Pareto Fronts for Compositionally Solving String Diagrams of Parity Games

TL;DR

This paper introduces Pareto fronts for open parity games within string diagrams, enabling efficient compositional solutions and a divide-and-conquer approach for large parity games.

Contribution

It presents a novel framework for multi-objective solutions of open parity games and a translation method to leverage existing algorithms.

Findings

Established positional determinacy of open parity games with Pareto fronts.

Developed a translation method converting open parity games into single-objective parity games.

Designed a compositional algorithm for solving string diagrams of parity games.

Abstract

Open parity games are proposed as a compositional extension of parity games with algebraic operations, forming string diagrams of parity games. A potential application of string diagrams of parity games is to describe a large parity game with a given compositional structure and solve it efficiently as a divide-and-conquer algorithm by exploiting its compositional structure. Building on our recent progress in open Markov decision processes, we introduce Pareto fronts of open parity games, offering a framework for multi-objective solutions. We establish the positional determinacy of open parity games with respect to their Pareto fronts through a novel translation method. Our translation converts an open parity game into a parity game tailored to a given single-objective. Furthermore, we present a simple algorithm for solving open parity games, derived from this translation that allows the application of existing efficient algorithms for parity games. Expanding on this foundation, we develop a compositional algorithm for string diagrams of parity games.