MDA’s Miscellany of Moore Rules

[MDA]

I created this topic mostly to incubate some of my in-progress INT rules (mostly in the Havenspace) until they have matured enough to be posted in the Haven thread or in whatever thread they belong in. Others are free to contribute to the development of these rules.

Some helpful terminology to understand the rules in this thread:

Havenspace: The rulespace starting at B3aijq4ant5ir6i/S2eik3ijnr and ending at B2i3-r4-i5-a67e8/S2-a3-a4-akrw5-ijqy6-ak.

Havenrule: Any rule in the Havenspace.

Havenoid: Any rule not in the Havenspace but which supports infinite families of OWSSes.

Proto-Haven: Any Havenrule or Havenoid which I developed before creating the Haven thread.

Rules

Proto-Haven 1a (B3-cenr4anrt5inr6i/S2eik34inrz5j)

Code: Select all

x = 0, y = 0, rule = B3-cenr4anrt5inr6i/S2eik34inrz5j
!
[[ RANDOMIZE ]]

Proto-Haven 1b (B3-cenr4anrt5inr6i/S2eik34inrz5aj)

Code: Select all

x = 0, y = 0, rule = B3-cenr4anrt5inr6i/S2eik34inrz5aj
!
[[ RANDOMIZE ]]

Proto-Haven 2 (B3-cenr4anrt5ijnr6ai8/S2eik34inrz5aj6ein)

Code: Select all

x = 0, y = 0, rule = B3-cenr4anrt5ijnr6ai8/S2eik34inrz5aj6ein
!
[[ RANDOMIZE ]]

Haven 0.0 (B3-ery4-ikz5-aj6-ac8/S2eik3-a4ceijnyz5-ijy6en)

Code: Select all

x = 0, y = 0, rule = B3-ery4-ikz5-aj6-ac8/S2eik3-a4ceijnyz5-ijy6en
!
[[ RANDOMIZE ]]

Haven 0.0C (B3-ery4-ikz5-aj6-ac/S2eik3-a4ceijnyz5-ijy6en)

Code: Select all

x = 0, y = 0, rule = B3-ery4-ikz5-aj6-ac/S2eik3-a4ceijnyz5-ijy6en
!
[[ RANDOMIZE ]]

Patterns

I found a puffer in a Havenrule and decided to corderize it for fun (6c/32o):

Code: Select all

x = 3, y = 4, rule = B3-r4acnqt5ceikr6i/S2-a3-a4cnq5aeknr6-ak
2bo$b2o$obo$bo!

Code: Select all

x = 120, y = 104, rule = B3-r4acnqt5ceikr6i/S2-a3-a4cnq5aeknr6-ak
9bo5bo5bo5bo$7b3o4b2o3b2o4b3o$9bo5bo5bo5bo$2bo$b2o$obo$bo$26bo$25b2o$
24bobo$25bo4$30bo5bo5bo$29b2o3b2o4b3o$30bo5bo5bo3$bo$obo$b2o38bo$2bo
37b2o$39bobo$40bo4$45bo5bo5bo$2bo41b2o3b2o4b3o$b2o42bo5bo5bo$obo$bo3$
56bo$55b2o$54bobo$55bo4$60bo5bo5bo$59b2o3b2o4b3o$60bo5bo5bo$bo$obo$b2o
$2bo$71bo$70b2o$69bobo$70bo3$2bo$b2o72bo5bo5bo$obo71b2o3b2o4b3o$bo73bo
5bo5bo5$86bo$85b2o$84bobo$85bo4$90bo5bo5bo$bo87b2o3b2o4b3o$obo87bo5bo
5bo$b2o$2bo3$101bo$100b2o$99bobo$100bo$2bo$b2o$obo$bo103bo5bo5bo$104b
2o3b2o4b3o$105bo5bo5bo5$116bo$115b2o$114bobo$115bo3$bo$obo$b2o$2bo114b
obo$9bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo$7b3o4b2o
3b2o4b3o4b2o3b2o4b3o4b2o3b2o4b3o4b2o3b2o4b3o4b2o3b2o4b3o4b2o3b2o5b3o$
9bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo5bo!

