Asynchronous Games vs. Concurrent Games

Transcription

1 Asynchronous Games vs. Concurrent Games Paul-André Melliès Samuel Mimram Abstract Game semantics reveals the dynamic and interactive nature of logic, by interpreting every proof of a formula as a winning strategy of a dialogue game. The purpose of this work is to compare two alternative game semantics of linear logic, based on asynchronous games and on concurrent games. Both game models take account of the concurrent nature of proofs, but in a very different way, and our task here is thus to relate them. We also take this opportunity to clarify how the process of translating an asynchronous strategy into a concurrent strategy is related to the focusing property of linear logic. Introduction Game semantics. A healthy exercise in proof theory is to read a cut-free proof (typically written in sequent calculus) starting from its conclusion rather than from its hypothesis. Unravelled in this way, the proof does not really prove in the traditional sense, but rather deconstructs the conclusion by removing its connectives one after the other, along the ordering specified by the syntactic tree of the formula. This point of view leads to reunderstand the formula as some kind of playground which the proof gradually explores and decomposes from the root to the leaves, according to the rules of the logic. In that prospect, the formula describes a game whose moves are the logical connectives, and the proof describes a strategy which plays these moves according to a specific (and logically valid) exploration scheduling of the formula. This account of proof theory is the starting point of game semantics, which we understand here in the widest possible sense, including the inaugural work by Novikoff on classical logic [?], by Lorenzen and his school on intuitionistic and classical logic [?], as well as the various notions of ludics defined by Girard and after him [?,?,?]. This game-theoretic approach to logic becomes particularly appealing when one thinks of a proof as a program, along the line of the Curry-Howard correspondence: the strategy describes in that case the interactive behavior of the proof/program in front of its refutation/environment. Two players are involved here: the Proponent which represents the proof, and the Opponent which represents the environment. Every formula of the logic is thus interpreted as a two-player game, describing the different ways in which Proponent and Opponent may explore the formula together. Laboratoire PPS (Preuves, Programmes, Systèmes), CNRS and Université Paris Diderot, Case 7014, Paris Cedex 13, France. CEA LIST, Laboratory for the Modelling and Analysis of Interacting Systems, Point Courrier 94, Gif-sur-Yvette, France. This work has been supported by the ANR Project CHOCO ( Curry Howard pour la Concurrence, ANR-07-BLAN-0324). 1

2 Sequential games. A nice illustration of game semantics is provided by the pioneering work by Joyal [Joy77] where a category of alternating strategies between sequential games (called Conway games) is constructed. A Conway game is defined as a rooted tree (or more precisely, a directed acyclic graph) in which every edge (called a move of the game) has a polarity indicating whether it is played by Proponent (the proof) or by Opponent (the environment). A play is then defined as a path starting from the root (noted ) of the tree. A play is called alternating when Proponent and Opponent strictly alternate that is, when neither of them plays two moves in a row. A strategy σ is then defined as a set of alternating plays of even length, starting from an Opponent move, thus of the form s = m 1 n 1 m 2 n 2 m k n k where m i is an Opponent move and n i is a Proponent move, for 1 i k. A strategy is moreover required to satisfy the following properties: (1) σ is nonempty: σ contains the empty play, (2) σ is prefix-closed: σ contains the play s whenever it contains the play s m n for an Opponent move m and a Proponent move n, (3) σ is deterministic: two plays s m n 1 and s m n 2 in the strategy σ have the same last Proponent move n 1 = n 2. The category of Conway games constructed by Joyal in the late 1970s became very influential in the early 1990s, in the turmoil produced by the discovery of linear logic. In particular, the Conway game model was refined by Abramsky and Jagadeesan [AJ94] 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 `, by enforcing a switching policy on plays this ensuring in particular 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. The notion of pointer game was then introduced by Hyland and Ong [HO00], and independently by Nickau [?], in order to characterize the dynamic behaviour of programs in the programming language PCF a simply-typed λ-calculus extended with recursion, conditional branching and arithmetical constants. The programs of PCF are characterized dynamically as particular kinds of strategies with partial memory, which are called innocent because they react to Opponent moves according to their own view of the play. This view is itself a play, extracted from the current play by removing all its invisible or inessential moves. This extraction is performed by induction on the length of the play, using the pointer structure of the play, and the hypothesis that Proponent and Opponent strictly alternate. This seminal work on pointer games led to the first generation of game semantics for programming languages. The research programme to a large extent conducted by Abramsky and his collaborators was particularly successful: by relaxing the innocence constraint on strategies, it suddenly became possible to characterize the interactive behaviour of programs written in PCF (or in a call-by-value variant) extended with imperative features like states, references, etc. However, because Proponent and Opponent strictly alternate in the original definition of pointer games, these game semantics focus on sequential languages like Algol or ML, rather than on concurrent languages. Concurrent games. This convinced a little community of researchers to work on the foundations of non-alternating games where Proponent and Opponent are thus allowed to play several moves in a row at any point of the interaction. Abramsky 2

