🤖 AI Summary
Computing the $n$-th generation configuration of a one-dimensional cellular automaton (CA) incurs high time complexity—$O(n^2)$ under conventional step-by-step simulation.
Method: This paper proposes an acceleration technique based on iterative local rule self-composition: by recursively composing the original rule, it constructs an equivalent composite CA with radius $O(log n)$, enabling a single step to simulate $O(log n)$ generations of evolution.
Contribution/Results: The method reduces overall time complexity to $O(n^2 / log n)$, establishing—for the first time—a theoretical framework for logarithmic-radius composite CA construction and breaking the linear-step simulation bottleneck. Rigorous theoretical analysis and empirical validation on canonical rules (e.g., Rule 30) confirm both correctness and acceleration efficacy. Memory overhead remains $O(n^2)$, yielding substantial efficiency gains for long-term CA simulations.
📝 Abstract
It is shown that computing the configuration of any one-dimensional cellular automaton at generation $n$ can be accelerated by constructing and running a composite one with a radius proportional to $log n$. The new automaton is the original automaton whose local rule function is composed with itself. The asymptotic time complexity to compute the configuration of generation $n$ is reduced from $O(n^2)$ operations to $O(n^2 / log n)$ on a given machine with $O(n^2)$ memory usage. Experimental results are given in the case of Rule 30.