Deciding game invariance

TL;DR

This paper investigates the decidability of whether a given sequence of positions can be generated by an invariant take-away game, expanding understanding of game invariance and its computational aspects.

Contribution

It demonstrates that for a broad class of sequences defined by infinite words, the existence of an invariant game with those positions is a decidable problem.

Findings

Decidability established for sequences defined by infinite words

Extension of invariance concepts to a large class of sequences

Connection between sequence properties and game invariance

Abstract

Duch\^ene and Rigo introduced the notion of invariance for take-away games on heaps. Roughly speaking, these are games whose rulesets do not depend on the position. Given a sequence $S$ of positive tuples of integers, the question of whether there exists an invariant game having $S$ as set of $\mathcal{P}$-positions is relevant. In particular, it was recently proved by Larsson et al. that if $S$ is a pair of complementary Beatty sequences, then the answer to this question is always positive. In this paper, we show that for a fairly large set of sequences (expressed by infinite words), the answer to this question is decidable.