Deciding game invariance

Eric Duchene (1) , Aline Parreau (1, 2) , Michel Rigo (2)

1 GOAL - Graphes, AlgOrithmes et AppLications
2 Département de Mathématiques [Liège]

Duchêne 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 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.

Domaines: Combinatoire [math.CO]; Complexité [cs.CC]

Eric Duchene, Aline Parreau, Michel Rigo. Deciding game invariance. Information and Computation, 2017, 253 (1), pp.127-142. ⟨10.1016/j.ic.2017.01.010⟩. ⟨hal-01283830⟩