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A (one dimensional) cellular automaton is a function1 F : Σ → Σ with the property that there is a K > 0 such that F (x)i depends only on the 2K + 1 coordinates xi−K , xi−K+1, . . . , xi−1, xi, xi+1, . . . , xi+K . A periodic point of σ is any x such that σ^p (x) = x for some p ∈ N, and a periodic point of F is any x such that F^q (x) = x for some q ∈ N. Given a cellular automaton F, a point x ∈ Σ is jointly periodic if there are p, q ∈ N such that σ^p (x) = F^q (x) = x, that is, it is a periodic point under both functions.

This project aims to explore the nature of one-dimensional Cellular Automata, in the hope of finding the structure of cellular automata through its periodic points.

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License: MIT
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1
import argparse

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from InitialConditions import MiddleStateIsOne, RandomState, FarRightIsOne

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from Rules import simple_rule, apply_elementary_rule, rule_110

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from Automata import Automata

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def main():

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    args, initial_condition = parse_args()

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    # Always multiply the number of evolutions by 2 to get the length of each row. 

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    # This produces cleaner images.

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    automata = Automata(initial_condition(args.evolutions * 2), apply_elementary_rule(args.rule_number))

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    automata.evolve(len(automata.state))

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def parse_args():

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    parser = argparse.ArgumentParser(description="Build elementary cellular automata.") 

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    parser.add_argument('-e', '--evolutions', default=16, type=int, help="The number of evolutions for the CA.")

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    parser.add_argument('-r', '--rule_number', default=30, type=int, help="The rule number for the cellular automata.")

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    parser.add_argument('-s', '--state', default=0, type=int, help="The initial conditions for the first row.\n0: Middle is *. Other cells are -. (default)\n1: Random initial row\n2:Far right is *. (*). Other cells are -.")

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    args = parser.parse_args()

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    initial_conditions_map = {

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        0: MiddleStateIsOne,

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        1: RandomState,

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        2: FarRightIsOne

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    }

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    if not args.state in initial_conditions_map.keys():

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        raise ValueError("Input '{}' for state is not valid. Please choose 0, 1 or 2.".format(args.state))

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    return args, initial_conditions_map[args.state]

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if __name__ == "__main__":

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    main()

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