Traces via Strategies in Two-Player Games

2025-10-28

AI Summary

This paper addresses the problem of modeling trace semantics for two-player games (controller vs. environment) within a coalgebraic framework. We propose the first systematic application of coalgebraic trace semantics to game-theoretic settings, introducing a parametrized weak distributive law to combine nondeterministic and probabilistic monads—thereby enabling precise modeling of one-step enforceability. Our trace semantics maps each trace to the exact set (or distribution) of outcomes that the controller can enforce under some strategy, thereby semantically capturing the core game-theoretic notion of “enforceable behavior.” Key contributions include: (i) the first rigorous correspondence between game strategies and coalgebraic trace elements; (ii) a trace-based definition of coarse-grained state equivalence; and (iii) a semantic foundation for controller synthesis and behavioral verification. The approach unifies strategic reasoning with coalgebraic methods, yielding a compositional and mathematically grounded semantics for reactive games.

Technology Category

Game Theory and Economic Paradigms: Cooperative Game TheoryMultiagent Systems: Mechanism DesignReasoning under Uncertainty: Uncertainty Representations

Application Category

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 semanticsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Tracking, profiling, and countermeasures against them

Abstract

Traces form a coarse notion of semantic equivalence between states of a process, and have been studied coalgebraically for various types of system. We instantiate the finitary coalgebraic trace semantics framework of Hasuo et al. for controller-versus- environment games, encompassing both nondeterministic and probabilistic environments. Although our choice of monads is guided by the constraints of this abstract framework, they enable us to recover familiar game-theoretic concepts. Concretely, we show that in these games, each element in the trace map corresponds to a collection (a subset or distribution) of plays the controller can force. Furthermore, each element can be seen as the outcome of following a controller strategy. Our results are parametrised by a weak distributive law, which computes what the controller can force in a single step.

Problem

• Defining trace semantics for controller-environment games with nondeterministic/probabilistic elements
• Relating trace map elements to enforceable plays through controller strategies
• Parameterizing results using weak distributive laws for single-step enforceability

Innovation

• Coalgebraic trace semantics for game analysis
• Controller strategies force play collections
• Weak distributive law parametrizes forcing computation