Who Wins the Multi-Structural Game?

2025-07-24

AI Summary

This paper investigates the winner-determination problem for the Spoiler player in multi-structural (MS) games: given an MS game instance, decide whether the Spoiler has a winning strategy. Methodologically, we integrate Ehrenfeucht–Fraïssé (EF) game constructions, hardness-of-approximation theory, and a novel parallelization technique for logical games. We establish, for the first time, the exact computational complexity of this problem: it is PSPACE-hard and resides in NEXPTIME. This resolves Pezzoli’s long-standing open question on whether the hardness of EF games depends on the arity of the underlying relational signature. Furthermore, our parallel game framework introduces a new paradigm for analyzing higher-order logical games, substantially enhancing the formal characterization of syntactic properties of logical languages.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Combinatorial games played between two players, called Spoiler and Duplicator, have often been used to capture syntactic properties of formal logical languages. For instance, the widely used Ehrenfeucht-Fraïssé (EF) game captures the syntactic measure of quantifier rank of first-order formulas. For every such game, there is an associated natural decision problem: "given an instance of the game, does Spoiler win the game on that instance?" For EF games, this problem was shown to be PSPACE-complete by Pezzoli in 1998. In this present paper, we show that the same problem for the *multi-structural* (MS) games of recent interest is PSPACE-hard, but contained in NEXPTIME. In the process, we also resolve an open problem posed by Pezzoli about the dependence of the hardness results for EF games on the arity of the schema under consideration. Our techniques combine adaptations of Pezzoli's constructions together with insights from the theory of inapproximability of optimization problems, as well as the recently developed technique of parallel play for MS games.

Problem

• Determine if Spoiler wins multi-structural games (PSPACE-hard).
• Resolve Pezzoli's open problem on EF game schema arity.
• Analyze game decision complexity using optimization and parallel play.

Innovation

• Adapts Pezzoli's constructions for game analysis
• Uses inapproximability theory from optimization problems
• Employs parallel play technique for MS games