[MDA]

I am currently trying to find a reasonably chaotic Havenrule with a spaceship shaped like a UWSS in one of its phases.

Progress:

HR-00 (+B5c)

Code: Select all

x = 0, y = 0, rule = B3-ry4acenqt5ceir6-ek/S2-a3-a4nq5aeknr6-ak
!
[[ RANDOMIZE ]]

Adds a small symmetric c/4d spaceship, into which the UWSS evolves.

HR-01 (+B5c, -B6c, -B6n)

The UWSS becomes a diehard in this rule.

HR-02 (-B3e, +B3y, +B5c, -B5e, +B5q, -B6a, -B6n, -S4q, +S4t)

Still fruitless in terms of UWSS-based spaceships, but it is the first axially-explosive Havenrule I’ve found, as demonstrated by this simple pattern:

Code: Select all

x = 23, y = 3, rule = B3-er4acenqt5ciqr6ci/S2-a3-a4nt5aeknr6-ak
bo19bo$3o17b3o$bo19bo!

HR-03 (-B3e, +B5c, -B5e, -B6n, +S4q)

An interesting dirty quadrep:

Code: Select all

x = 3, y = 4, rule = B3-ery4acenqt5cir6aci/S2-a3-a4nqt5aeknr6-ak
bo$3o$obo$3o!

[MDA]

A breeder in a wiring rule I made:

Code: Select all

x = 30, y = 25, rule = Absolute48
11A$10.A$8.2AMCAKCAKCAOCAQCAQCAQC$8.A20.A$8.A.CKACMACKACQACUACKACM$8.
A.A$8.A.UCAQCAKCAMCAKCAUCAQC$8.A20.A$8.A.CMACKACQACUACKACMACK$8.A.A$
8.A.KCAUCAQCAKCAMCAKCAUC$8.A20.A$8.A.CMACQACpLACMACQACQACQ$8.A.A$8.A.
pLCAQCAQCAQCAQCAMCAKC$8.A20.A$8.A.CQACpRACBACPACKACIACK$8.A.A$8.A.QCA
QCAQCAQCAQCAQCAQC$8.A20.A$8.A.CKACKACKACKACQACQACQ$8.A.A$8.A.KCAICAKC
AKCAKCAKCAKC$8.A20.A$8.11ACKACKACKACK!

Edit (August 16th, 2026, 6:10 PM): Smaller:

Code: Select all

x = 22, y = 13, rule = Absolute48
2.A$2.A$8AKCAKCAICAKCAKC$A20.A$A.CKACKACIACQACQACPACO$A.A$A.KCAKCAUCA
UCAUCAUCAQC$A20.A$A.CpLACpLACKACKACKACIACQ$A.A$A.QCAQCAKCAKCAICABCApR
C$A20.A$14ACKACKACQ!

[MDA]

I’m going to start work on a small script to systematically search for Havenrules with UWSS-shaped spaceships. I’ll post any results I find here.

[MDA]

I just finished writing a simplified, von Neumann version of my wiring rule in 32 states:

Code: Select all

x = 11, y = 10, rule = AbsoluteVN
ACMACMACMA$M8.C$C8.M$A8.A$M8.C$C8.M$A8.A$J8.C$C8.M$AWCAMCABC2A!

@RULE AbsoluteVN

@TABLE

neighborhood: vonNeumann
symmetries: rotate4
n_states: 32

var a={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31} # anything
var b=a
var c=a
var d=a

var x=a

var e={12,15,18} # terminal builders
var f=e
var g=e
var h=e

var y=e

var i={2,10,13,16,19,20,21,23,26,31} # signals
var j=i
var k=i
var l=i

var z=i

var m={0,4,5,6,7,22,24,25} # opaque cells + 0
var n=m
var o=m
var p=m

var w=m

var va={8,24} # 8 + sheath-8
var vb=va
var vc=va
var vd=va

var v=va

var ua={9,25} # 9 + sheath-9
var ub=ua
var uc=ua
var ud=ua

var u=ua

var qa={0,1,4,5,6,7,22,24,25,30} # some special cases (non-signal)
var qb=qa
var qc=qa
var qd=qa

