2g8421yng8421zxk21yng8421zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 64 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzycg842zy41248gyzy2g8421zy91248gyvg8421zye1248gylg8421zyj1248gybg8421zyo1248gy1g8421zyrg8l8421zymg8ka521zyhg8ka521zycg8ka521zy7g8ka521zy2g8ka521zx4a521|patternname=A p64 spaceship}}<br /> |-<br /> | 65 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzyeg85zy41248gyzy4g8421zy91248gyxg8421zye1248gyng8421zyj1248gydg8421zyo1248gy3g8421zyrg850g8421zymg842h8421zyhg842h8421zycg842h8421zy7g842h8421zy2g842h8421zxk2h8421|patternname=A p64 spaceship}}<br /> |-<br /> | 66 || A p64 puffer, 41 xp2_12, 8 xp6_1248, 6 xp4_25ak8, 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 67 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_xk2101248gzy01248gx1248gzy51248gx1248gzya1248gx1248gzyf1248gx1248gzyk1248gx1248gzyp1248gx1248gzyqg842y41248gzylg8421yd1248gzygg8421yn1248gzybg8421yx1248gzy6g8421yzy41248gzy1g8421yzye12kzxahyzygg8421zy11248gyzy6g8421zy61248gyzg8421zyb12kyrg8421zy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p64 spaceship}}<br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Several trends can be found from the list for positive lengths.<br /> <br /> * Regardless of the type, a length-''n'' extended photon has a [[period]] of 2&lt;sup&gt;floor(log₂(''n''))&lt;/sup&gt;, i.e. Sloane's {{OEIS|A053644}}(''n'').<br /> * A length-''n'' extended photon will converge to a spaceship if ''n'' is in Sloane's {{OEIS|A099627}}. A vanishing T-square fractal&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/T-square_%28fractal%29 T-square fractal] at Wikipedia&lt;/ref&gt; can be seen at the tail for large enough ''n'' close to some power of 2.<br /> * A length-''n'' extended photon will converge to a [[clean]] puffer leaving one oscillator per period if ''n'' not in A099627 and {{nowrap|{{times|3|2&lt;sup&gt;''k''&lt;/sup&gt;}} ≤ ''n'' ≤ 2&lt;sup&gt;''k'' + 2&lt;/sup&gt;}} for some positive integer ''k''. Its trailing end is divided into two halves, one becomes a disappearing [[spark]] from a diagonal line of {{nowrap|2&lt;sup&gt;''k'' + 1&lt;/sup&gt; - 1}} cells while another evolves into the oscillator.<br /> ** The oscillators are square and oscillate according to a [[block cellular automaton]] on the [[Margolus neighbourhood]]. Previously the phenomenon was observed and studied in an outer-totalistic rule called [[2×2]], where the oscillators are made up of {{times|2|2}} [[block]]s, followed by several other non-totalistic rules later.&lt;ref name=&quot;post54201&quot; /&gt;&lt;ref name=&quot;post54268&quot; /&gt; Here the oscillators can be seen as composed of [[tub]]s, though it is not the smallest element because width-1 forms as labelled above exist. A width-1 diagonal line of 2''m'' cells has a period of Sloane's {{OEIS|A160657}}(''m''), and appears in the list above at length ''n'' = {{OEIS|A079946}}(''m''), or Sloane's {{OEIS|A049039}}(''m'')-th place among oscillators. Other diagonal widths are resulted from [[inflation]] by a factor of {{times|2&lt;sup&gt;''k''&lt;/sup&gt;|2&lt;sup&gt;''k''&lt;/sup&gt;}}, which also multiplies the period by 2&lt;sup&gt;''k''&lt;/sup&gt;.<br /> * An extended photon is [[dirty]] if its length does not satisfy any requirements above. In particular, a period-2&lt;sup&gt;''k''&lt;/sup&gt; extended photon produces the largest amount of [[ash]] if its length is {{nowrap|2&lt;sup&gt;''k''&lt;/sup&gt; + 2}}. Behind a puffer of this kind, straight diagonal lines replicate and decay in a complex manner leading to configurations with some self-similarity.<br /> <br /> [[Image:b2aes_complex_puffer.png|framed|center|The tail of a westbound length-16386 extended photon at generation 40960, showing both decaying and settled ash fields. Scale is 2^5:1.]]<br /> <br /> : However, no matter how dirty the ash is, it is guaranteed not to emit photons or replicators because B2e-governed evolution will only have new cells born at the same checkerboard parity, therefore prohibiting domino frontend from forming.<br /> * Regardless of the type, the reaction [[envelope]] of the tail is limited to one side of the extended photon. This implies that it can have two halves extended independently; not only their length can be different, but also they can operate at different periods with any [[phase]] difference.<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 3, rule = B2ae/S<br /> 2o$2bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Apart from the long extended photons above, the emergence of patterns may also be observed with smaller seeds, for instance the diagonally [[symmetry|symmetric]] [[tetraplet]] shown to the right. If random [[soup]]s are put behind the long diagonal line after a while to perturb the [[wavestretcher]], the outcome will be more complicated.&lt;ref name=&quot;post39221&quot; /&gt;<br /> <br /> ==Notes==<br /> &lt;references group=note&gt;<br /> &lt;/references&gt;<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54201&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54201<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 25, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54268&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54268<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 26, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post39221&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 39221<br /> |title = Re: Thread for basic questions<br /> |author = blah<br /> |date = January 10, 2017<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - note: the author claims to have all of the files for this web site, but it's not currently hosted anywhere. NOTE: The files for the original website are now permanently stored on a Google shared folder.<br /> * [https://drive.google.com/drive/folders/11x-Z3CzKPumuI5tqHWfWvY__7Yxq0urY Google Shared Folder] where all of the files for the original Logic Rule web page are stored.<br /> * {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> * {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where the author originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.<br /> <br /> [[Category:Isotropic non-totalistic rules]]<br /> [[Category:Rules without survival conditions]]</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=121051 2022-12-17T21:34:35Z

