Deterministic concurrent strategies

Abstract

Nondeterministic concurrent strategies—those strategies compatible with copy-cat behaving as identity w.r.t. composition—have been characterised as certain maps of event structures. This leads to a bicategory of general concurrent games in which the maps are nondeterministic concurrent strategies. This paper explores the important sub-bicategory of deterministic concurrent strategies. It is shown that deterministic strategies in a game can be identified with certain subgames, with the benefit that the bicategory of deterministic games becomes equivalent to a technically-simpler order-enriched category. Via a characterisation, deterministic strategies are shown to coincide with the receptive ingenuous strategies of Melliès and Mimram. Deterministic strategies determine closure operators, in accord with an early definition of Abramsky and Melliès. Known subcategories appear as special cases: Berry’s order-enriched category of dI-domains and stable functions arises as a full subcategory in which the games comprise solely of Player moves; the `simple games’ of Hyland et al., a basis for much of game semantics, form a subcategory in which the games permit no concurrency, Player-Opponent moves alternate and Opponent always moves first.