var q=qa

var ra={1,3} # other special cases
var rb=ra
var rc=ra
var rd=ra

var r=ra

var sa={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,30,31} # anything except for recovery state
var sb=sa
var sc=sa
var sd=sa

var s=sa

# Converter logic

1,2,30,c,d,10
1,2,b,c,30,10

1,10,30,c,d,13
1,10,b,c,30,13

1,13,30,c,d,16
1,13,b,c,30,16

1,16,30,c,d,19
1,16,b,c,30,19

1,19,30,c,d,20
1,19,b,c,30,20

1,20,30,c,d,21
1,20,b,c,30,21

1,21,30,c,d,23
1,21,b,c,30,23

1,23,30,c,d,26
1,23,b,c,30,26

1,26,30,c,d,31
1,26,b,c,30,31

1,31,30,c,d,2
1,31,b,c,30,2

# Signal propagation

1,i,b,c,d,i

y,i,b,c,d,i

# NOT gates

1,5,b,c,d,2

2,4,b,c,d,3

4,7,b,c,d,4
5,7,b,c,d,4

4,6,b,c,d,5 # Requires all NOT gates to have at least one input

6,2,b,c,d,7

7,2,b,c,d,7
7,a,b,c,d,6

# Fading

10,3,n,o,p,11
11,a,b,c,d,12
12,a,b,c,d,1

13,3,n,o,p,14
14,a,b,c,d,15
15,a,b,c,d,1

16,3,n,o,p,17
17,a,b,c,d,18
18,a,b,c,d,1

19,3,0,0,0,0

20,3,n,o,p,4

21,3,n,o,p,6

23,3,n,o,p,22

26,3,n,o,p,27

31,3,n,o,p,30

i,3,b,c,d,3

3,a,b,c,d,1

# Sheath logic

22,2,b,c,d,2

# Construction

# - Left

0,10,b,c,d,8
22,10,b,c,d,24

0,12,ub,qc,qd,1
22,12,ub,qc,qd,22

# - Forward

0,13,b,c,d,8
22,13,b,c,d,24

8,14,qb,qc,qd,9
24,14,qb,qc,qd,25

9,15,qb,qc,qd,1
25,15,qb,qc,qd,22

# - Right

0,16,b,c,d,8
22,16,b,c,d,24

0,ua,18,qc,qd,1
22,ua,18,qc,qd,22

# - Delete

0,26,b,c,d,8
22,26,b,c,d,24

8,27,qb,qc,qd,9
24,27,qb,qc,qd,25

9,27,0,c,0,28

# General

8,a,b,c,d,0
24,a,b,c,d,22

0,va,rb,c,d,9
22,va,rb,c,d,25

0,rb,vb,c,d,9
22,rb,va,c,d,25

9,a,b,c,d,0
25,a,b,c,d,22

27,8,b,c,d,27

27,a,b,c,d,29
28,a,b,c,d,0
29,a,b,c,d,1

s,28,b,c,d,0

@COLORS

# Basic states

0 000 000 000
1 064 064 064
2 000 192 255
3 255 255 255

# NOT gate

4 255 064 064
5 255 192 064

# Input for NOT

6 064 255 064
7 128 255 064

# Marker

8 192 192 192
9 160 160 160

# Left

10 064 064 255
11 128 128 255
12 192 192 255

# Forward

13 128 064 255
14 192 160 255
15 224 208 255

# Right

16 255 064 192
17 255 160 224
18 255 208 240

# Backward

19 032 032 032

# Mergers

20 192 064 064
21 064 192 064

# Sheath

22 128 128 160
23 128 128 192

# Sheath II

24 128 128 240
25 128 128 224

# Delete

26 032 032 032
27 016 016 016
28 008 008 008
29 240 240 240

# Convert

30 192 160 064
31 224 208 160

[MDA]

A glide-symmetric 2c/6d in Haven (found using the C++ version of LLSSS):

Code: Select all

x = 10, y = 9, rule = B3-ry4acenqt5eir6-ek/S2-a3-a4nq5aeknr6-ak
2b3o$b2ob3o$2b3o2bobo$o2b2o4bo$3bo3bobo$b3o5bo$3b3obobo$4b2o$5bobo!