<p>DavidConant: /* External links */</p> <hr /> <div>{{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation {{iff}} it is currently dead and exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With [[Hensel notation]]&lt;ref group=note&gt;[[Hensel notation]] is a [[rulestring]] notation for isotropic rules, supported in CA simulation software including [[Golly]] and [[LifeViewer]].&lt;/ref&gt; this is expressed as B2ae/S. The rule was included as a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Signal logic==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of Margolus oscillators (see [[#Other patterns|the following section]]), the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit stream generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 11, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo2$13bo39bo$12bo39bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = IBSG and a p8 banana gun based on it<br /> |style = width:300px;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form {{nowrap|8''n'' + 1}} are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is {{eq|{{times|3|8}}|24}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 5, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo2$21bo$20bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The #17 BSG<br /> |style = width:300px;<br /> }}<br /> <br /> It appears that the period of {{nowrap|''n''-th}} BSG follows Sloane's {{OEIS|A086839}} and {{nowrap|log₂(period + 1)}} almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> From another point of view, any BSG with width {{nowrap|8 ''n'' + 1}} generates the output of a linear-feedback shift register&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/Linear-feedback_shift_register Linear-feedback shift register] at Wikipedia&lt;/ref&gt; (LFSR). Only numbers from Sloane's {{OEIS|A123399}} generate maximum length LFSRs with periods of {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt; - 1}}.<br /> <br /> The highest number listed is 419 which has a period of {{nowrap|2&lt;sup&gt;419&lt;/sup&gt; - 1}}, about {{times|1.35|10&lt;sup&gt;126&lt;/sup&gt;}} before the output repeats.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,247<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,991<br /> |}<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., {{nowrap|2&lt;sup&gt;''n''&lt;/sup&gt;-th}} place and the rest is sea of 0's. This is listed in Sloane's {{OEIS|A036987}}, the Fredholm-Rueppel sequence. An equivalent device akin to a [[caber tosser]] can be designed with only one unbounded replicator and two IBSG's, so that signals from the other stationary end can be fed into circuitry easily; the {{nowrap|''n''-th}} output starting from 0 arrives at the shown location at generation {{nowrap|2&lt;sup&gt;''n'' + 3&lt;/sup&gt; - 8}}.<br /> <br /> {{EmbedViewer<br /> |rle = x = 6, y = 5, rule = B2ae/S<br /> o4bo$o4bo2$3bo$2bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = XBSG, or Fredholm-Rueppel sequence generator<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 35, y = 34, rule = B2ae/S<br /> 21bo$20bo6$18b2o3$11bo$10bo7b2obo$10bo7b2o2bo2$15bo$16bo$18b2o6$20bo$<br /> 21bo2$11b2o5bo15bo$9bo3bo4bo15bo$8bo2$3bo3bo2bobo3bo3bo$3bo3bo8bo3bo$b<br /> o20bo$o11bo10bo$11bo!<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |position = center<br /> |caption = The equivalent device with only one side of the [[bounding box]] growing infinitely<br /> |style = width:300px;<br /> }}<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an [[intermittent stream|intermittent]] p8 input stream with an IBSG coming orthogonally at certain timing. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> {{EmbedViewer<br /> |rle = x = 98, y = 8, rule = B2ae/S<br /> 18bo47bo$17bo47bo2$o17bo4bo17bo24bo4bo24bo$bo16bo4bo16bo25bo4bo25bo2$<br /> 21bo47bo$20bo47bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = The BSGs used as signal sources are: {{eq|A|BSG2 width 17}}, {{eq|B|BSG3 width 25}}<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 77, y = 45, rule = B2ae/S<br /> 3bo$4bo16$6bo$3b2o2bo2$o$bo2$3b2o14$21bo31bo$20bo31bo2$4bo7bo8bo4bo8b<br /> o17bo4bo17bo$3bo7bo9bo4bo9bo16bo4bo16bo2$24bo31bo$23bo31bo!<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |position = center<br /> |caption = NOT A<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 179, y = 67, rule = B2ae/S<br /> 3bo$4bo8$6bo$3b2o2bo2$o59bo$bo59bo2$3b2o9$4bo$3bo4$21bo$20bo2$4bo16bo<br /> 4bo16bo$3bo17bo4bo17bo2$24bo$23bo23$51bo$50bo2$26bo24bo4bo24bo17bo4bo<br /> 17bo24bo4bo24bo$25bo25bo4bo25bo16bo4bo16bo25bo4bo25bo2$54bo47bo47bo$53b<br /> o47bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A AND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 205, y = 103, rule = B2ae/S<br /> 3bo$4bo23$53bo$6bo47bo$3b2o2bo2$o$bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o<br /> 2bo$3bo$30bo$31bo2$33b2o14$51bo$50bo2$34bo7bo8bo4bo8bo$33bo7bo9bo4bo9b<br /> o2$54bo$53bo13$71bo53bo47bo$70bo53bo47bo2$62bo8bo4bo8bo22bo16bo4bo16b<br /> o25bo4bo25bo$61bo9bo4bo9bo20bo17bo4bo17bo24bo4bo24bo2$74bo53bo47bo$73b<br /> o53bo47bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NAND B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 219, y = 107, rule = B2ae/S<br /> 3bo$4bo7$53bo$54bo$3b2o2$bo$o2$3b2o2bo$6bo8$4bo$3bo4$26bo$27bo2$4bo16b<br /> o4bo16bo$3bo17bo4bo17bo2$23bo$24bo23$56bo$57bo2$26bo24bo4bo24bo$25bo25b<br /> o4bo25bo2$53bo$54bo33$96bo34bo55bo$97bo32bo55bo2$82bo8bo4bo8bo8bo16bo<br /> 4bo16bo8bo24bo4bo24bo$81bo9bo4bo9bo6bo17bo4bo17bo6bo25bo4bo25bo2$93bo<br /> 40bo55bo$94bo38bo55bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A OR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 176, y = 83, rule = B2ae/S<br /> 3bo$4bo24$6bo$3b2o2bo2$o56bo$bo56bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o2b<br /> o$3bo$30bo$31bo2$33b2o14$51bo39bo52bo$50bo39bo52bo2$34bo7bo8bo4bo8bo8b<br /> o16bo4bo16bo5bo24bo4bo24bo$33bo7bo9bo4bo9bo6bo17bo4bo17bo3bo25bo4bo25b<br /> o2$54bo39bo52bo$53bo39bo52bo!<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |position = center<br /> |caption = A NOR B<br /> |style = width:300px;<br /> }}<br /> <br /> <br /> {{EmbedViewer<br /> |rle = x = 250, y = 188, rule = B2ae/S<br /> 64bo$63bo2$35bo25bo4bo25bo$36bo24bo4bo24bo2$67bo$66bo6$33bo$34bo16$36b<br /> o$33b2o2bo2$30bo38bo$31bo36bo2$33b2o15$3bo$4bo$34bo$33bo6$6bo$3b2o2bo<br /> 2$o$bo2$3b2o9$4bo$3bo19$63bo$64bo24$66bo$63b2o2bo2$60bo38bo$61bo38bo2$<br /> 63b2o24$91bo$64bo25bo$63bo$74bo16bo4bo16bo$73bo17bo4bo17bo2$94bo$93bo<br /> 23$121bo39bo56bo$120bo39bo56bo2$112bo8bo4bo8bo8bo16bo4bo16bo9bo24bo4b<br /> o24bo$111bo9bo4bo9bo6bo17bo4bo17bo7bo25bo4bo25bo2$124bo39bo56bo$123bo<br /> 39bo56bo!<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |position = center<br /> |caption = A XOR B<br /> |style = width:300px;<br /> }}<br /> <br /> ==Other patterns==<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 5, rule = B2ae/S<br /> bo$o$o$bo$2bo!<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> |position = right<br /> |style = width:300px;<br /> }}<br /> <br /> Despite its name, there are more &quot;naturalistic&quot; patterns of note beyond logic circuitry in Logic Rule. Consider a class of extended photons similar to the moon, with one &quot;wing&quot; fixed and the other of variable length. The first two cases are the banana (with length 0) and the moon (with length 1). At length 2 as shown to the right, the pattern is a [[predecessor]] of a period-2 spaceship. Depending on the length, these extended photons have different [[fate]]s and may become either a [[spaceship]] or a [[puffer]]. The first few cases are tabulated below, with links to the corresponding objects on [[catagolue]] at their first occurence.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view the fates of this class of patterns.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;wikitable sortable&quot;<br /> ! Length !! Fate<br /> |-<br /> | 0 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_16|patternname=A p2 spaceship (banana)}}<br /> |-<br /> | 1 || {{LinkCatagolue|style=raw|rule=b2aes|xq1_69|patternname=A p1 spaceship (moon)}}<br /> |-<br /> | 2 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_21ic|patternname=A p2 spaceship}}<br /> |-<br /> | 3 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_211ak8|patternname=A p2 spaceship}}<br /> |-<br /> | 4 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_8kahzw12442|patternname=A p4 spaceship}}<br /> |-<br /> | 5 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_0g8ka5ak8z34|patternname=A p4 spaceship}}<br /> |-<br /> | 6 || A p4 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2_12}} (duoplet, width 1) per period<br /> |-<br /> | 7 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_69g8g0g8gzx125ala52|patternname=A p4 spaceship}}<br /> |-<br /> | 8 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211ak8gzx4alalak8zy0g8521zx1|patternname=A p8 spaceship}}<br /> |-<br /> | 9 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_x25ak8gzxg8kalala4zx8la521z1221|patternname=A p8 spaceship}}<br /> |-<br /> | 10 || A p8 puffer, 2 xp2_12 per period<br /> |-<br /> | 11 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211248gzxlx1248gzy0hay2hazx58g0g8421|patternname=A p8 spaceship}}<br /> |-<br /> | 12 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp6_1248}} (width 1) per period<br /> |-<br /> | 13 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4_25ak8}} (width 2) per period<br /> |-<br /> | 14 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_0g8421z1}} (width 1) per period<br /> |-<br /> | 15 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_y1g842101248gzxahy7hazxg85y1g8421zxawg8421z488421|patternname=A p8 spaceship}}<br /> |-<br /> | 16 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy7hazy21248wg8421zy1g8ka521zx2521|patternname=A p16 spaceship}}<br /> |-<br /> | 17 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy712kzy21248y0g8421zy1g842h8421zxa18421|patternname=A p16 spaceship}}<br /> |-<br /> | 18 || A p16 puffer, 5 xp2_12 and 1 xp6_1248 per period<br /> |-<br /> | 19 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y4g842zy1g8421xg8421zxahxg8421zy1101|patternname=A p16 spaceship}}<br /> |-<br /> | 20 || A p16 puffer, 3 xp6_1248 per period<br /> |-<br /> | 21 || A p16 puffer, 2 xp4_25ak8 per period<br /> |-<br /> | 22 || A p16 puffer, 2 xp14_0g8421z1 per period<br /> |-<br /> | 23 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248zy1g8421y7g8421zxahy7g8421zy11248g0g8421|patternname=A p16 spaceship}}<br /> |-<br /> | 24 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_xg8421z421}} (width 1) per period<br /> |-<br /> | 25 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp12_xg8ka52z4a521}} (width 2) per period<br /> |-<br /> | 26 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248g}} (width 1) per period<br /> |-<br /> | 27 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp8_xg8kalak8z4alala521zx1}} (width 4) per