Impartial games emulating one-dimensional cellular automata and undecidability

TL;DR

This paper demonstrates that certain impartial take-away games can simulate one-dimensional cellular automata like rule 110, leading to undecidability results for game outcomes similar to those in cellular automata.

Contribution

It introduces a class of impartial games that emulate cellular automata, establishing undecidability of their outcome determination, paralleling the Turing-completeness of rule 110.

Findings

Games can simulate cellular automata behavior

Outcome questions for these games are undecidable

Results connect game theory with cellular automata complexity

Abstract

We study two-player \emph{take-away} games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its behavior are algorithmically undecidable. We show that similar questions are undecidable for our \emph{rule 110} game.