Focusing in Asynchronous Games

Game semantics provides an interactive point of view on proofs, which enables one to describe precisely their dynamical behavior during cut elimination, by considering formulas as games on which proofs induce strategies. We are specifically interested here in relating two such semantics of linear logic, of very different flavor, which both take in account concurrent features of the proofs: asynchronous games and concurrent games. Interestingly, we show that associating a concurrent strategy to an asynchronous strategy can be seen as a semantical counterpart of the focusing property of linear logic. A cut-free proof in sequent calculus, when read from bottom up, progressively introduces the connectives of the formula that it proves, in the order specified by the syntactic tree constituting the formula, following the conventions induced by the logical rules. In this sense, a formula can be considered as a playground that the proof will explore. The formula describes the rules that this exploration should obey, it can thus be abstractly considered as a game, whose moves are its connectives, and a proof as a strategy to play on this game. If we follow the principle given by the Curry-Howard correspondence, and see a proof as some sort of program, this way of considering proof theory is particularly interesting because the strategies induced by proofs describe very precisely the interactive behavior of the corresponding program in front of its environment. This point of view is at the heart of game semantics and has proved to be very successful in order to provide denotational semantics which are able to describe precisely the dynamics of proofs and programs. In this interactive perspective, two players are involved: the Proponent, which represents the proof, and the Opponent, which represents its environment. A formula induces a game which is to be played by the two players, consisting of a set of moves together with the rules of the game, which are formalized by the polarity of the moves (the player which should play a move) and the order in which the moves should be played. The interaction between the two players is formalized by a play, which is a sequence of moves corresponding to the part of the formula being explored during the cut-elimination of the proof with another proof. A proof is thus described in this setting by a strategy which corresponds to the set of interactions that the proof is willing to have with its environment. ? CEA LIST, Laboratory for the Modelling and Analysis of Interacting Systems, Point Courrier 94, 91191 Gif-sur-Yvette, France. E-mail: [email protected]. This work has been supported by the CHOCO (“Curry Howard pour la Concurrence”, ANR-07BLAN-0324) French ANR project. This approach has been fruitful for modeling a wide variety of logics and programming languages. By refining Joyal’s category of Conway games [12] and Blass’ games [7], Abramsky and Jagadeesan were able to give the first fully complete game model of the multiplicative fragment of linear logic extended with the MIX rule [3], which was later refined into a fully abstract model of PCF (Programming Language of Computable Functions) [4]. Here, “fully complete” and “fully abstract” essentially mean that the model is very precise, in the sense that every strategy is definable (i.e. is the interpretation of a proof or a program); more details can be found in Curien’s survey on the subject [9]. Giving such a precise model of this language, introduced by Plotkin [18], was considered as a corner stone in computer science because it is a prototypical programming language, consisting of the λ-calculus extended with base data types and a fixpoint operator. At exactly the same time, Hyland and Ong gave another fully abstract model of PCF based on a variant of game semantics called pointer games [11]. In this model, definable strategies are characterized by two conditions imposed to strategies (well-bracketing and innocence). This setting was shown to be extremely expressive: relaxing in various ways these conditions gave rise to fully abstract models of a wide variety of programming languages with diverse features such as references, control, etc. Game semantics is thus helpful to understand how logic and typing regulate computational processes. It also provides ways to analyze them (for example by doing model checking [2]) or to properly extend them with new features [9], and this methodology should be helpful to understand better concurrent programs. Namely, concurrency theory being relatively recent, there is no consensus about what a good process calculus should be (there are dozens of variants of the π-calculus and only one λ-calculus) and what a good typing system for process calculus should be: we believe that the study of denotational semantics of those languages is necessary in order to reveal their fundamental structures, with a view to possibly extending the Curry-Howard correspondence to programming languages with concurrent features. A few game models of concurrent programming languages have been constructed and studied. In particular, Ghica and Murawski have built a fully abstract model of Idealized Algol (an imperative programming language with references) extended with parallel composition and mutexes [10] and Laird a game semantics of a typed asynchronous π-calculus [13]. In this paper, we take a more logical point of view and are specifically interested concurrent denotational models of linear logic. The idea that multiplicative connectives express concurrent behaviors is present since the beginnings of linear logic: it is namely very natural to see a proof of A` B or A⊗ B as a proof A in “parallel” with a proof of B, the corresponding introduction rules being ` Γ,A,B ` Γ,A`B [`] and ` Γ,A ` ∆,B ` Γ,∆,A⊗B [⊗] with the additional restriction that the two proofs should be “independent” in the case of the tensor, since the corresponding derivations in premise of the rule are two disjoint subproofs. Linear logic is inherently even more parallel: it has the focusing property [6] which implies that every proof can be reorganized into one in which all the connectives of the same polarity at the root of a formula are introduced at once (this is sometimes also formulated using synthetic connectives). This property, originally discovered in order to ease proof-search has later on revealed to be fundamental in semantics and type theory. Two game models of linear logic have been developed in order to capture this intuition. The first, by Abramsky and Melliès, called concurrent games, models strategies as closure operators [5] following the domain-theoretic principle that computations add information to the current state of the program (by playing moves). It can be considered as a big-step semantics because concurrency is modeled by the ability that strategies have to play multiple moves at once. The other one is the model of asynchronous games introduced by Melliès [14] where, in the spirit of “true concurrency”, playing moves in parallel is modeled by the possibility for strategies to play any interleaving of those moves and these interleavings are considered to be equivalent. We recall here these two models and explain here that concurrent games can be related to asynchronous games using a semantical counterpart of focusing. A detailed presentation of these models together with the proofs of many properties evoked in this paper can be found in [16,17]. 1 Asynchronous games Recall that a graph G = (V,E, s, t) consists of a set V of vertices (or positions), a set E of edges (or transitions) and two functions s, t : E → V which to every transition associate a position which is called respectively its source and its target. We write m : x −→ y to indicate that m is a transition with x as source and y as target. A path is a sequence of consecutive transitions and we write t : x −→ y to indicate that t is a path whose source is x and target is y. The concatenation of two consecutive paths s : x −→ y and t : y −→ z is denoted s · t. An asynchronous graph G = (G, ) is a graph G together with a tiling relation , which relates paths of length two with the same source and the same target. If m : x −→ y1, n : x −→ y2, p : y1 −→ z and q : y2 −→ z are four transitions, we write diagrammatically