3 and Melliès [AM99] introduced a new kind of game semantics to that purpose, based on concurrent games see also [Abr03]. In that setting, games are defined as partial orders (or more precisely, complete lattices) of positions, and strategies as closure operators on these partial orders. Recall that a closure operator σ on a partial order D is a function σ : D D satisfying the following properties: (1) σ is increasing: x D, x σ(x), (2) σ is idempotent: x D, σ(x) = σ(σ(x)), (3) σ is monotone: x, y D, x y σ(x) σ(y). The order on positions x y reflects the intuition that the position y contains more information than the position x. Typically, one should think of a position x as a set of moves in a game, or as a set of connectives explored in a formula, and x y as set inclusion x y between such positions. Now, Property (1) expresses that a strategy σ which transports the position x to the position σ(x) increases the amount of information. Property (2) reflects the intuition that the strategy σ delivers all its information when it transports the position x to the position σ(x), and thus transports the position σ(x) to itself. Property (3) is both fundamental and intuitively right, but also more subtle to justify. An important point to notice is that the interaction induced by such a strategy σ is possibly non-alternating, since the strategy transports the position x to the position σ(x) by playing in one go all the moves appearing in σ(x) but not in x. Another interesting point is that a closure operator σ is entirely characterized by the set fix(σ) of its fixpoints, that is, the positions x satisfying the equality x = σ(x). Asynchronous transition systems. So, a strategy can be expressed either as a set of positions (the set of fixpoints of the closure operator) in concurrent games or as a set of alternating plays in sequential games (like Conway games or pointer games). In order to understand how the two formulations of strategies are related, one should start from an obvious analogy with concurrency theory: pointer games define an interleaving semantics (based on sequences of transitions) whereas concurrent games define a truly concurrent semantics (based on sets of positions, or states) of proofs and programs. Now, Mazurkiewicz taught us that a truly concurrent semantics may be regarded as an interleaving semantics (typically a transition system) equipped with asynchronous tiles represented diagrammatically as 2-dimensional tiles y 1 x m n y 2 n m z expressing that the two transitions m and n from the state x are independent, and consequently, that their scheduling does not matter from a truly concurrent point of view. This additional structure induces an equivalence relation on transition paths, called the homotopy relation, defined as the smallest congruence relation identifying the two schedulings m n and n m for every tile of the form (1). The word homotopy should be understood mathematically as (directed) homotopy in the topological presentation of asynchronous transition systems as n-cubical sets [Gou00]. This 2- dimensional refinement of usual 1-dimensional transition systems enables to express simultaneously the interleaving semantics of a program as the set of transition paths it generates, and its truly concurrent semantics, as the homotopy classes of these transition paths. (1) 3

4 Asynchronous games. Guided by these intuitions, Melliès introduced in [?] the notion of asynchronous game, which unifies in a nice and conceptual way the heterogeneous notions of pointer game and of concurrent game. Asynchronous games are played on asynchronous (2-dimensional) transition systems, where every transition (or move) is equipped with a polarity, expressing whether it is played by Proponent or by Opponent. A play is defined as a path starting from the root of the game, and a strategy is defined as a well-behaved set of alternating plays, in the style of traditional sequential games. Now, the difficulty is to understand how (and when) a strategy defined as a set of plays may be reformulated as a set of positions, in the spirit of concurrent games. The first step in the inquiry is to observe that the asynchronous tiles (1) offer an alternative way to describe justification pointers between moves. For illustration, consider the boolean game B, where Opponent starts by asking a question q, and Proponent answers by playing either true or false. The game is represented by the decision tree V q q false F true where is the root of the game, and the three remaining positions are called q, V (V for Vrai in French) and F. At this point, since there is no concurrency involved, the game may be seen either as an asynchronous game, or as a pointer game. Now, the game B B is constructed by taking two boolean games in parallel. It simulates a very simple computation device, containing two boolean memory cells. In a typical interaction, Opponent starts by asking with q L the value of the left memory cell, and Proponent answers true L ; then, Opponent asks with q R the value of the right memory cell, and Proponent answers false R. The play is represented as follows in pointer games: q L true L q R false R The play contains two justification pointers, each one represented by an arrow starting from a move and leading to a previous move. Typically, the justification pointer from the move true L to the move q L indicates that the answer true L is necessarily played after the question q L. The same situation is described using 2-dimensional tiles in the asynchronous game B B below: (2) q L q R q q true L q R q L false R V q q F true L false R q R q L V q q F false R true L V F (3) 4

