Lattice games without rational strategies

TL;DR

This paper demonstrates that certain lattice games can perform universal computation, providing a counterexample to the conjecture that all such games have rational strategies, by exhibiting a three-dimensional game with non-rational winning positions.

Contribution

The paper proves that some lattice games support universal computation and presents an explicit counterexample to the conjecture that all lattice games have rational strategies.

Findings

Lattice games can support universal computation.

Counterexample: a 3D lattice game with non-rational winning positions.

Disproves the conjecture that all lattice games have rational strategies.

Abstract

We show that the lattice games of Guo and Miller support universal computation, disproving their conjecture that all lattice games have rational strategies. We also state an explicit counterexample to that conjecture: a three dimensional lattice game whose set of winning positions does not have a rational generating function.