Schema Redescription in Cellular Automata: Revisiting Emergence in Complex Systems

TL;DR

This paper introduces a schema redescription method for Boolean automata that reduces redundancy in transition tables, enabling better comparison of cellular automata rules beyond emergent behaviors.

Contribution

The authors develop a two-symbol schema redescription technique to characterize canalization in Boolean functions, facilitating rule comparison independent of emergent patterns.

Findings

GKL CA is a special case of GP-derived CA.

Schema redescription reveals similarities not evident from emergent behavior.

Method improves understanding of local interactions in cellular automata.

Abstract

We present a method to eliminate redundancy in the transition tables of Boolean automata: schema redescription with two symbols. One symbol is used to capture redundancy of individual input variables, and another to capture permutability in sets of input variables: fully characterizing the canalization present in Boolean functions. Two-symbol schemata explain aspects of the behaviour of automata networks that the characterization of their emergent patterns does not capture. We use our method to compare two well-known cellular automata for the density classification task: the human engineered CA GKL, and another obtained via genetic programming (GP). We show that despite having very different collective behaviour, these rules are very similar. Indeed, GKL is a special case of GP. Therefore, we demonstrate that it is more feasible to compare cellular automata via schema redescriptions of their rules, than by looking at their emergent behaviour, leading us to question the tendency in complexity research to pay much more attention to emergent patterns than to local interactions.