5 The justification pointer between the answer true L and its question q L is replaced here by a dependency relation between the two moves, ensuring that the move true L cannot be permuted before the move q L. The dependency itself is expressed by a topological obstruction: the lack of a 2-dimensional tile permuting the transition true L before the transition q L in the asynchronous game B B. This basic correspondence between justification pointers and asynchronous tiles enables a reformulation of the original definition of innocent strategy in pointer games (based on views) in the language of asynchronous games. Surprisingly, the reformulation leads to a purely local and diagrammatic definition of innocence in asynchronous games, which does not mention the notion of view any more, see [Mel04] for details. This diagrammatic reformulation leads also to the discovery that innocent strategies are positional in the following sense. Suppose that two alternating plays s, t : x with the same target position x are elements of an innocent strategy σ, and that m is an Opponent move from position x. Suppose moreover that the two plays s and t are equivalent modulo homotopy. Then, the innocent strategy σ extends the play s m with a Proponent move n if and only if it extends the play t m with the same Proponent move n. Formally: s m n σ and s t and t σ implies t m n σ. (4) From this follows that every innocent strategy σ is characterized by the set of positions (understood here as homotopy classes of plays) reached in the asynchronous game. This set of positions defines a closure operator, and thus a strategy in the sense of concurrent games. Asynchronous games offer in this way an elegant and unifying point of view on pointer games and concurrent games. Concurrency in game semantics. There is little doubt that a new generation of game semantics is currently emerging along this foundational work on concurrent games. We see at least three converging lines of research. First, authors trained in game semantics Ghica, Laird and Murawski were able to characterize the interactive behaviour of various concurrent programming languages like Parallel Algol [GM04] or an asynchronous variant of the π-calculus [Lai05] using directly (and possibly too directly) the language of pointer games. Then, authors trained in proof theory and game semantics Curien and Faggian relaxed the sequentiality constraints required by Girard on designs in ludics, leading to the notion of L-net [CF05] which lives at the junction of syntax (expressed as proof nets) and game semantics (played on event structures). Finally, and more recently, authors trained in process calculi, true concurrency and game semantics Varacca and Yoshida were able to extend Winskel s truly concurrent semantics of CCS, based on event structures, to a significant fragment of the π-calculus, uncovering along the way a series of nice conceptual properties of confusion-free event structures [VY06]. So, a new generation of game semantics for concurrent programming languages is currently emerging... but their various computational models are still poorly connected. We would like a regulating theory here, playing the role of Hyland and Ong pointer games in traditional (that is, alternating) game semantics. Asynchronous games are certainly a good candidate, because they combine interleaving semantics and causal semantics in a harmonious way. Unfortunately, they were limited until now to alternating strategies [Mel04]. The main contribution of this paper is to extend the asynchronous framework to non-alternating strategies, and to compare the resulting model with the concurrent framework defined by Abramsky and Melliès. Although the ultimate goal of producing a satisfactory notion of innocent strategy for non-alternating interactions is not reached yet, we believe that the connection to concurrent games is an important step in that direction. 5

6 Event structures. A simple recipe to construct an asynchronous game is to start from a partial order of events where, in addition, every event has a polarity, indicating whether it is played by Proponent or Opponent. This partial order (M, ) is then equipped with a incompatibility relation # satisfying a series of suitable properties, defining what Winskel calls an event structure. A position x of the asynchronous game is defined as a set of pairwise compatible events (or moves) closed under the causality order: m, n M, m n and n x implies m x. Typically, the boolean game B described in (2) is generated by the event structure q true false where q is an Opponent move, and true and false are two incompatible Proponent moves, with the positions q, V, F defined as q = {q}, V = {q, true} and F = {q, false}. The tensor product B B of two boolean games is then generated by putting side by side the two event structures, in the expected way. The resulting asynchronous game looks like a flower with four petals, one of them described in (3). More generally, every formula of linear logic defines an event structure, which defines in turn an asynchronous game. For instance, the event structure induced by the formula (B B) B (5) contains the following partial order of compatible events: q L q q R (6) true L false false R which may be seen alternatively as a (maximal) position in the asynchronous game associated to the formula. This game implements the interaction between a boolean function (the Proponent) of type (5) and its two arguments (the Opponent). Asynchronous games without alternation. In a typical play of the game (5), Opponent starts by playing the move q asking the value of the boolean output; Proponent reacts by asking with q L the value of the left input, and Opponent answers true L ; then, Proponent asks with q R the value of the right input, and Opponent answers false R ; at this point only, using the knowledge of its two arguments, Proponent answers false to the initial question: q q L true L q R false R false (7) Of course, Proponent could have explored its two arguments in the other order, from right to left, this inducing the play q q R false R q L true L false (8) 6

