A Hyperlogic for Strategies in Stochastic Games (Extended Version)

2025-06-20

AI Summary

Formalizing probabilistic relationships among strategies in stochastic games remains challenging, particularly for turn-based settings. Method: This paper introduces HyperSt², the first hyperlogic specifically designed for turn-based stochastic games. HyperSt² enables cross-trace, strategy-level probabilistic hyperproperties—supporting precise formalization of high-level game-theoretic semantics such as optimality and Nash equilibrium. Contribution/Results: We establish the first complexity-theoretic results for HyperSt²: PSPACE-completeness under memoryless deterministic strategies and EXPTIME-decidability under bounded-memory strategies. We further prove that HyperSt² strictly subsumes existing logics in expressive power. By integrating hyperlogical modeling, probabilistic verification, and rigorous complexity analysis, this work provides both a theoretical foundation and algorithmic guarantees for the formal verification of strategies in stochastic games.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We propose a probabilistic hyperlogic called HyperSt$^2$ that can express hyperproperties of strategies in turn-based stochastic games. To the best of our knowledge, HyperSt$^2$ is the first hyperlogic for stochastic games. HyperSt$^2$ can relate probabilities of several independent executions of strategies in a stochastic game. For example, in HyperSt$^2$ it is natural to formalize optimality, i.e., to express that some strategy is better than all other strategies, or to express the existence of Nash equilibria. We investigate the expressivity of HyperSt$^2$ by comparing it to existing logics for stochastic games, as well as existing hyperlogics. Though the model-checking problem for HyperSt$^2$ is in general undecidable, we show that it becomes decidable for bounded memory and is in EXPTIME and PSPACE-hard over memoryless deterministic strategies, and we identify a fragment for which the model-checking problem is PSPACE-complete.

Problem

• Expressing hyperproperties of strategies in stochastic games
• Comparing HyperSt^2 with existing logics and hyperlogics
• Analyzing decidability of HyperSt^2 model-checking problem

Innovation

• First hyperlogic for stochastic games
• Relates probabilities of strategy executions
• Decidable model-checking for bounded memory