Edit (September 8th, 2026, 1:01 PM): And a bilateral c/4d:

Code: Select all

x = 28, y = 28, rule = B3-ry4acenqt5eir6-ek/S2-a3-a4nq5aeknr6-ak
5b4o$5bob5o$4bo5bo2bo$4b2o2b2o$2b3o3b2o4bo$2obob3o2bobobo$o4b5o4b2o$2o
3b3obob2obo2bo$2ob2obobob3obo3b2o$bob2ob2obobo3b3obo$b2o2bo2bo7b3o$bo
5b3ob3o2b2o2bo$5bob2o2bo4bobobo$2bo8bo2b2obobob2o$4b5o4b2obobob4o$6bo
2bo3bob5obob2o$9b4ob4o4bo$7bob3obob5ob2o$8bobobob2obob3o$8b2o3bobob3o
2b2o$11b2obo3bobo3bobo$13b3ob2o3bob4o$13b2ob2obobobob3o$14b2o3bo2b3o$
15bo4b2obo$21b2o3b2o$20b3o2b3o$21b2o2b2o!

Edit (September 8th, 2026, 4:41 PM): An eater for the 2c/6d:

Code: Select all

x = 22, y = 39, rule = B3-ry4acenqt5eir6-ek/S2-a3-a4nq5aeknr6-ak
3o2$2bobo2$4bobo2$6bobo$8bo$8bo6$6b3o16$14b3o$13b2ob3o$14b3o2bobo$12bo
2b2o4bo$15bo3bobo$13b3o5bo$15b3obobo$16b2o$17bobo!

[MDA]

After a month of manual searching, I found this Havenrule:

Code: Select all

x = 0, y = 0, rule = B2i3-r4acenqrt5-ae6i8/S2-ac3-a4-akrw5-ijqy6-ak
!
[[ RANDOMIZE ]]

Besides having the UWSS and not having one-sided exploding OWSSes, this rule eliminates all cis-odd OWSS spacefillers and “trans-odd OWSS spacefiller 1”.

This rule has many D2 methuselahs and a small c/2o spaceship.

Edit (September 19th, 2026, 3:31 PM): More active variant:

Code: Select all

x = 0, y = 0, rule = B2i3-r4-eikr5-ae6ain8/S2-ac3-a4-akrw5-ijqy6-ak
!
[[ RANDOMIZE ]]

This rule could be a good starting point to try to find Haven 2.

More patterns to come soon.

[MDA]

I'm going to attempt to work on a proof against the existence of spaceships in LongLife.

The version included in this post is just a basic outline and may differ from the final proof.

All contributions to this proof are welcome.

OUTLINE:

For a pattern in LongLife to be a spaceship, it needs to be able to permanently escape its bounding diamond and needs to make use of the B3a condition.

We can represent the edge of such a pattern's bounding diamond like this:

Code: Select all

x = 4, y = 4, rule = B345/S5Super
3.B$2.AB$.2AB$4B!

To allow an eventual B3a birth outside this bounding diamond, such a configuration must

a) happen immediately after the first B3a birth or

b) remain supported by S5 for a variable number of generations.

We begin by examining case a:

For a B3a birth to happen immediately after another, the first B3a birth must generate all the cells necessary for the second B3a birth. We then can disprove any isolated B3a from causing a subsequent birth under case a:

Code: Select all

#C The B3a birth in the second pattern cannot be caused using only one B3a birth in the first pattern.
x = 20, y = 5, rule = B345/S5Super
10.D$3.B7.D6.DB$2.AB3.6D4.ADB$.2AB7.D5.3B$4B6.D5.4B!