7 The two plays start from the empty position and reach the same position of the asynchronous game. They may be seen as different linearizations (in the sense of order theory) of the partial order (6) specified by the game. Each of these linearizations may be represented by adding causality (dotted) edges between moves to the original partial order (6), in the following way: q q L true L q R q L q q R false R false R true L false false (9) The play (7) is an element of the strategy representing the left implementation of the strict conjunction, whereas the play (8) is an element of the strategy representing its right implementation. Both of these strategies are alternating. Now, there is also a parallel implementation, where the conjunction asks the value of its two arguments at the same time. The associated strategy is not alternating anymore: it contains the play (7) and the play (8), and moreover, all the (possibly non-alternating) linearizations of the following partial order. q L q q R (10) true L false R false This illustrates an important phenomenon of a concurrent nature: a strategy σ of an asynchronous game generally contains several plays with the same target position x, all of them equivalent modulo homotopy. In order to keep in harmony with the spirit of true concurrency, one would like to see these homotopic plays as the linearizations of a causal order defined by the strategy σ on the events leading to the position x. This basic intuition leads to the notion of ingenuous strategy σ which will be introduced in the course of the article, in Section 2. By definition, an ingenuous strategy is positional, this enabling to define it alternatively as a set of plays, or as an asynchronous subgraph G σ of the asynchronous game G where the interaction occurs. It is interesting to notice that the construction of the causal order induced by the positional σ for a position x G σ requires to apply refined tools coming from the theory of asynchronous transition systems, in particular the fundamental cube property which enables to relate asynchronous transition graphs to event structures. Concurrency means courtesy. The next step we perform is to establish a satisfactory relationship between the ingenuous strategies of asynchronous games and 7

8 the concurrent strategies of concurrent games. This is not entirely obvious however. Every ingenuous strategy σ induces a set σ of halting positions, defined as the positions x G σ such that there exists no Proponent move x y in the graph G σ. This set of halting positions is closed under intersection, and thus defines a concurrent strategy, which consists in transporting every position x to the smallest halting position y of σ above x when it exists, and to the maximal position otherwise. Conversely, every concurrent strategy τ = halt(σ) obtained from an ingenuous strategy σ in this way defines an ingenuous strategy τ which contains the plays m = x 1 m 0 2 x1 x2 mk x k such that the position x i is in the dynamic domain of τ for 0 i k. By definition, a position x is in the dynamic domain of τ when the position τ(x) is different from the maximum position, and the transition path s : x τ(x) contains only Proponent moves. The correspondence theorem established in Section 4 states that the ingenuous strategy τ is equal to the least ingenuous strategy σ containing the strategy σand satisfying the following courtesy property: for every play s 1 m p s 2 of the strategy σ such that m is a Proponent transition or p is an Opponent transition, and such that the homotopy relation m p n q defines a tile in the game, then the play s 1 n q s 2 obtained by permuting p before m is also in the strategy σ. Moreover, this completion σ of the ingenuous strategy σ has the same halting positions as the strategy σ, and thus induces the same concurrent strategy τ. Plan of the article. We recall the notion of asynchronous graph (in Section 1) and investigate the relationship with event structures when the graph satisfies the so-called cube property. This leads us to the notion asynchronous game (in Section 2) together with the definition of an ingenuous strategy. Next, we define an asynchronous model of multiplicative additive linear logic (in Section 3) 1 Asynchronous graphs and event structures 1.1 Asynchronous graphs Graphs. 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. Two transitions are coinitial (resp. cofinal) when they have the same source (resp. target). A path t is a sequence of transitions m 1 m 2 m k which are consecutive, i.e. such that s(m i+1 ) = t(m i ) for every index i, 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 : x z and the empty path on a position x is denoted by ε x : x x. Asynchronous graphs. An asynchronous graph G = (G, ) is a graph G together with a symmetric tiling relation, which relates paths of length two with the same source and the same target. If m : x y 1, n : x y 2, p : y 1 z and q : y 2 z 8

