Revisiting the Rice Theorem of Cellular Automata

TL;DR

This paper refines Rice's theorem for cellular automata with binary states, proving that most properties of their limit sets are undecidable, except for surjectivity, thus deepening understanding of their computational limits.

Contribution

It extends Rice's theorem specifically to binary-state cellular automata, identifying which properties are decidable and which are not.

Findings

All properties of limit sets are undecidable except surjectivity.

Refinement of Kari's classical Rice Theorem for binary-state automata.

Clarifies the computational boundaries of cellular automata limit sets.

Abstract

A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e. the infinite sequences of cell states. The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution. In this paper, we prove that all properties of limit sets of cellular automata with binary-state cells are undecidable, except surjectivity. This is a refinement of the classical "Rice Theorem" that Kari proved on cellular automata with arbitrary state sets.