Fast Simulation of Cellular Automata by Self-Composition

TL;DR

This paper introduces a method to accelerate the simulation of one-dimensional cellular automata by using self-composition of rules, reducing computation time from quadratic to near-quadratic with a logarithmic factor, demonstrated on Rule 30.

Contribution

The paper presents a novel self-composition technique for cellular automata that significantly speeds up simulation while highlighting a time-memory tradeoff.

Findings

Reduced time complexity from O(n^2) to O(n^2 / log n)

Demonstrated effectiveness on Rule 30 automaton

Established a tradeoff between computation time and memory usage

Abstract

Computing the configuration of any one-dimensional cellular automaton at generation $n$ can be accelerated by constructing and running a composite rule with a radius proportional to $\log n$. The new automaton is the original one, but with its local rule function composed with itself. Consequently, the asymptotic time complexity to compute the configuration of generation $n$ is reduced from $O(n^2)$-time to $O(n^2 / \log n)$, but with $O(n^2/(\log n)^3)$-space, demonstrating a time-memory tradeoff. Experimental results are given in the case of Rule 30.