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how many rule strings are there S/B
Posted: March 31st, 2024, 3:57 pm
by kmadpar
I'm wondering how many possible S/B rule strings possibilities there are.
Re: how many rule strings are there S/B
Posted: March 31st, 2024, 4:25 pm
by confocaloid
The set of all possible rulestrings is larger than the set of possible cellular automata that can be defined by a rulestring. For example, Conway's Life can be equivalently defined by the following rulestrings:
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B3/S23
B3/S32
23/3
B3cekainyqjr/S2aceikn3aceijknqry
The set of acceptable rulestrings that can be entered into Golly/LifeViewer is much larger:
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#C This is Conway's Life
x = 3, y = 3, rule = B3ceikanqjry/S2aceikn34-aceiknyqjrtwz5-aceiknyqjr6-cenkai7-ce
b2o$2o$bo!
However, if you just wanted to know the number of cellular automata that can be defined, then here are sizes of some rulespaces:
- there are 2^18 = 262144 Life-like cellular automata. Every cellular automaton of this type can be defined by writing down a birth/survival rulestring with neighbour-counting conditions. Example: B357/S238.
- there are 2^102 = 5070602400912917605986812821504 two-state isotropic cellular automata with range-1 Moore neighbourhood. Every such cellular automaton can be defined by a rulestring written in Hensel notation. Example: B2n3/S23-q.
- there are 2^14 = 16384 two-state outer-totalistic cellular automata with range-1 hexagonal neighbourhood. Example: B25/S34H.
- there are 2^26 = 67108864 two-state isotropic cellular automata with range-1 hexagonal neighbourhood. Every such cellular automaton can be defined by a rulestring written in extended Callahan's notation. Example: B2o/S2m34H.
Re: how many rule strings are there S/B
Posted: April 12th, 2024, 12:05 am
by H. H. P. M. P. Cole
In each finite 2-state rulespace, there are a finite number of conditions that can be toggled. The rulespace is taken to be the power set of the set of toggleable conditions.
As an example, consider the outer-totalistic von Neumann rulespace. The set of conditions is {B0,B1,B2,B3,B4,S0,S1,S2,S3,S4}, and the rulespace is the power set of these conditions, giving a total of 1024 rules in this rulespace.
Re: how many rule strings are there S/B
Posted: April 12th, 2024, 12:10 am
by confocaloid
Things become more interesting with three or more cellstates, because the simple representation "set of living conditions" is lost.
Instead of a power set, one has the set of all possible rulesets (where a ruleset can be viewed as a mapping from the set of "toggleable" conditions to the set of cellstates).