period<br /> |-<br /> | 28 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_y3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 29 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y3g8ka52zwg8ka521z2521}} (width 2) per period<br /> |-<br /> | 30 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_1248gzy11248gzy61248}} (width 1) per period<br /> |-<br /> | 31 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahynhazy11248gydg8421zy61248gy3g8421zyb1240421|patternname=A p16 spaceship}}<br /> |-<br /> | 32 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn12kzy11248gyfg8421zy61248gy5g8421zyb12k0g8421zy8g8ka521zy3g8ka521zx8ka521|patternname=A p32 spaceship}}<br /> |-<br /> | 33 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_x58h248gzy21248h248gzy71248h248gzychaw1248gzy7g8421y51248gzy2g8421yf1248gzxk21yng85zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 34 || A p32 puffer, 14 xp2_12, 4 xp6_1248 and 1 xp4_25ak8 per period<br /> |-<br /> | 35 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gylg85zy61248gybg8421zyb12ky3g8421zy8g8421xg8421zy3g8421xg8421zx8421xg8421zx1240421|patternname=A p32 spaceship}}<br /> |-<br /> | 36 || A p32 puffer, 5 xp2_12, 4 xp6_1248, 1 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 37 || A p32 puffer, 5 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 38 || A p32 puffer, 4 xp2_12, 4 xp14_0g8421z1 and 1 xp14_xg8421z421 per period<br /> |-<br /> | 39 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_y1g842101248gzxahy71248gzy11248gy71248gzy61248gy71248gzyb12ky91248gzy8g8421yf1248gzy3g8421yng85zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p32 spaceship}}<br /> |-<br /> | 40 || A p32 puffer, 3 xp14_xg8421z421 and 1 xp2_12 per period<br /> |-<br /> | 41 || A p32 puffer, 3 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 42 || A p32 puffer, 2 xp2_12, 2 xp6_1248, 2 xp62_1248gzy11248g and 1 xp4_25ak8 per period<br /> |-<br /> | 43 || A p32 puffer, 2 xp8_xg8kalak8z4alala521zx1 per period<br /> |-<br /> | 44 || A p32 puffer, 2 xp126_y3g8421zwg8421z21 and 1 xp14_0g8kala4z12521 per period<br /> |-<br /> | 45 || A p32 puffer, 2 xp28_y3g8ka52zwg8ka521z2521 per period<br /> |-<br /> | 46 || A p32 puffer, 2 xp30_1248gzy11248gzy61248 per period<br /> |-<br /> | 47 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp12kzy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy41248gy7g8421zy91248g0g8421|patternname=A p32 spaceship}}<br /> |-<br /> | 48 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_y7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 49 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y7g8ka52zy2g8ka521z0g8ka521z121}} (width 2) per period<br /> |-<br /> | 50 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_y9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 51 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp24_8kalak8gzw125alalak8gzy3125alalak8gzy8125a521}} (width 4) per period<br /> |-<br /> | 52 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_1248gzy11248gzy61248gzyb1248g}} (width 1) per period<br /> |-<br /> | 53 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp124_ybg8ka52zy6g8ka521zy1g8ka521zg8ka521z01}} (width 2) per period<br /> |-<br /> | 54 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4094_ydg8421zy8g8421zy3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 55 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp16_wg842101248gzahy71248gzw1248gy71248gzy31248gy3g842zy81240421}} (width 8) per period<br /> |-<br /> | 56 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2046_1248gzy11248gzy61248gzyb1248gzyg1248}} (width 1) per period<br /> |-<br /> | 57 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp252_25ak8gzy0125ak8gzy5125ak8gzya125ak8gzyf125ak8}} (width 2) per period<br /> |-<br /> | 58 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_yhg8421zycg8421zy7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 59 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp56_8kalak8gzw125alalak8gzy3125alalak8gzy8125alalak8gzyd125alalak8zyi121}} (width 4) per period<br /> |-<br /> | 60 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp32766_yjg8421zyeg8421zy9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 61 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp60_yjg8ka52zyeg8ka521zy9g8ka521zy4g8ka521zxg8ka521z4a521}} (width 2) per period<br /> |-<br /> | 62 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248gzy61248gzyb1248gzyg1248gzyl1248g}} (width 1) per period<br /> |-<br /> | 63 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_ypg842101248gzykg8421y71248gzyfg8421yh1248gzyag8421yr1248gzy5g8421yzw1248gzy0g8421yzy81248gzx58gyzyeg85zy21248gyzy4g8421zy71248gyxg8421zychayqg8421zy7g8421yng8421zy2g8421yng8421zxk21yng8421zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 64 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzycg842zy41248gyzy2g8421zy91248gyvg8421zye1248gylg8421zyj1248gybg8421zyo1248gy1g8421zyrg8l8421zymg8ka521zyhg8ka521zycg8ka521zy7g8ka521zy2g8ka521zx4a521|patternname=A p64 spaceship}}<br /> |-<br /> | 65 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzyeg85zy41248gyzy4g8421zy91248gyxg8421zye1248gyng8421zyj1248gydg8421zyo1248gy3g8421zyrg850g8421zymg842h8421zyhg842h8421zycg842h8421zy7g842h8421zy2g842h8421zxk2h8421|patternname=A p64 spaceship}}<br /> |-<br /> | 66 || A p64 puffer, 41 xp2_12, 8 xp6_1248, 6 xp4_25ak8, 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 67 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_xk2101248gzy01248gx1248gzy51248gx1248gzya1248gx1248gzyf1248gx1248gzyk1248gx1248gzyp1248gx1248gzyqg842y41248gzylg8421yd1248gzygg8421yn1248gzybg8421yx1248gzy6g8421yzy41248gzy1g8421yzye12kzxahyzygg8421zy11248gyzy6g8421zy61248gyzg8421zyb12kyrg8421zy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p64 spaceship}}<br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Several trends can be found from the list for positive lengths.<br /> <br /> * Regardless of the type, a length-''n'' extended photon has a [[period]] of 2&lt;sup&gt;floor(log₂(''n''))&lt;/sup&gt;, i.e. Sloane's {{OEIS|A053644}}(''n'').<br /> * A length-''n'' extended photon will converge to a spaceship if ''n'' is in Sloane's {{OEIS|A099627}}. A vanishing T-square fractal&lt;ref group=note&gt;[https://en.wikipedia.org/wiki/T-square_%28fractal%29 T-square fractal] at Wikipedia&lt;/ref&gt; can be seen at the tail for large enough ''n'' close to some power of 2.<br /> * A length-''n'' extended photon will converge to a [[clean]] puffer leaving one oscillator per period if ''n'' not in A099627 and {{nowrap|{{times|3|2&lt;sup&gt;''k''&lt;/sup&gt;}} ≤ ''n'' ≤ 2&lt;sup&gt;''k'' + 2&lt;/sup&gt;}} for some positive integer ''k''. Its trailing end is divided into two halves, one becomes a disappearing [[spark]] from a diagonal line of {{nowrap|2&lt;sup&gt;''k'' + 1&lt;/sup&gt; - 1}} cells while another evolves into the oscillator.<br /> ** The oscillators are square and oscillate according to a [[block cellular automaton]] on the [[Margolus neighbourhood]]. Previously the phenomenon was observed and studied in an outer-totalistic rule called [[2×2]], where the oscillators are made up of {{times|2|2}} [[block]]s, followed by several other non-totalistic rules later.&lt;ref name=&quot;post54201&quot; /&gt;&lt;ref name=&quot;post54268&quot; /&gt; Here the oscillators can be seen as composed of [[tub]]s, though it is not the smallest element because width-1 forms as labelled above exist. A width-1 diagonal line of 2''m'' cells has a period of Sloane's {{OEIS|A160657}}(''m''), and appears in the list above at length ''n'' = {{OEIS|A079946}}(''m''), or Sloane's {{OEIS|A049039}}(''m'')-th place among oscillators. Other diagonal widths are resulted from [[inflation]] by a factor of {{times|2&lt;sup&gt;''k''&lt;/sup&gt;|2&lt;sup&gt;''k''&lt;/sup&gt;}}, which also multiplies the period by 2&lt;sup&gt;''k''&lt;/sup&gt;.<br /> * An extended photon is [[dirty]] if its length does not satisfy any requirements above. In particular, a period-2&lt;sup&gt;''k''&lt;/sup&gt; extended photon produces the largest amount of [[ash]] if its length is {{nowrap|2&lt;sup&gt;''k''&lt;/sup&gt; + 2}}. Behind a puffer of this kind, straight diagonal lines replicate and decay in a complex manner leading to configurations with some self-similarity.<br /> <br /> [[Image:b2aes_complex_puffer.png|framed|center|The tail of a westbound length-16386 extended photon at generation 40960, showing both decaying and settled ash fields. Scale is 2^5:1.]]<br /> <br /> : However, no matter how dirty the ash is, it is guaranteed not to emit photons or replicators because B2e-governed evolution will only have new cells born at the same checkerboard parity, therefore prohibiting domino frontend from forming.<br /> * Regardless of the type, the reaction [[envelope]] of the tail is limited to one side of the extended photon. This implies that it can have two halves extended independently; not only their length can be different, but also they can operate at different periods with any [[phase]] difference.<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 3, rule = B2ae/S<br /> 2o$2bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Apart from the long extended photons above, the emergence of patterns may also be observed with smaller seeds, for instance the diagonally [[symmetry|symmetric]] [[tetraplet]] shown to the right. If random [[soup]]s are put behind the long diagonal line after a while to perturb the [[wavestretcher]], the outcome will be more complicated.&lt;ref name=&quot;post39221&quot; /&gt;<br /> <br /> ==Notes==<br /> &lt;references group=note&gt;<br /> &lt;/references&gt;<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54201&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54201<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 25, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post54268&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 54268<br /> |title = Re: Thread for basic questions<br /> |author = Dave Greene<br /> |date = December 26, 2017<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post39221&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 39221<br /> |title = Re: Thread for basic questions<br /> |author = blah<br /> |date = January 10, 2017<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - note: the author claims to have all of the files for this web site, but it's not currently hosted anywhere.<br /> * [https://drive.google.com/drive/folders/11x-Z3CzKPumuI5tqHWfWvY__7Yxq0urY Google Shared Folder] where all of the files for the original Logic Rule web page are stored.<br /> * {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> * {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where the author originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.<br /> <br /> [[Category:Isotropic non-totalistic rules]]<br /> [[Category:Rules without survival conditions]]</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82748 2021-09-21T13:37:32Z

