Solving Parity Games on Integer Vectors

TL;DR

This paper introduces a framework for solving parity games on infinite graphs with integer vectors, demonstrating the computability of winning sets and initial credits, and resolving open problems in the field.

Contribution

The paper shows the minimal elements of winning sets are computable for a class of single-sided parity games on vass, enabling solutions for multidimensional energy parity games.

Findings

Computability of minimal elements of upward-closed winning sets.

Decidability of Pareto frontiers for initial credits in energy parity games.

Decidability of weak simulation and model checking for vass.

Abstract

We consider parity games on infinite graphs where configurations are represented by control-states and integer vectors. This framework subsumes two classic game problems: parity games on vector addition systems with states (vass) and multidimensional energy parity games. We show that the multidimensional energy parity game problem is inter-reducible with a subclass of single-sided parity games on vass where just one player can modify the integer counters and the opponent can only change control-states. Our main result is that the minimal elements of the upward-closed winning set of these single-sided parity games on vass are computable. This implies that the Pareto frontier of the minimal initial credit needed to win multidimensional energy parity games is also computable, solving an open question from the literature. Moreover, our main result implies the decidability of weak simulation preorder/equivalence between finite-state systems and vass, and the decidability of model checking vass with a large fragment of the modal mu-calculus.