A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata

TL;DR

This paper introduces a meta-algorithm that efficiently computes the distribution of residue classes of polynomial coefficients raised to a power modulo a prime, enabling fast counting of ON cells in odd-rule cellular automata.

Contribution

It presents a novel meta-algorithm that transforms polynomial exponentiation into a logarithmic-time process for residue class counting, improving computational efficiency.

Findings

Achieves logarithmic time complexity for residue counting

Applicable to polynomials in multiple variables

Enhances analysis of odd-rule cellular automata

Abstract

We develop a meta-algorithm that, given a polynomial (in one or more variables), and a prime p, produces a fast (logarithmic time) algorithm that takes a positive integer n and outputs the number of times each residue class modulo p appears as a coefficient when the polynomial is raised to the power n and the coefficients are read modulo p.