<p>DavidConant: /* Logic gates */ Added logic gates</p> <hr /> <div>{{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation {{iff}} it is currently dead and exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S, which can be used by modern cellular automata simulators such as [[Golly]] and [[LifeViewer]]. Previously, the rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Signal logic==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of Margolus oscillators (see [[OCA:Logic_Rule#Other_patterns|the following section]]), the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 11, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo2$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 5, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo2$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> From another point of view, any BSG with width 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a [https://en.wikipedia.org/wiki/Linear-feedback_shift_register Linear Feedback Shift Register]. Only numbers from Sloane's {{OEIS|A123399}} generate maximum length LFSRs with periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;419&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;126&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,247<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,991<br /> |}<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in Sloane's {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an [[intermittent stream|intermittent]] p8 input stream with an IBSG coming orthogonally at certain timing. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> {{EmbedViewer<br /> |rle = x = 98, y = 8, rule = B2ae/S<br /> 18bo47bo$17bo47bo2$o17bo4bo17bo24bo4bo24bo$bo16bo4bo16bo25bo4bo25bo2$<br /> 21bo47bo$20bo47bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |caption = The Gates used: A = BSG2 width 17, BSG3 width 25&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 77, y = 45, rule = B2ae/S<br /> 3bo$4bo16$6bo$3b2o2bo2$o$bo2$3b2o14$21bo31bo$20bo31bo2$4bo7bo8bo4bo8b<br /> o17bo4bo17bo$3bo7bo9bo4bo9bo16bo4bo16bo2$24bo31bo$23bo31bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 4 ]]<br /> |caption = NOT A&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 179, y = 67, rule = B2ae/S<br /> 3bo$4bo8$6bo$3b2o2bo2$o59bo$bo59bo2$3b2o9$4bo$3bo4$21bo$20bo2$4bo16bo<br /> 4bo16bo$3bo17bo4bo17bo2$24bo$23bo23$51bo$50bo2$26bo24bo4bo24bo17bo4bo<br /> 17bo24bo4bo24bo$25bo25bo4bo25bo16bo4bo16bo25bo4bo25bo2$54bo47bo47bo$53b<br /> o47bo47bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |caption = A AND B&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 205, y = 103, rule = B2ae/S<br /> 3bo$4bo23$53bo$6bo47bo$3b2o2bo2$o$bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o<br /> 2bo$3bo$30bo$31bo2$33b2o14$51bo$50bo2$34bo7bo8bo4bo8bo$33bo7bo9bo4bo9b<br /> o2$54bo$53bo13$71bo53bo47bo$70bo53bo47bo2$62bo8bo4bo8bo22bo16bo4bo16b<br /> o25bo4bo25bo$61bo9bo4bo9bo20bo17bo4bo17bo24bo4bo24bo2$74bo53bo47bo$73b<br /> o53bo47bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |caption = A NAND B&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 219, y = 107, rule = B2ae/S<br /> 3bo$4bo7$53bo$54bo$3b2o2$bo$o2$3b2o2bo$6bo8$4bo$3bo4$26bo$27bo2$4bo16b<br /> o4bo16bo$3bo17bo4bo17bo2$23bo$24bo23$56bo$57bo2$26bo24bo4bo24bo$25bo25b<br /> o4bo25bo2$53bo$54bo33$96bo34bo55bo$97bo32bo55bo2$82bo8bo4bo8bo8bo16bo<br /> 4bo16bo8bo24bo4bo24bo$81bo9bo4bo9bo6bo17bo4bo17bo6bo25bo4bo25bo2$93bo<br /> 40bo55bo$94bo38bo55bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |caption = A OR B&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 176, y = 83, rule = B2ae/S<br /> 3bo$4bo24$6bo$3b2o2bo2$o56bo$bo56bo2$3b2o7$33bo$34bo16$36bo$4bo28b2o2b<br /> o$3bo$30bo$31bo2$33b2o14$51bo39bo52bo$50bo39bo52bo2$34bo7bo8bo4bo8bo8b<br /> o16bo4bo16bo5bo24bo4bo24bo$33bo7bo9bo4bo9bo6bo17bo4bo17bo3bo25bo4bo25b<br /> o2$54bo39bo52bo$53bo39bo52bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 3 ]]<br /> |caption = A NOR B&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 250, y = 188, rule = B2ae/S<br /> 64bo$63bo2$35bo25bo4bo25bo$36bo24bo4bo24bo2$67bo$66bo6$33bo$34bo16$36b<br /> o$33b2o2bo2$30bo38bo$31bo36bo2$33b2o15$3bo$4bo$34bo$33bo6$6bo$3b2o2bo<br /> 2$o$bo2$3b2o9$4bo$3bo19$63bo$64bo24$66bo$63b2o2bo2$60bo38bo$61bo38bo2$<br /> 63b2o24$91bo$64bo25bo$63bo$74bo16bo4bo16bo$73bo17bo4bo17bo2$94bo$93bo<br /> 23$121bo39bo56bo$120bo39bo56bo2$112bo8bo4bo8bo8bo16bo4bo16bo9bo24bo4b<br /> o24bo$111bo9bo4bo9bo6bo17bo4bo17bo7bo25bo4bo25bo2$124bo39bo56bo$123bo<br /> 39bo56bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 2 ]]<br /> |caption = A XOR B&lt;br&gt;<br /> }}<br /> &lt;br&gt;<br /> {{EmbedViewer<br /> |rle = x = 6, y = 5, rule = B2ae/S<br /> o4bo$o4bo2$3bo$2bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = Fredholm-Rueppel sequence, OEIS A036987&lt;br&gt;<br /> }}<br /> <br /> ==Other patterns==<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 5, rule = B2ae/S<br /> bo$o$o$bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Despite its name, there are more &quot;naturalistic&quot; patterns of note beyond logic circuitry in Logic Rule. Consider a class of extended photons similar to the moon, with one &quot;wing&quot; fixed and the other of variable length. The first two cases are the banana (with length 0) and the moon (with length 1). At length 2 as shown to the right, the pattern is a [[predecessor]] of a period-2 spaceship. Depending on the length, these extended photons have different [[fate]]s and may become either a [[spaceship]] or a [[puffer]]. The first few cases are tabulated below, with links to the corresponding objects on [[catagolue]] at their first occurence.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view the fates of this class of patterns.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;wikitable sortable&quot;<br /> ! Length !! Fate<br /> |-<br /> | 0 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_16|patternname=A p2 spaceship (banana)}}<br /> |-<br /> | 1 || {{LinkCatagolue|style=raw|rule=b2aes|xq1_69|patternname=A p1 spaceship (moon)}}<br /> |-<br /> | 2 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_21ic|patternname=A p2 spaceship}}<br /> |-<br /> | 3 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_211ak8|patternname=A p2 spaceship}}<br /> |-<br /> | 4 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_8kahzw12442|patternname=A p4 spaceship}}<br /> |-<br /> | 5 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_0g8ka5ak8z34|patternname=A p4 spaceship}}<br /> |-<br /> | 6 || A p4 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2_12}} (duoplet, width 1) per period<br /> |-<br /> | 7 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_69g8g0g8gzx125ala52|patternname=A p4 spaceship}}<br /> |-<br /> | 8 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211ak8gzx4alalak8zy0g8521zx1|patternname=A p8 spaceship}}<br /> |-<br /> | 9 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_x25ak8gzxg8kalala4zx8la521z1221|patternname=A p8 spaceship}}<br /> |-<br /> | 10 || A p8 puffer, 2 xp2_12 per period<br /> |-<br /> | 11 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211248gzxlx1248gzy0hay2hazx58g0g8421|patternname=A p8 spaceship}}<br /> |-<br /> | 12 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp6_1248}} (width 1) per period<br /> |-<br /> | 13 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4_25ak8}} (width 2) per period<br /> |-<br /> | 14 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_0g8421z1}} (width 1) per period<br /> |-<br /> | 15 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_y1g842101248gzxahy7hazxg85y1g8421zxawg8421z488421|patternname=A p8 spaceship}}<br /> |-<br /> | 16 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy7hazy21248wg8421zy1g8ka521zx2521|patternname=A p16 spaceship}}<br /> |-<br /> | 17 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy712kzy21248y0g8421zy1g842h8421zxa18421|patternname=A p16 spaceship}}<br /> |-<br /> | 18 || A p16 puffer, 5 xp2_12 and 1 xp6_1248 per period<br /> |-<br /> | 19 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y4g842zy1g8421xg8421zxahxg8421zy1101|patternname=A p16 spaceship}}<br /> |-<br /> | 20 || A p16 puffer, 3 xp6_1248 per period<br /> |-<br /> | 21 || A p16 puffer, 2 xp4_25ak8 per period<br /> |-<br /> | 22 || A p16 puffer, 2 xp14_0g8421z1 per period<br /> |-<br /> | 23 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248zy1g8421y7g8421zxahy7g8421zy11248g0g8421|patternname=A p16 spaceship}}<br /> |-<br /> | 24 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_xg8421z421}} (width 1) per period<br /> |-<br /> | 25 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp12_xg8ka52z4a521}} (width 2) per period<br /> |-<br /> | 26 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248g}} (width 1) per period<br /> |-<br /> | 27 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp8_xg8kalak8z4alala521zx1}} (width 4) per period<br /> |-<br /> | 28 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_y3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 29 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y3g8ka52zwg8ka521z2521}} (width 2) per period<br /> |-<br /> | 30 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_1248gzy11248gzy61248}} (width 1) per period<br /> |-<br /> | 31 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahynhazy11248gydg8421zy61248gy3g8421zyb1240421|patternname=A p16 spaceship}}<br /> |-<br /> | 32 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn12kzy11248gyfg8421zy61248gy5g8421zyb12k0g8421zy8g8ka521zy3g8ka521zx8ka521|patternname=A p32 spaceship}}<br /> |-<br /> | 33 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_x58h248gzy21248h248gzy71248h248gzychaw1248gzy7g8421y51248gzy2g8421yf1248gzxk21yng85zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 34 || A p32 puffer, 14 xp2_12, 4 xp6_1248 and 1 xp4_25ak8 per period<br /> |-<br /> | 35 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gylg85zy61248gybg8421zyb12ky3g8421zy8g8421xg8421zy3g8421xg8421zx8421xg8421zx1240421|patternname=A p32 spaceship}}<br /> |-<br /> | 36 || A p32 puffer, 5 xp2_12, 4 xp6_1248, 1 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 37 || A p32 puffer, 5 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 38 || A p32 puffer, 4 xp2_12, 4 xp14_0g8421z1 and 1 xp14_xg8421z421 per period<br /> |-<br /> | 39 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_y1g842101248gzxahy71248gzy11248gy71248gzy61248gy71248gzyb12ky91248gzy8g8421yf1248gzy3g8421yng85zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p32 spaceship}}<br /> |-<br /> | 40 || A p32 puffer, 3 xp14_xg8421z421 and 1 xp2_12 per period<br /> |-<br /> | 41 || A p32 puffer, 3 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 42 || A p32 puffer, 2 xp2_12, 2 xp6_1248, 2 xp62_1248gzy11248g and 1 xp4_25ak8 per period<br /> |-<br /> | 43 || A p32 puffer, 2 xp8_xg8kalak8z4alala521zx1 per period<br /> |-<br /> | 44 || A p32 puffer, 2 xp126_y3g8421zwg8421z21 and 1 xp14_0g8kala4z12521 per period<br /> |-<br /> | 45 || A p32 puffer, 2 xp28_y3g8ka52zwg8ka521z2521 per period<br /> |-<br /> | 46 || A p32 puffer, 2 xp30_1248gzy11248gzy61248 per period<br /> |-<br /> | 47 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp12kzy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy41248gy7g8421zy91248g0g8421|patternname=A p32 spaceship}}<br /> |-<br /> | 48 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_y7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 49 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y7g8ka52zy2g8ka521z0g8ka521z121}} (width 2) per period<br /> |-<br /> | 50 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_y9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 51 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp24_8kalak8gzw125alalak8gzy3125alalak8gzy8125a521}} (width 4) per period<br /> |-<br /> | 52 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_1248gzy11248gzy61248gzyb1248g}} (width 1) per period<br /> |-<br /> | 53 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp124_ybg8ka52zy6g8ka521zy1g8ka521zg8ka521z01}} (width 2) per period<br /> |-<br /> | 54 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4094_ydg8421zy8g8421zy3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 55 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp16_wg842101248gzahy71248gzw1248gy71248gzy31248gy3g842zy81240421}} (width 8) per period<br /> |-<br /> | 56 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2046_1248gzy11248gzy61248gzyb1248gzyg1248}} (width 1) per period<br /> |-<br /> | 57 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp252_25ak8gzy0125ak8gzy5125ak8gzya125ak8gzyf125ak8}} (width 2) per period<br /> |-<br /> | 58 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_yhg8421zycg8421zy7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 59 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp56_8kalak8gzw125alalak8gzy3125alalak8gzy8125alalak8gzyd125alalak8zyi121}} (width 4) per period<br /> |-<br /> | 60 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp32766_yjg8421zyeg8421zy9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 61 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp60_yjg8ka52zyeg8ka521zy9g8ka521zy4g8ka521zxg8ka521z4a521}} (width 2) per period<br /> |-<br /> | 62 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248gzy61248gzyb1248gzyg1248gzyl1248g}} (width 1) per period<br /> |-<br /> | 63 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_ypg842101248gzykg8421y71248gzyfg8421yh1248gzyag8421yr1248gzy5g8421yzw1248gzy0g8421yzy81248gzx58gyzyeg85zy21248gyzy4g8421zy71248gyxg8421zychayqg8421zy7g8421yng8421zy2g8421yng8421zxk21yng8421zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 64 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzycg842zy41248gyzy2g8421zy91248gyvg8421zye1248gylg8421zyj1248gybg8421zyo1248gy1g8421zyrg8l8421zymg8ka521zyhg8ka521zycg8ka521zy7g8ka521zy2g8ka521zx4a521|patternname=A p64 spaceship}}<br /> |-<br /> | 65 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzyeg85zy41248gyzy4g8421zy91248gyxg8421zye1248gyng8421zyj1248gydg8421zyo1248gy3g8421zyrg850g8421zymg842h8421zyhg842h8421zycg842h8421zy7g842h8421zy2g842h8421zxk2h8421|patternname=A p64 spaceship}}<br /> |-<br /> | 66 || A p64 puffer, 41 xp2_12, 8 xp6_1248, 6 xp4_25ak8, 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 67 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_xk2101248gzy01248gx1248gzy51248gx1248gzya1248gx1248gzyf1248gx1248gzyk1248gx1248gzyp1248gx1248gzyqg842y41248gzylg8421yd1248gzygg8421yn1248gzybg8421yx1248gzy6g8421yzy41248gzy1g8421yzye12kzxahyzygg8421zy11248gyzy6g8421zy61248gyzg8421zyb12kyrg8421zy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p64 spaceship}}<br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Several trends can be found from the list for positive lengths.<br /> <br /> * Regardless of the type, a length-&lt;i&gt;n&lt;/i&gt; extended photon has a [[period]] of 2&lt;sup&gt;floor(log₂(&lt;i&gt;n&lt;/i&gt;))&lt;/sup&gt;, i.e. Sloane's {{OEIS|A053644}}(&lt;i&gt;n&lt;/i&gt;).<br /> * A length-&lt;i&gt;n&lt;/i&gt; extended photon will converge to a spaceship if &lt;i&gt;n&lt;/i&gt; is in Sloane's {{OEIS|A099627}}. A vanishing [https://en.wikipedia.org/wiki/T-square_%28fractal%29 T-square fractal] can be seen at the tail for large enough &lt;i&gt;n&lt;/i&gt; close to some power of 2.<br /> * A length-&lt;i&gt;n&lt;/i&gt; extended photon will converge to a [[clean]] puffer leaving one oscillator per period if &lt;i&gt;n&lt;/i&gt; not in A099627 and 3×2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt; ≤ &lt;i&gt;n&lt;/i&gt; ≤ 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;+2&lt;/sup&gt; for some positive integer &lt;i&gt;k&lt;/i&gt;. Its trailing end is divided into two halves, one becomes a disappearing [[spark]] from a diagonal line of 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;+1&lt;/sup&gt;-1 cells while another evolves into the oscillator.<br /> ** The oscillators are square and oscillate according to a [[block cellular automaton]] on the [[Margolus neighbourhood]]. Previously the phenomenon was observed and studied in a outer-totalistic rule called [[2×2]], where the oscillators are made up of 2×2 [[block]]s. Here the oscillators can be seen as composed of [[tub]]s, though it is not the smallest element because width-1 forms as labelled above exist. A width-1 diagonal line of 2&lt;i&gt;m&lt;/i&gt; cells has a period of Sloane's {{OEIS|A160657}}(&lt;i&gt;m&lt;/i&gt;), and appears in the list above at length &lt;i&gt;n&lt;/i&gt; = {{OEIS|A079946}}(&lt;i&gt;m&lt;/i&gt;), or Sloane's {{OEIS|A049039}}(&lt;i&gt;m&lt;/i&gt;)-th place among oscillators. Other diagonal widths are resulted from [[inflation]] by a factor of 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;×2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;, which also multiplies the period by 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;.<br /> * An extended photon is [[dirty]] if its length does not satisfy any requirements above. In particular, a period-2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt; extended photon produces the largest amount of [[ash]] if its length is 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;+2. Behind a puffer of this kind, straight diagonal lines replicate and decay in a complex manner leading to configurations with some self-similarity.<br /> <br /> [[Image:b2aes_complex_puffer.png|framed|center|The tail of a westbound length-16386 extended photon at generation 40960, showing both decaying and settled ash fields. Scale is 2^5:1.]]<br /> <br /> : However, no matter how dirty the ash is, it is guaranteed not to emit photons or replicators because B2e-governed evolution will only have new cells born at the same checkerboard parity, therefore prohibiting domino frontend from forming.<br /> * Regardless of the type, the reaction [[envelope]] of the tail is limited to one side of the extended photon. This implies that it can have two halves extended independently; not only their length can be different, but also they can operate at different periods with any [[phase]] difference.<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 3, rule = B2ae/S<br /> 2o$2bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Apart from the long extended photons above, the emergence of patterns may also be observed with smaller seeds, for instance the diagonally [[symmetry|symmetric]] [[polyplet#Tetraplets|tetraplet]] shown to the right. If random [[soup]]s are put behind the long diagonal line after a while to perturb the [[wavestretcher]], the outcome will be more complicated.&lt;ref name=&quot;post39221&quot; /&gt;<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post39221&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 39221<br /> |title = Re: Thread for basic questions<br /> |author = blah<br /> |date = January 10, 2017<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - note: the author claims to have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where the author originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82745 2021-09-21T13:12:31Z

