Asynchronous Games: Innocence Without Alternation

Abstract

The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of λ-terms and PCF programs. An innocent strategy is defined as an alternating strategy with partial memory, in which the strategy plays according to its view. Extending the definition to nonalternating strategies is problematic, because the traditional definition of views is based on the hypothesis that Opponent and Proponent alternate during the interaction. Here, we take advantage of the diagrammatic reformulation of alternating innocence in asynchronous games, in order to provide a tentative definition of innocence in non-alternating games. The task is interesting, and far from easy. It requires the combination of true concurrency and game semantics in a clean and organic way, clarifying the relationship between asynchronous games and concurrent games in the sense of Abramsky and Melliès. It also requires an interactive reformulation of the usual acyclicity criterion of linear logic, as well as a directed variant, as a scheduling criterion. IntroductionThe alternating origins of game semantics. Game semantics was invented (or reinvented) at the beginning of the 1990s in order to describe the dynamics of proofs and programs. It proceeds according to the principles of trace semantics in concurrency theory: every program and proof is interpreted by the sequences of interactions, called plays, that it can have with its environment. The novelty of game semantics is that this set of plays defines a strategy which reflects the interactive behaviour of the program inside the game specified by the type of the program.Game semantics was originally influenced by a pioneering work by Joyal [16] building a category of games (called Conway games) and alternating strategies. In this setting, a game is defined as a decision tree (or more precisely, a dag) in which every edge, called move, has a polarity indicating whether it is played by the program, called Proponent, or by the environment, called Opponent. A play is alternating when Proponent and Opponent alternate strictly -that is, when neither of them plays two moves in a row. A strategy is alternating when it contains only alternating plays.The category of alternating strategies introduced by Joyal was later refined by Abramsky and Jagadeesan [2] in order to characterize the dynamic behaviour of proofs in (multiplicative) linear logic. The key idea is that the tensor product of linear logic, noted ⊗, may be distinguished from its dual, noted Γ, by enforcing a switching policy on plays -ensuring for instance that a strategy of A ⊗ B reacts to an Opponent move played in the subgame A by playing a Proponent move in the same subgame A. ⋆ This work has been supported by the ANR Invariants algébriques des systèmes informatiques (INVAL). Physical address: Équipe PPS, CNRS and Université Paris 7, 2 place Jussieu, case 7017, 75251 Paris cedex 05, France.

Borel Determinacy of Concurrent Games

“…For instance, the games in [1], [8] only allow deterministic strategies-the reason why they are not determined, cf. [4]; Games in logic and formal verification such as those in [2], [3] are not determined with respect to pure strategies.…”

The Winning Ways of Concurrent Games

“…Via Lemma 6, deterministic strategies in a game correspond to certain subfamilies of configurations of the game. A characterisation of those subfamilies which correspond to deterministic strategies [11] shows them to coincide with the receptive ingenuous strategies of Mimram and Melliès [10]. Via the presentation of deterministic strategies as families DGames is equivalent to an order-enriched category.…”

“…Note that some of the previous approaches to concurrent games [1], [10] were restricted to deterministic strategies, hence by the example above could not enjoy determinacy. In our setting, the ability to handle nondeterminism permits a determinacy result.…”

“…Melliès et al [1], [9], [10] have done extensive work on concurrent games on asynchronous transition systems; however, determinacy issues were not addressed. Concurrent games on graphs [3], [4] have also been studied in order to solve verification problems for open systems; such games are undetermined in the general case and as a consequence stochastic strategies are used.…”

On the Compositionality of Round Abstraction

“…Using a similar approach, Laird gave a game semantics for Idealized CSP [Lai01b] and later to the asynchronous π-calculus [Lai05b]. Finally, Melliès and Mimram combined the two approaches (interleaved and non-interleaved) in their work on asynchronous games [MM07]. Melliès later demonstrated how the simply typed lambda calculus can be modelled in this setting [Mel06].…”