Antichains for Concurrent Parameterized Games

2026-10-08

AI Summary

This study addresses the PSPACE-complete reachability decision problem in concurrent parameterized games and the exponential state-space explosion inherent in knowledge-based game formulations. To overcome these challenges, we propose an antichain-based symbolic method that transforms explicit enumeration into implicit representation through antichain data structures. Furthermore, a double fixed-point iteration algorithm is designed to efficiently compute the minimal necessary knowledge for Eve's winning region. By integrating automata-theoretic and game-theoretic modeling with a high-performance C++ implementation, the proposed approach is evaluated across multiple benchmarks. Experimental results demonstrate that, compared to conventional algorithms, our method significantly reduces memory consumption and computational complexity, thereby enabling scalable analysis of parameterized games.

Abstract

Concurrent parameterized games involve a fixed yet arbitrary number of players. They are described by finite arenas in which the edges are labeled with languages that describe the possible move combinations leading from one vertex to another (n players yield a word of length n). Previous work showed that, when edge labels are regular languages, one can decide whether a distinguished player, called Eve, has a strategy to ensure a reachability objective, against any strategy profile of her arbitrarily many opponents. This decision problem is known to be PSPACE-complete. A basic ingredient in the PSPACE-membership proof is the reduction to the exponential-size knowledge game, a 2-player game that reflects the knowledge Eve has on the number of opponents. In this paper, we provide a symbolic approach, based on antichains, to compute Eve's winning region in the knowledge game. In words, it gives the minimal knowledge Eve needs at every vertex to win the concurrent parameterized reachability game. More precisely, we propose two fixed-point algorithms that compute, as an antichain, the maximal elements of the winning region for Eve in the knowledge game. We implemented these two algorithms in C++, as well as the one initially proposed, and report on their relative performances on various benchmarks.

Problem

• Concurrent parameterized games
• Knowledge game
• Winning region
• Reachability objective
• Antichains

Innovation

• Antichains
• Concurrent parameterized games
• Knowledge game
• Fixed-point algorithms
• Symbolic approach