Traces via Strategies in Two-Player Games

TL;DR

This paper develops a coalgebraic framework for analyzing traces in two-player controller-environment games, connecting trace semantics with game strategies and outcomes in nondeterministic and probabilistic settings.

Contribution

It extends coalgebraic trace semantics to controller-environment games, linking trace elements to strategies and outcomes, and introduces a parametrized approach using weak distributive laws.

Findings

Trace elements correspond to sets or distributions of forced plays.

Each trace element reflects a controller strategy outcome.

Framework applies to nondeterministic and probabilistic environments.

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.