We can conclude that, for any pattern making use of case a, more than one B3a birth is necessary.

An example frontend making use of 2 B3a births is shown below:

Code: Select all

#C However, this frontend is invalid, because it does not create a B3a birth, but rather A2e.
#C The inner cell is unnecessary, and thus marked as light blue.
x = 22, y = 5, rule = B345/S5Super
4.B6.D9.B$3.AB7.D6.A2B$2.2AB3.6D4.AD2B$.2A2B7.D5.2BIB$5B6.D5.5B!

For the corner cell of a B3a condition to live, the center cell of our frontend must survive, and must be supported by S5a (which is the only S5 condition which is valid here):

Code: Select all

x = 22, y = 5, rule = B345/S5Super
4.B6.D9.B$3.AB7.D6.A2B$2.2AB3.6D4.2A2B$.3AB7.D5.4B$5B6.D5.5B!

However, the resulting pattern after advancing is not valid, as already proven.

We can conclude that, for any pattern making use of case a, more than two B3a births are necessary.

A frontend making use of 3 B3a births is shown below:

Code: Select all

#C This frontend is a simple extension of the previous one.
x = 24, y = 6, rule = B345/S5Super
5.B17.B$4.AB6.D8.A2B$3.2AB7.D6.2A2B$2.3AB3.6D4.2AD2B$.3A2B7.D5.5B$6B
6.D5.6B!

However, this frontend is invalid, since the central cell dies under D6n.

We can also prove that no cell around this frontend can be removed (to cause S5) without breaking it, thus rendering this frontend unusable and proving no spaceship frontends in LongLife can exist under case a.

We then move on to case b.

Assuming any cells outside a pattern's bounding diamond are allowed (under S5a, which is the only valid condition here) to survive at least one generation, this contradicts the assumption that this frontend constitutes an example of case b, since S5a cannot exist on the edge of a pattern's bounding diamond without causing B3a births.

Having ruled out cases a and b, we can "prove" (this is just a draft and very likely to be incorrect) that no spaceships exist in LongLife.

[May13]

I'm going to attempt to work on a proof against the existence of spaceships in LongLife.

c/9 diagonal frontend in LongLife:

Code: Select all

x = 18, y = 18, rule = B345/S5
9bobobo3bo$7bobob2o3b2o$5bobob8o$4b3o2bo3b3obo$3b4ob3o4bobo$2b4obob3o
2b2o$3b2o3b3ob2o$b2o2bobo3b3o$4bobobob2obo$7o2b4o$2bob3ob2ob2o$3o2bob
4o$b2o3b2ob2o$ob2o2b3o$2b2obo$2b4o$b2o$2ob2o!

[MDA]

[...]

c/9 diagonal frontend in LongLife:

Code: Select all

...

Good point. With the existence of frontends like that one, it seems that proving that there are no spaceships in LongLife will become harder (or impossible if a counterexample is found). I hadn't considered the existence of frontends where one cell can cause births behind it. I'll probably work on a new version of the proof which takes frontends like yours into account (unless, of course, we find a spaceship soon).

[NNlk05]

I'm going to attempt to work on a proof against the existence of spaceships in LongLife.

llsss'd c/2 disproof.

Code: Select all

./rlife llsss-recentering-wao --rule B345/S5 --filters bcaf c2-f2b '@bg' XX > longlife_c2.txt

[MDA]

[...]

llsss'd c/2 disproof.

[...]

Thank you. I had, coincidentally, proven the same result by hand using a similar technique to my Holstein c/2 disproof.

[Ninedlingss]

I'm going to attempt to work on a proof against the existence of spaceships in LongLife.

obviously not the best c/5o frontend:

Code: Select all

 x = 27, y = 12, rule = B345/S5
13bo2$10b7o2$8b5ob5o2$5b17o2$3b3obob9obob3o$6bo4bobobo4bo$7ob4o3b4ob7o
$b7o2b3ob3o2b7o!