Reintroducing the Second Player in EPR

2026-02-18

AI Summary

This study investigates a PSPACE-complete fragment of the Bernays–Schönfinkel class (effectively propositional logic, or EPR), aiming to establish a complexity characterization aligned with the game semantics of Quantified Boolean Formulas (QBF). By constructing an EPR subfragment that preserves the two-player game semantics of QBF, and leveraging logical restrictions, a complexity-preserving translation from QBF to EPR, and analysis of the polynomial hierarchy, the work presents the first EPR fragment shown to be PSPACE-complete via a direct reduction from QBF. The paper not only proves the PSPACE-completeness of the proposed fragment but also demonstrates that further syntactic restrictions yield fragments capturing multiple completeness results across the polynomial hierarchy. Moreover, it successfully identifies several problem instances in the TPTP library belonging to this hierarchy, thereby validating the fragment’s expressiveness and practical relevance.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this work we investigate the computational complexity of the satisfiability problem of sub-fragments of the Bernays-Schoenfinkel class of first-order logic, also known as EPR (Effectively Propositional). While Bernays-Schoenfinkel is NEXPTIME-complete, we already can obtain fragments that are PSPACE-complete by restricting our clauses to DET-HORN or KROM. However such restrictions yield very different formulas to the canonical PSPACE-complete language of Quantified Boolean Formulas (QBF). This is despite Bernays-Schoenfinkel having a natural connection to an extension of QBF known as Dependency QBF. Our main contribution is the definition of a PSPACE-complete sub-fragment of Bernays-Schoenfinkel that extends from a translation of QBF, retains a similar two-player game evaluation for its semantics and can be restricted in various ways to obtain other complete problems, particularly those at different levels in the polynomial hierarchy. We use this definition to identify problems in the TPTP library that fall into this fragment and their level in the polynomial hierarchy.

Problem

• EPR
• Bernays-Schoenfinkel
• QBF
• PSPACE-complete
• two-player game

Innovation

• Bernays-Schoenfinkel
• PSPACE-complete
• Quantified Boolean Formulas
• two-player game semantics
• polynomial hierarchy