Towards a Noether-like conservation law theorem for one dimensional reversible cellular automata

TL;DR

This paper explores a Noether-like theorem for conservation laws in one-dimensional reversible cellular automata, linking conservation laws to maximal congruences and null spaces of specific matrices, suggesting a fundamental correspondence.

Contribution

It introduces a novel connection between conservation laws and maximal congruences in 1D reversible cellular automata, extending the theoretical framework beyond automorphisms.

Findings

Conservation laws correspond to null spaces of structured matrices.

Conservation laws relate to maximal congruences of index 2.

In studied examples, all conservation laws are accounted for by these congruences.

Abstract

Evidence and results suggesting that a Noether--like theorem for conservation laws in 1D RCA can be obtained. Unlike Noether's theorem, the connection here is to the maximal congruences rather than the automorphisms of the local dynamics. We take the results of Takesue and Hattori (1992) on the space of additive conservation laws in one dimensional cellular automata. In reversible automata, we show that conservation laws correspond to the null spaces of certain well-structured matrices. It is shown that a class of conservation laws exist that correspond to the maximal congruences of index 2. In all examples investigated, this is all the conservation laws. Thus we conjecture that there is an equality here, corresponding to a Noether--like theorem.