Counting and Hardness-of-Finding Fixed Points in Cellular Automata on Random Graphs

TL;DR

This paper investigates the fixed points of cellular automata on random graphs, analyzing their existence, structure, and computational hardness using advanced statistical physics methods and experimental algorithms.

Contribution

It applies the cavity method to characterize fixed point structures and links solution clustering to algorithmic difficulty in cellular automata on random graphs.

Findings

Cavity method reveals solution space structure and clustering.

Frozen variables in solutions correlate with computational hardness.

Belief-propagation reinforcement algorithm performance aligns with theoretical predictions.

Abstract

We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.