From heaps of matches to the limits of computability

TL;DR

This paper investigates invariant matchstick games with multiple heaps, demonstrating their computational universality and proving that fundamental questions about their equivalence are algorithmically undecidable.

Contribution

It introduces a formal framework for invariant games and proves their capacity for universal computation, leading to undecidability results for game equivalence.

Findings

Invariant games can simulate universal computation.

Deciding game equivalence is algorithmically undecidable.

The framework generalizes classical combinatorial game theory results.

Abstract

We study so-called invariant games played with a fixed number $d$ of heaps of matches. A game is described by a finite list $\mathcal{M}$ of integer vectors of length $d$ specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in $\mathcal{M}$, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap game, the vector $(1,-2)$ would mean adding one match to the first heap and removing two matches from the second heap. If $(1,-2) \in \mathcal{M}$, such a move would be permitted provided there are at least two matches in the second heap. Two players take turns, and a player unable to make a move loses. We show that these games embrace computational universality, and that therefore a number of basic questions about them are algorithmically undecidable. In particular, we prove that there is no algorithm that takes two games $\mathcal{M}$ and $\mathcal{M}'$ (with the same number of heaps) as input, and determines whether or not they are equivalent in the sense that every starting-position which is a first player win in one of the games is a first player win in the other.