<p>DavidConant: /* Bit Stream Generators */ Corrected values 51 and 53 in the table</p> <hr /> <div>{{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation {{iff}} it is currently dead and exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S, which can be used by modern cellular automata simulators such as [[Golly]] and [[LifeViewer]]. Previously, the rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Signal logic==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of Margolus oscillators (see [[OCA:Logic_Rule#Other_patterns|the following section]]), the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 11, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo2$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 5, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo2$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> From another point of view, any BSG with width 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a [https://en.wikipedia.org/wiki/Linear-feedback_shift_register Linear Feedback Shift Register]. Only numbers from Sloane's {{OEIS|A123399}} generate maximum length LFSRs with periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;419&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;126&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,247<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,991<br /> |}<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in Sloane's {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an [[intermittent stream|intermittent]] p8 input stream with an IBSG coming orthogonally at certain timing. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ==Other patterns==<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 5, rule = B2ae/S<br /> bo$o$o$bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Despite its name, there are more &quot;naturalistic&quot; patterns of note beyond logic circuitry in Logic Rule. Consider a class of extended photons similar to the moon, with one &quot;wing&quot; fixed and the other of variable length. The first two cases are the banana (with length 0) and the moon (with length 1). At length 2 as shown to the right, the pattern is a [[predecessor]] of a period-2 spaceship. Depending on the length, these extended photons have different [[fate]]s and may become either a [[spaceship]] or a [[puffer]]. The first few cases are tabulated below, with links to the corresponding objects on [[catagolue]] at their first occurence.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view the fates of this class of patterns.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;wikitable sortable&quot;<br /> ! Length !! Fate<br /> |-<br /> | 0 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_16|patternname=A p2 spaceship (banana)}}<br /> |-<br /> | 1 || {{LinkCatagolue|style=raw|rule=b2aes|xq1_69|patternname=A p1 spaceship (moon)}}<br /> |-<br /> | 2 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_21ic|patternname=A p2 spaceship}}<br /> |-<br /> | 3 || {{LinkCatagolue|style=raw|rule=b2aes|xq2_211ak8|patternname=A p2 spaceship}}<br /> |-<br /> | 4 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_8kahzw12442|patternname=A p4 spaceship}}<br /> |-<br /> | 5 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_0g8ka5ak8z34|patternname=A p4 spaceship}}<br /> |-<br /> | 6 || A p4 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2_12}} (duoplet, width 1) per period<br /> |-<br /> | 7 || {{LinkCatagolue|style=raw|rule=b2aes|xq4_69g8g0g8gzx125ala52|patternname=A p4 spaceship}}<br /> |-<br /> | 8 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211ak8gzx4alalak8zy0g8521zx1|patternname=A p8 spaceship}}<br /> |-<br /> | 9 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_x25ak8gzxg8kalala4zx8la521z1221|patternname=A p8 spaceship}}<br /> |-<br /> | 10 || A p8 puffer, 2 xp2_12 per period<br /> |-<br /> | 11 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_211248gzxlx1248gzy0hay2hazx58g0g8421|patternname=A p8 spaceship}}<br /> |-<br /> | 12 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp6_1248}} (width 1) per period<br /> |-<br /> | 13 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4_25ak8}} (width 2) per period<br /> |-<br /> | 14 || A p8 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_0g8421z1}} (width 1) per period<br /> |-<br /> | 15 || {{LinkCatagolue|style=raw|rule=b2aes|xq8_y1g842101248gzxahy7hazxg85y1g8421zxawg8421z488421|patternname=A p8 spaceship}}<br /> |-<br /> | 16 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy7hazy21248wg8421zy1g8ka521zx2521|patternname=A p16 spaceship}}<br /> |-<br /> | 17 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy712kzy21248y0g8421zy1g842h8421zxa18421|patternname=A p16 spaceship}}<br /> |-<br /> | 18 || A p16 puffer, 5 xp2_12 and 1 xp6_1248 per period<br /> |-<br /> | 19 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y4g842zy1g8421xg8421zxahxg8421zy1101|patternname=A p16 spaceship}}<br /> |-<br /> | 20 || A p16 puffer, 3 xp6_1248 per period<br /> |-<br /> | 21 || A p16 puffer, 2 xp4_25ak8 per period<br /> |-<br /> | 22 || A p16 puffer, 2 xp14_0g8421z1 per period<br /> |-<br /> | 23 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248zy1g8421y7g8421zxahy7g8421zy11248g0g8421|patternname=A p16 spaceship}}<br /> |-<br /> | 24 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp14_xg8421z421}} (width 1) per period<br /> |-<br /> | 25 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp12_xg8ka52z4a521}} (width 2) per period<br /> |-<br /> | 26 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248g}} (width 1) per period<br /> |-<br /> | 27 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp8_xg8kalak8z4alala521zx1}} (width 4) per period<br /> |-<br /> | 28 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_y3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 29 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y3g8ka52zwg8ka521z2521}} (width 2) per period<br /> |-<br /> | 30 || A p16 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_1248gzy11248gzy61248}} (width 1) per period<br /> |-<br /> | 31 || {{LinkCatagolue|style=raw|rule=b2aes|xq16_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahynhazy11248gydg8421zy61248gy3g8421zyb1240421|patternname=A p16 spaceship}}<br /> |-<br /> | 32 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn12kzy11248gyfg8421zy61248gy5g8421zyb12k0g8421zy8g8ka521zy3g8ka521zx8ka521|patternname=A p32 spaceship}}<br /> |-<br /> | 33 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_x58h248gzy21248h248gzy71248h248gzychaw1248gzy7g8421y51248gzy2g8421yf1248gzxk21yng85zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 34 || A p32 puffer, 14 xp2_12, 4 xp6_1248 and 1 xp4_25ak8 per period<br /> |-<br /> | 35 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gylg85zy61248gybg8421zyb12ky3g8421zy8g8421xg8421zy3g8421xg8421zx8421xg8421zx1240421|patternname=A p32 spaceship}}<br /> |-<br /> | 36 || A p32 puffer, 5 xp2_12, 4 xp6_1248, 1 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 37 || A p32 puffer, 5 xp4_25ak8 and 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 38 || A p32 puffer, 4 xp2_12, 4 xp14_0g8421z1 and 1 xp14_xg8421z421 per period<br /> |-<br /> | 39 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_y1g842101248gzxahy71248gzy11248gy71248gzy61248gy71248gzyb12ky91248gzy8g8421yf1248gzy3g8421yng85zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p32 spaceship}}<br /> |-<br /> | 40 || A p32 puffer, 3 xp14_xg8421z421 and 1 xp2_12 per period<br /> |-<br /> | 41 || A p32 puffer, 3 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 42 || A p32 puffer, 2 xp2_12, 2 xp6_1248, 2 xp62_1248gzy11248g and 1 xp4_25ak8 per period<br /> |-<br /> | 43 || A p32 puffer, 2 xp8_xg8kalak8z4alala521zx1 per period<br /> |-<br /> | 44 || A p32 puffer, 2 xp126_y3g8421zwg8421z21 and 1 xp14_0g8kala4z12521 per period<br /> |-<br /> | 45 || A p32 puffer, 2 xp28_y3g8ka52zwg8ka521z2521 per period<br /> |-<br /> | 46 || A p32 puffer, 2 xp30_1248gzy11248gzy61248 per period<br /> |-<br /> | 47 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp12kzy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy41248gy7g8421zy91248g0g8421|patternname=A p32 spaceship}}<br /> |-<br /> | 48 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp30_y7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 49 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp28_y7g8ka52zy2g8ka521z0g8ka521z121}} (width 2) per period<br /> |-<br /> | 50 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_y9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 51 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp24_8kalak8gzw125alalak8gzy3125alalak8gzy8125a521}} (width 4) per period<br /> |-<br /> | 52 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp126_1248gzy11248gzy61248gzyb1248g}} (width 1) per period<br /> |-<br /> | 53 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp124_ybg8ka52zy6g8ka521zy1g8ka521zg8ka521z01}} (width 2) per period<br /> |-<br /> | 54 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp4094_ydg8421zy8g8421zy3g8421zwg8421z21}} (width 1) per period<br /> |-<br /> | 55 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp16_wg842101248gzahy71248gzw1248gy71248gzy31248gy3g842zy81240421}} (width 8) per period<br /> |-<br /> | 56 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp2046_1248gzy11248gzy61248gzyb1248gzyg1248}} (width 1) per period<br /> |-<br /> | 57 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp252_25ak8gzy0125ak8gzy5125ak8gzya125ak8gzyf125ak8}} (width 2) per period<br /> |-<br /> | 58 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp1022_yhg8421zycg8421zy7g8421zy2g8421z0g8421z1}} (width 1) per period<br /> |-<br /> | 59 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp56_8kalak8gzw125alalak8gzy3125alalak8gzy8125alalak8gzyd125alalak8zyi121}} (width 4) per period<br /> |-<br /> | 60 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp32766_yjg8421zyeg8421zy9g8421zy4g8421zxg8421z421}} (width 1) per period<br /> |-<br /> | 61 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp60_yjg8ka52zyeg8ka521zy9g8ka521zy4g8ka521zxg8ka521z4a521}} (width 2) per period<br /> |-<br /> | 62 || A p32 puffer, 1 {{LinkCatagolue|style=raw|rule=b2aes|xp62_1248gzy11248gzy61248gzyb1248gzyg1248gzyl1248g}} (width 1) per period<br /> |-<br /> | 63 || {{LinkCatagolue|style=raw|rule=b2aes|xq32_ypg842101248gzykg8421y71248gzyfg8421yh1248gzyag8421yr1248gzy5g8421yzw1248gzy0g8421yzy81248gzx58gyzyeg85zy21248gyzy4g8421zy71248gyxg8421zychayqg8421zy7g8421yng8421zy2g8421yng8421zxk21yng8421zy01248gyfg8421zy3g85y9g8421zx8421y7g8421zx12ky3g8421zxa1xg8421zgwh8421z011|patternname=A p32 spaceship}}<br /> |-<br /> | 64 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzycg842zy41248gyzy2g8421zy91248gyvg8421zye1248gylg8421zyj1248gybg8421zyo1248gy1g8421zyrg8l8421zymg8ka521zyhg8ka521zycg8ka521zy7g8ka521zy2g8ka521zx4a521|patternname=A p64 spaceship}}<br /> |-<br /> | 65 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_211248gzxlx1248gzy0hay21248gzx58gy71248gzy21248y81248gzy1g8421yd1248gzxahyn1248gzy11248gyn1248gzy61248gyn1248gzyb12kyp1248gzy8g8421yv1248gzy3g8421yzy21248gzx8421yzyc1248gzx1248gyzyeg85zy41248gyzy4g8421zy91248gyxg8421zye1248gyng8421zyj1248gydg8421zyo1248gy3g8421zyrg850g8421zymg842h8421zyhg842h8421zycg842h8421zy7g842h8421zy2g842h8421zxk2h8421|patternname=A p64 spaceship}}<br /> |-<br /> | 66 || A p64 puffer, 41 xp2_12, 8 xp6_1248, 6 xp4_25ak8, 1 xp12_xg8ka52z4a521 per period<br /> |-<br /> | 67 || {{LinkCatagolue|style=raw|rule=b2aes|xq64_xk2101248gzy01248gx1248gzy51248gx1248gzya1248gx1248gzyf1248gx1248gzyk1248gx1248gzyp1248gx1248gzyqg842y41248gzylg8421yd1248gzygg8421yn1248gzybg8421yx1248gzy6g8421yzy41248gzy1g8421yzye12kzxahyzygg8421zy11248gyzy6g8421zy61248gyzg8421zyb12kyrg8421zy8g8421yng8421zy3g8421yng8421zx8421yng8421zx1248gyhg8421zy4hayag8421zxg8421y7g8421zx248y4g8421zxk21xg8421zx2g8421z1221|patternname=A p64 spaceship}}<br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Several trends can be found from the list for positive lengths.<br /> <br /> * Regardless of the type, a length-&lt;i&gt;n&lt;/i&gt; extended photon has a [[period]] of 2&lt;sup&gt;floor(log₂(&lt;i&gt;n&lt;/i&gt;))&lt;/sup&gt;, i.e. Sloane's {{OEIS|A053644}}(&lt;i&gt;n&lt;/i&gt;).<br /> * A length-&lt;i&gt;n&lt;/i&gt; extended photon will converge to a spaceship if &lt;i&gt;n&lt;/i&gt; is in Sloane's {{OEIS|A099627}}. A vanishing [https://en.wikipedia.org/wiki/T-square_%28fractal%29 T-square fractal] can be seen at the tail for large enough &lt;i&gt;n&lt;/i&gt; close to some power of 2.<br /> * A length-&lt;i&gt;n&lt;/i&gt; extended photon will converge to a [[clean]] puffer leaving one oscillator per period if &lt;i&gt;n&lt;/i&gt; not in A099627 and 3×2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt; ≤ &lt;i&gt;n&lt;/i&gt; ≤ 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;+2&lt;/sup&gt; for some positive integer &lt;i&gt;k&lt;/i&gt;. Its trailing end is divided into two halves, one becomes a disappearing [[spark]] from a diagonal line of 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;+1&lt;/sup&gt;-1 cells while another evolves into the oscillator.<br /> ** The oscillators are square and oscillate according to a [[block cellular automaton]] on the [[Margolus neighbourhood]]. Previously the phenomenon was observed and studied in a outer-totalistic rule called [[2×2]], where the oscillators are made up of 2×2 [[block]]s. Here the oscillators can be seen as composed of [[tub]]s, though it is not the smallest element because width-1 forms as labelled above exist. A width-1 diagonal line of 2&lt;i&gt;m&lt;/i&gt; cells has a period of Sloane's {{OEIS|A160657}}(&lt;i&gt;m&lt;/i&gt;), and appears in the list above at length &lt;i&gt;n&lt;/i&gt; = {{OEIS|A079946}}(&lt;i&gt;m&lt;/i&gt;), or Sloane's {{OEIS|A049039}}(&lt;i&gt;m&lt;/i&gt;)-th place among oscillators. Other diagonal widths are resulted from [[inflation]] by a factor of 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;×2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;, which also multiplies the period by 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;.<br /> * An extended photon is [[dirty]] if its length does not satisfy any requirements above. In particular, a period-2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt; extended photon produces the largest amount of [[ash]] if its length is 2&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;&lt;/sup&gt;+2. Behind a puffer of this kind, straight diagonal lines replicate and decay in a complex manner leading to configurations with some self-similarity.<br /> <br /> [[Image:b2aes_complex_puffer.png|framed|center|The tail of a westbound length-16386 extended photon at generation 40960, showing both decaying and settled ash fields. Scale is 2^5:1.]]<br /> <br /> : However, no matter how dirty the ash is, it is guaranteed not to emit photons or replicators because B2e-governed evolution will only have new cells born at the same checkerboard parity, therefore prohibiting domino frontend from forming.<br /> * Regardless of the type, the reaction [[envelope]] of the tail is limited to one side of the extended photon. This implies that it can have two halves extended independently; not only their length can be different, but also they can operate at different periods with any [[phase]] difference.<br /> <br /> {{EmbedViewer<br /> |rle = x = 3, y = 3, rule = B2ae/S<br /> 2o$2bo$2bo!<br /> |position = right<br /> |viewerconfig = #C [[ THUMBSIZE 4 ]]<br /> }}<br /> <br /> Apart from the long extended photons above, the emergence of patterns may also be observed with smaller seeds, for instance the diagonally [[symmetry|symmetric]] [[polyplet#Tetraplets|tetraplet]] shown to the right. If random [[soup]]s are put behind the long diagonal line after a while to perturb the [[wavestretcher]], the outcome will be more complicated.&lt;ref name=&quot;post39221&quot; /&gt;<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;ref name=&quot;post39221&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 39221<br /> |title = Re: Thread for basic questions<br /> |author = blah<br /> |date = January 10, 2017<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - note: the author claims to have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where the author originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82126 2021-08-20T11:34:28Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I plan to include some examples of gates using LifeViewer. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 18:33, 19 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82112 2021-08-20T01:49:56Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I plan to include some examples of gates using LifeViewer windows. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 18:33, 19 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82111 2021-08-20T01:46:30Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I do plan to add the 'gate schematics' using LifeViewer windows. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 18:33, 19 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82110 2021-08-19T18:47:37Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I do plan to add the 'gate schematics' as I call it, to show how signal circuitry would be done in simple CAs. That is part of the information that is in the old web page. My next step is to embed the LifeViewer windows. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 18:33, 19 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82109 2021-08-19T18:33:02Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I do plan to add the 'gate schematics' as I call it, to show how signal circuitry would be done in simple CAs. That is part of the information that is in the old web page. Most of them are animated GIF files. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 18:33, 19 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82108 2021-08-19T18:32:29Z

<p>DavidConant: </p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)<br /> <br /> : You're welcome :) I added a few material for curiosity just now; hopefully there is only the part for logic gates that needs expanding in details. Have fun! [[User:GUYTU6J|GUYTU6J]] ([[User talk:GUYTU6J|talk]]) 16:17, 19 August 2021 (UTC)<br /> <br /> Thanks for cleaning up the article too, I'm still learning about Wiki editing and protocols. I do plan to add the 'gate schematics' as I call it, to show how signal circuitry would be done in simple CAs. That is part of the information that is in the old web page. Most of them are animated GIF files.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82077 2021-08-18T13:08:42Z

<p>DavidConant: /* Linear feedback shift registers */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==Linear feedback shift registers==<br /> Any BSG, with width is 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a Linear Feedback Shift Register. Only numbers from {{OEIS|A123399}} generate maximum length LFSRs with periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;&lt;i&gt;419&lt;/i&gt;&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;&lt;i&gt;126&lt;/i&gt;&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,240<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,990<br /> |}<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82076 2021-08-18T13:07:44Z

<p>DavidConant: /* Linear feedback shift registers */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==Linear feedback shift registers==<br /> Any BSG, with width is 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a Linear Feedback Shift Register. Only numbers from {{OEIS|A123399}} generate maximum length LFSRs, wnth periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;&lt;i&gt;419&lt;/i&gt;&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;&lt;i&gt;126&lt;/i&gt;&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,240<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,990<br /> |}<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82073 2021-08-18T00:00:17Z

<p>DavidConant: </p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==Linear feedback shift registers==<br /> Any BSG, with width is 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a Linear Feedback Shift Register. Only numbers from {{OEIS|A123399}} generate maximum length LFSRs, and have periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;&lt;i&gt;419&lt;/i&gt;&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;&lt;i&gt;126&lt;/i&gt;&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,240<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,990<br /> |}<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82072 2021-08-17T23:57:25Z

<p>DavidConant: </p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ===Linear feedback shift registers===<br /> Any BSG, with width is 8&lt;i&gt;n&lt;/i&gt;+1 generates the output of a Linear Feedback Shift Register. Only numbers from {{OEIS|A123399}} generate maximum length LFSRs, and have periods of 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-1.&lt;br&gt;<br /> <br /> The highest number listed is 419 which has a period of 2&lt;sup&gt;&lt;i&gt;419&lt;/i&gt;&lt;/sup&gt;-1, about 1.35 x 10&lt;sup&gt;&lt;i&gt;126&lt;/i&gt;&lt;/sup&gt; before the output repeats.&lt;br&gt;<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! | {{OEIS|A123399}} !! Width !! Period<br /> |-<br /> | 1 || 9 || 1<br /> |-<br /> | 2 || 17 || 3<br /> |-<br /> | 3 || 25 || 7<br /> |-<br /> | 5 || 41 || 31<br /> |-<br /> | 6 || 49 || 63<br /> |-<br /> | 9 || 73 || 511<br /> |-<br /> | 11 || 89 || 2,047<br /> |-<br /> | 14 || 113 || 16,383<br /> |-<br /> | 23 || 185 || 8,388,607<br /> |-<br /> | 26 || 209 || 67,108,863<br /> |-<br /> | 29 || 233 || 536,870,911<br /> |-<br /> | 30 || 241 || 1,073,741,824<br /> |-<br /> | 33 || 265 || 8,589,934,592<br /> |-<br /> | 35 || 281 || 34,359,738,368<br /> |-<br /> | 39 || 313 || 549,755,813,887<br /> |-<br /> | 41 || 329 || 2,199,023,255,551<br /> |-<br /> | 51 || 409 || 2,251,799,813,685,240<br /> |-<br /> | 53 || 425 || 9,007,199,254,740,990<br /> |}<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82071 2021-08-17T23:47:45Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit BSG info] Google spreadsheet<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82070 2021-08-17T23:45:55Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit Google spreadsheet] with information about BSG periods.<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.<br /> * [https://drive.google.com/file/d/1UMSTvG-z0h5Pf0RVrznSGAHDtZpihrJY/view?usp=sharing BSG.c] Original C program used to generate output and period.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82068 2021-08-17T23:13:15Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit Google spreadsheet] with information about BSG periods.<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82067 2021-08-17T23:12:10Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit Google spreadsheet] with information about BSG periods.<br /> * [https://drive.google.com/file/d/1bOS52ssJ_yo2UFBwNVktexf04eUlsm4-/view?usp=sharing LIFE050.COM] Zip file with the original DOS program where I originally discovered the Logic Rule. You will need DOSBox or another DOS emulator to run it.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82066 2021-08-17T22:51:33Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}<br /> * [https://docs.google.com/spreadsheets/d/1z6mX6dWofmo7KwOuY-4U-ixorAnaTdfZtAPwCcs-_gY/edit Google spreadsheet] with information about BSG periods.</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA_talk:Logic_Rule&diff=82065 2021-08-17T22:35:13Z

<p>DavidConant: Created page with &quot;Thanks to User:GUYTU6J for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found...&quot;</p> <hr /> <div>Thanks to [[User:GUYTU6J]] for creating this page. I discovered this rule in 1993 and have been working on it off and on ever since then. I will update here what I have found out since the web page was originally published. [[User:DavidConant|DavidConant]] ([[User talk:DavidConant|talk]]) 22:35, 17 August 2021 (UTC)</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82064 2021-08-17T22:27:48Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but it's not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=82063 2021-08-17T21:53:56Z

<p>DavidConant: /* External Links */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down) - I have all of the files for this web site, but is not currently hosted anywhere.<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=81959 2021-08-16T20:52:02Z

<p>DavidConant: /* Bit Stream Generators */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down)<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}</div>

DavidConant https://conwaylife.com/w/index.php?title=OCA:Logic_Rule&diff=81958 2021-08-16T20:51:10Z

<p>DavidConant: /* Bit Stream Generators */</p> <hr /> <div>{{stub}}<br /> {{Rule<br /> |name = Logic Rule<br /> |b = 2ae<br /> }}<br /> '''Logic Rule''' (or '''LogicRule''') is a minimalistic [[isotropic non-totalistic Life-like cellular automaton]] devised by David Conant in March 2001. It is one of the early demonstrations showing how [[signal]] [[circuitry]] can be built in simple cellular automata.<br /> <br /> In this rule, a cell is alive the next generation if exactly two neighbouring cells are alive, and those two cells are connected to each other either orthogonally or diagonally. With Hensel notation this is expressed as B2ae/S. The rule is a built-in &quot;general binary&quot; rule in [[Mirek's Cellebration]] along with several patterns, but the program uses another notation which turned out to be inaccurate later.&lt;ref name=&quot;post66037&quot; /&gt;<br /> <br /> ==Patterns==<br /> <br /> The transition B2a indicates that a [[domino]] is a linear [[replicator]] at [[c orthogonal]] following Wolfram's [[Rule 90]], whereas B2e allows for the existence of a family of oscillators, the smallest of which is the period-2 [[duoplet]] (originally named &quot;#2 Oscillator&quot;). Meanwhile, there are two small [[photon]]s (lightspeed [[spaceship|glider]]s): the first of which is the period-1 [[moon]] as in [[seeds]], and the second is the period-2 [[banana spark|banana]]. Serving as the basic unit of information in signal circuitry, any of them may also be called a &quot;bit&quot; and the presence/absence in a &quot;bit stream&quot; can be encoded in binary as 1/0.<br /> <br /> ===Bit Stream Generators===<br /> <br /> [[Gun]]s for both of the photons can easily be constructed from dominoes and duoplets, as an interaction between two dominoes gives two moons as well as two shifted dominoes, and duoplets can suppress dominoes from replicating, [[eater|eat]] a moon, or convert a moon to a banana. The smallest known moon gun operates at period 8, and is called an &quot;Index Bit Stream Generator&quot; (IBSG).<br /> <br /> {{EmbedViewer<br /> |rle = x = 66, y = 12, rule = B2ae/S<br /> 12b2o38b2o$11bo2bo39bo2$50bo$49bo2$bo8bo4bo8bo16bo8bo4bo8bo$o9bo4bo9bo<br /> 14bo9bo4bo9bo3$13bo39bo$12bo39bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = IBSG and a p8 banana gun based on it&lt;br&gt;<br /> }}<br /> <br /> IBSG is systematically called #9 BSG, where the index #9 indicates a distance of 9 cells between the domino and the duoplet. A number of BSGs with larger indices of the form 8&lt;i&gt;n&lt;/i&gt;+1 are documented by comparing their output streams with the IBSG. For example, the next smallest member is #17 BSG that emits a string of 110's with period 3. The actual period measured in [[tick]]s is 3×8 = 24.<br /> <br /> {{EmbedViewer<br /> |rle = x = 42, y = 6, rule = B2ae/S<br /> bo16bo4bo16bo$o17bo4bo17bo3$21bo$20bo!<br /> |position = center<br /> |viewerconfig = #C [[ ZOOM 8 ]]<br /> |caption = The #17 BSG<br /> }}<br /> <br /> It appears that the period of &lt;i&gt;n&lt;/i&gt;-th BSG follows Sloane's {{OEIS|A086839}} and log₂(period+1) almost follows Sloane's {{OEIS|A003558}} except for the #145 BSG in the following list. Here the contents of bit outputs are calculated with a C program.<br /> <br /> &lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;<br /> Click on &quot;Expand&quot; to the right to view statistics and bit outputs for selected BSGs. Only the first eight and last eight bits are shown for long strings; click on them to toggle the view of full strings.<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> {| class=&quot;sortable wikitable&quot; style=&quot;margin-left:auto;margin-right:auto&quot;<br /> |-<br /> ! rowspan=&quot;2&quot; | Index !! colspan=&quot;2&quot; | Statistics !! colspan=&quot;2&quot; width=300px | Content<br /> |-<br /> ! Period !! log₂(Period+1) !! width=150px | Brief !! width=150px | Full<br /> |-<br /> | 9 || 1 || 1 || colspan=&quot;2&quot; | 1 <br /> |-<br /> | 17 || 3 || 2 || colspan=&quot;2&quot; | 110<br /> |-<br /> | 25 || 7 || 3 || colspan=&quot;2&quot; | 1101001<br /> |-<br /> | 33 || 7 || 3 || colspan=&quot;2&quot; | 1101000<br /> |-<br /> | 41 || 31 || 5 || &lt;span class=&quot;mw-customtoggle-41&quot;&gt;11010001...00110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-41&quot;&gt;1101000100101011&lt;br&gt;000011100110111&lt;/div&gt;<br /> |-<br /> | 49 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-49&quot;&gt;11010001...01111110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-49&quot;&gt;1101000100001011&lt;br&gt;0010101001001111&lt;br&gt;0000011011100110&lt;br&gt;001110101111110&lt;/div&gt;<br /> |-<br /> | 57 || 15 || 4 || colspan=&quot;2&quot; | 110100010000001<br /> |-<br /> | 65 || 15 || 4 || colspan=&quot;2&quot; | 110100010000000<br /> |-<br /> | 73 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-73&quot;&gt;11010001...01111111&lt;/span&gt; ||&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-73&quot;&gt;1101000100000001&lt;br&gt;0010101000001011&lt;br&gt;0000010001001110&lt;br&gt;0010101011101101&lt;br&gt;0000001010110010&lt;br&gt;0001000011110100&lt;br&gt;1010011001001100&lt;br&gt;0111111011111011&lt;br&gt;0000101000101110&lt;br&gt;0100010100101110&lt;br&gt;1010001100101000&lt;br&gt;0101111100010010&lt;br&gt;0100011010110110&lt;br&gt;1011100011011000&lt;br&gt;1011011101110100&lt;br&gt;1101010101001111&lt;br&gt;0000000011100110&lt;br&gt;0000011011111100&lt;br&gt;0011101000011001&lt;br&gt;1010010011111110&lt;br&gt;0110111000001111&lt;br&gt;1010110001100010&lt;br&gt;0011101111010101&lt;br&gt;1010100100000110&lt;br&gt;0001101000111100&lt;br&gt;1110010110011110&lt;br&gt;1110011111001010&lt;br&gt;1111000111000010&lt;br&gt;0110110110010111&lt;br&gt;1011011110010010&lt;br&gt;1101001110110011&lt;br&gt;001110101111111&lt;/div&gt;<br /> |-<br /> | 81 || 63 || 6 || &lt;span class=&quot;mw-customtoggle-81&quot;&gt;11010001...00110110&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-81&quot;&gt;1101000100000001&lt;br&gt;0000101000101011&lt;br&gt;0000000001101110&lt;br&gt;000011100110110&lt;/div&gt;<br /> |-<br /> | 89 || 2047 || 11 || &lt;span class=&quot;mw-customtoggle-89&quot;&gt;11010001...01101001&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-89&quot;&gt;1101000100000001&lt;br&gt;0000001010100011&lt;br&gt;0000000010100110&lt;br&gt;0100011010100111&lt;br&gt;0000101000101010&lt;br&gt;0100010000111110&lt;br&gt;0001000010110110&lt;br&gt;0110110010010111&lt;br&gt;0000000001001110&lt;br&gt;0010111001001110&lt;br&gt;1011001000010010&lt;br&gt;0010111101110110&lt;br&gt;0000101100101000&lt;br&gt;1110011100111110&lt;br&gt;1011011011110000&lt;br&gt;1100101110011101&lt;br&gt;0010101000001010&lt;br&gt;0001000001011110&lt;br&gt;0000010001111110&lt;br&gt;1011100001110110&lt;br&gt;0100010100000110&lt;br&gt;1010100110001100&lt;br&gt;1010010011011111&lt;br&gt;1011110110010110&lt;br&gt;0000001011101101&lt;br&gt;0010111011101000&lt;br&gt;1111010010110101&lt;br&gt;0010010101011100&lt;br&gt;0100111100010110&lt;br&gt;1111011110001000&lt;br&gt;1101101001100111&lt;br&gt;1100101111010011&lt;br&gt;0000010001000100&lt;br&gt;1010001001001100&lt;br&gt;0010101100001000&lt;br&gt;1011001101011110&lt;br&gt;1010001000111000&lt;br&gt;0001111101111100&lt;br&gt;0110111101000010&lt;br&gt;1001011110011100&lt;br&gt;0001001010110011&lt;br&gt;0010101010010110&lt;br&gt;0100110011000011&lt;br&gt;0110000001011010&lt;br&gt;1110011010011010&lt;br&gt;0101001101010111&lt;br&gt;0110011111110001&lt;br&gt;1100100100111110&lt;br&gt;0010101010101100&lt;br&gt;0101011011111001&lt;br&gt;0000111001010100&lt;br&gt;1111110001001000&lt;br&gt;0101010110110000&lt;br&gt;1100010100011011&lt;br&gt;1010010010010001&lt;br&gt;1001001111011000&lt;br&gt;1011000011111111&lt;br&gt;0000000110011110&lt;br&gt;1111111110011101&lt;br&gt;1100001001100010&lt;br&gt;1111100111100110&lt;br&gt;0011110000010101&lt;br&gt;1111000001101101&lt;br&gt;1101101110001101&lt;br&gt;0000000000111010&lt;br&gt;0001101000111010&lt;br&gt;0110111000001110&lt;br&gt;0001101011010010&lt;br&gt;0000011011100111&lt;br&gt;1010001011101010&lt;br&gt;0110110110101111&lt;br&gt;1011100101110100&lt;br&gt;1110011000000110&lt;br&gt;0000111111001010&lt;br&gt;0000001111010101&lt;br&gt;1001011111010010&lt;br&gt;0011110011111101&lt;br&gt;1001100010000100&lt;br&gt;0110001110110101&lt;br&gt;0110101101110010&lt;br&gt;0000000110100100&lt;br&gt;1110010110100111&lt;br&gt;1010110001101100&lt;br&gt;1110001100110100&lt;br&gt;0011101011110010&lt;br&gt;0101001010000111&lt;br&gt;1011011000100010&lt;br&gt;1011100101001110&lt;br&gt;1111110000111100&lt;br&gt;0110000111000100&lt;br&gt;0001100100000111&lt;br&gt;1001000100110101&lt;br&gt;1001111000010111&lt;br&gt;1111010100101011&lt;br&gt;1101101011000001&lt;br&gt;1000110101110100&lt;br&gt;0000111001101110&lt;br&gt;1110011001110010&lt;br&gt;0011101110111110&lt;br&gt;1101111111001001&lt;br&gt;1010001001110110&lt;br&gt;0011000100110010&lt;br&gt;1101110101010000&lt;br&gt;1011100011101010&lt;br&gt;0001100110011011&lt;br&gt;1100110110101000&lt;br&gt;1111101000110011&lt;br&gt;1010101111000111&lt;br&gt;1100110010010000&lt;br&gt;0100001100001001&lt;br&gt;0110001110001111&lt;br&gt;0111000101001000&lt;br&gt;0110111110101010&lt;br&gt;1111111101110101&lt;br&gt;1010101010001011&lt;br&gt;0100000111011110&lt;br&gt;0101011101011101&lt;br&gt;1110101111110011&lt;br&gt;0101000000100100&lt;br&gt;1011011010000100&lt;br&gt;1111111111101001&lt;br&gt;1111011000010110&lt;br&gt;0010010111111010&lt;br&gt;0000100110110001&lt;br&gt;1111110110100010&lt;br&gt;1001111001011001&lt;br&gt;1101101101100101&lt;br&gt;011010001101001&lt;/div&gt;<br /> |-<br /> | 97 || 1023 || 10 || &lt;span class=&quot;mw-customtoggle-97&quot;&gt;11010001...11101000&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-97&quot;&gt;1101000100000001&lt;br&gt;0000000010100011&lt;br&gt;0000001010100010&lt;br&gt;0100010010100111&lt;br&gt;0000000000101110&lt;br&gt;0100111000101110&lt;br&gt;0001001010110010&lt;br&gt;0100111010010111&lt;br&gt;0010101000001010&lt;br&gt;0000010001011110&lt;br&gt;0001000001010110&lt;br&gt;1010110001110110&lt;br&gt;0000000100101110&lt;br&gt;1110110100101100&lt;br&gt;1011000011110110&lt;br&gt;1110100110010111&lt;br&gt;0000010001000100&lt;br&gt;0010101001001100&lt;br&gt;1010001000011000&lt;br&gt;0011101101011100&lt;br&gt;0000101100101010&lt;br&gt;1011001100111100&lt;br&gt;1110011001011010&lt;br&gt;1001111110010110&lt;br&gt;0010101010101001&lt;br&gt;0000011011111100&lt;br&gt;0101010011110001&lt;br&gt;1010110001011000&lt;br&gt;0100111100000000&lt;br&gt;1111111110011110&lt;br&gt;1111111001100001&lt;br&gt;1100001110011101&lt;br&gt;0000000000011010&lt;br&gt;0011101000011010&lt;br&gt;0000111001101110&lt;br&gt;0011101001110010&lt;br&gt;1110011000000110&lt;br&gt;0000001111001010&lt;br&gt;0000111111001101&lt;br&gt;1001101111010010&lt;br&gt;0000000011100101&lt;br&gt;1010010011100100&lt;br&gt;0110111110101101&lt;br&gt;1010011101110010&lt;br&gt;1111110000111100&lt;br&gt;0001100111000100&lt;br&gt;0110000111110111&lt;br&gt;1110100100110100&lt;br&gt;0000011011100110&lt;br&gt;0110111011101011&lt;br&gt;1010001000110110&lt;br&gt;0111010101110010&lt;br&gt;0001100110011000&lt;br&gt;1111110110101011&lt;br&gt;1100110001010000&lt;br&gt;1001101111001000&lt;br&gt;0011101011111111&lt;br&gt;1010101010001010&lt;br&gt;0101010111011111&lt;br&gt;0100000101110100&lt;br&gt;1111111111110110&lt;br&gt;0001011000001001&lt;br&gt;1111101000100101&lt;br&gt;111010011101000&lt;/div&gt;<br /> |-<br /> | 105 || 511 || 9 || &lt;span class=&quot;mw-customtoggle-105&quot;&gt;11010001...10110111&lt;/span&gt; || &lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-105&quot;&gt;1101000100000001&lt;br&gt;0000000000101011&lt;br&gt;0000101000101010&lt;br&gt;0000010001101111&lt;br&gt;0000000001000100&lt;br&gt;1010011001000110&lt;br&gt;0000101000111000&lt;br&gt;1011011001111101&lt;br&gt;0000001010100010&lt;br&gt;0001000010110010&lt;br&gt;0100010000010110&lt;br&gt;1011001010111110&lt;br&gt;0010101001001110&lt;br&gt;1010011101001000&lt;br&gt;0100111100111100&lt;br&gt;1001110111011011&lt;br&gt;0000000000001110&lt;br&gt;0110111000001110&lt;br&gt;0000000110100010&lt;br&gt;0110110110101100&lt;br&gt;0011101000011010&lt;br&gt;0000001110111110&lt;br&gt;1110011110100100&lt;br&gt;0110110101010110&lt;br&gt;0000011011100110&lt;br&gt;0001101011011100&lt;br&gt;1110011000111111&lt;br&gt;1011100011000010&lt;br&gt;0011101001110101&lt;br&gt;1010111111010010&lt;br&gt;1110100111110011&lt;br&gt;110110110110111&lt;/div&gt;<br /> |-<br /> | 113 || 16383 || 14 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215120130/http://logic-ca.com/output_113.html archieve])<br /> |-<br /> | 121 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 129 || 31 || 5 || colspan=&quot;2&quot; | <br /> |-<br /> | 137 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 145 || 87381 || 16.3244 || colspan=&quot;2&quot; | (See [https://web.archive.org/web/20041215112447/http://logic-ca.com/output_145.html archieve])<br /> |-<br /> | 153 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 161 || 1023 || 10 || colspan=&quot;2&quot; | <br /> |-<br /> | 169 || 127 || 7 || colspan=&quot;2&quot; | <br /> |-<br /> | 177 || 4095 || 12 || colspan=&quot;2&quot; | <br /> |-<br /> | 185 || 8388607 || 23 || colspan=&quot;2&quot; | <br /> |-<br /> | 193 || 2097151 || 21|| colspan=&quot;2&quot; | <br /> |-<br /> | 201 || 255 || 8 || colspan=&quot;2&quot; | <br /> |-<br /> | 209 || 67108863 || 26 || colspan=&quot;2&quot; | <br /> |-<br /> | 217 || 1048575 || 20 || colspan=&quot;2&quot; | <br /> |-<br /> | 225 || 511 || 9 || colspan=&quot;2&quot; | <br /> |}<br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> When removing the two terminal duoplets of a BSG, the dominoes replicate unboundedly and result in a non-repeating bit stream generator (XBSG). Its output string is predictable, though; the 1's appear at 1, 2, 4, 8, ..., 2&lt;sup&gt;&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;-th place and the rest is sea of 0's. This is listed in {{OEIS|A0036987}}, the Fredholm-Rueppel sequence.<br /> <br /> ===Logic gates===<br /> <br /> Photon interactions can be used to demonstrate logic gates, and arranging input(s) and IBSG(s) suffices to construct any of the gates. The simplest case, a NOT gate forms by colliding an input stream with an IBSG coming orthogonally. For each of the 1's in the input stream, a 2-banana annihilation occurs, while each of the 0's allows for a banana to pass through the intersection, thus effectively carrying out inversion.<br /> <br /> ===Other patterns===<br /> <br /> Thanks to B2e, the [[Margolus]] oscillators as in the [[OCA:Banks-I#Related rules|black-white reversal of Banks-I]] exist.<br /> <br /> ==References==<br /> &lt;references&gt;<br /> &lt;ref name=&quot;post66037&quot;&gt;{{LinkForumThread<br /> |format = ref<br /> |p = 66037<br /> |title = Interpreting MCell's &quot;general binary&quot; rule family<br /> |author = Jeremy Tan<br /> |date = November 30, 2018<br /> }}&lt;/ref&gt;<br /> &lt;/references&gt;<br /> <br /> ==External Links==<br /> * [https://web.archive.org/web/20050122081406/http://logic-ca.com/ Cellular Automata Logic Rule] at logic-ca.com (Jan 2005 archive on the Wayback Machine; currently down)<br /> {{LinkCatagolueRule|b2aes|rulename=LogicRule}}<br /> {{LinkMirek|rullex_gebi.html#LogicRule|title=MCell built-in General binary rules: LogicRule}}</div>

DavidConant https://conwaylife.com/w/index.php?title=User:DavidConant&diff=81957 2021-08-16T18:31:57Z

<p>DavidConant: Created page with &quot;Updating Logic Rule.&quot;</p> <hr /> <div>Updating Logic Rule.</div>

DavidConant https://conwaylife.com/w/index.php?title=User_talk:DavidConant&diff=81956 2021-08-16T18:30:42Z

<p>DavidConant: </p> <hr /> <div>I discovered this ca rule in 1993 using the DOS program Life050.com. Namely, it can simulate Linear Feedback Shift Registers</div>

DavidConant https://conwaylife.com/w/index.php?title=User_talk:DavidConant&diff=81955 2021-08-16T18:30:16Z

<p>DavidConant: Additional information will be added to this page.</p> <hr /> <div>I discovered this ca rule in 1973 using the DOS program Life050.com. Namely, it can simulate Linear Feedback Shift Registers</div>

DavidConant