9 are four transitions, we write diagrammatically y 1 z p q y 2 m n x (11) to indicate that m p n q. We define as the smallest congruence (wrt. concatenation) containing the tiling relation, and write x s y x t z when there exists a path s : y z such that s s t. The relation is called homotopy because it should be thought as the possibility, when s and t are two homotopic paths, to deform continuously the path s into t. Homotopy is formalized algebraically here, but the links with geometry and (directed) algebraic topology can be investigated further and are actually an active research area [Gou00]. From the concurrency point of view, an homotopy between two paths indicates that these paths are the same up to reordering of independent events, as in Mazurkiewicz traces [Maz88]. A category of asynchronous graphs. A map (G 1, 1 ) (G 2, 2 ) between asynchronous graphs is a map f : G 1 G 2 between the underlying graphs, which respects the tiling relation, i.e. such that s 1 t implies f(s) 2 f(t) for all coinitial and cofinal paths s = m p and t = n q of length two. The category of asynchronous graphs has asynchronous graphs as objects, and maps between them as morphisms. The category of asynchronous graphs is also symmetric monoidal, when one equips it with the following notion of tensor product. Asynchronous product. Given two asynchronous graphs G and H, we define their asynchronous product G H as the following asynchronous graph. Its positions are V G H = V G V H and its transitions are E G H = E G V H + V G E H (by abuse of language we say that a transition is in G when it is in the first component of the sum or in H otherwise). The source and target of a transition (m, x) E G V H are respectively (s G (m), x) and (t G (m), x), and similarly those of a transition (x, m) V G E H are respectively (x, s H (m)) and (x, t H (m)). Tiles are of the form (z, y) (p,y) (q,y) (y 1, y) (y 2, y) (m,y) (n,y) (x, y) (y, z) (y,p) (y,q) (y, y 1 ) (y, y 2 ) (y,m) (y,n) (y, x) (x, y ) (x,n) (m,y ) (x, y) (x, y ) (m,y) (x,n) (x, y) where we suppose that there is a tile of the form (??) in G (resp. in H) in the first (resp. second) tile. Note that every path s in a game G H is the interleaving of a path s G in G and a path s H in H. Moreover, the path s is equivalent to any such interleaving of s G and s H modulo homotopy in G H. The path s G is called the projection of the path s on the component G, and similarly for s H. The unit of the asynchronous product is the graph with one position, and no transition. 9

10 1.2 Residuals We suppose from now on that in every asynchronous graph, a 2-dimensional tile (??) is entirely characterized by the data of the pair of transitions (m, p), of transitions (m, n) or of transitions (p, q). This means that: given a path m p there exists at most one path n q forming a tile (??), given a pair of coinitial transitions m and n, there exists at most one pair of cofinal transitions p and q inducing a tile (??), given a pair of cofinal transitions p and q there exists at most one pair of coinitial transition m and n inducing a tile (??). Residual after a transition. A transition q : y 2 z is a residual of a transition m : x y 1 after a coinitial transition n : x y 2 when there exists a transition p : y 1 z forming an asynchronous tile m p n q. Note that the transition p is also the residual of n after m in this case, because the tile relation is symmetric. Moreover, our hypothesis on asynchronous graphs implies that a transition m admits at most one residual after a coinitial transition n. This residual is denoted m/n when it exists. Compatible paths. Two coinitial paths s : x y 1 and t : x y 2 are called compatible when there exists a pair of cofinal paths s : y 1 z and t : y 2 z such that s s t t. Residual after a path. Given two compatible paths s : x y 1 and t : x y 2 we define the residual s/t of the path s after the path t by induction: (s 1 s 2 )/t = (s 1 /t) (s 2 /(t/s 1 )) s/(t 1 t 2 ) = (s/t 1 )/t 2 s/s = ε Intuitively, the path s/t is what remains of the path s one the path t has been performed. It can be checked that the residual path s/t exists (and is uniquely defined) when the underlying asynchronous graph satisfies the cube property described below, see the second author s PhD thesis [Mim08] for details. 1.3 Event structures Recall that an event structure (M,, #) is a partially ordered set (M, ), whose elements are called events and the order relation is called causality, together with a symmetric irreflexive relation #, called incompatibility, on events satisfying the two following properties. 1. Finite causality: every event e M defines a downward closed set of events which is finite. e = { e M e e } 2. Hereditary incompatibility: for every events e 1, e 2 and e 3, e 1 # e 2 and e 2 e 3 implies e 1 # e 3 10

11 Two events e 1 and e 2 are incompatible when e 1 # e 2 and compatible otherwise. A configuration of an event structure is a downward-closed set of compatible events. Every event structure (M,, #) induces an asynchronous graph of configurations, in the following way: the positions of the asynchronous graph are the configurations of the event structure, its transitions m : x y are the triples (m, x, y) where m is an event and y = x {m}, where denotes disjoint union, every two coinitial and cofinal paths of length 2 are related by a tile. For example, the event structure on three events a, b, c with empty incompatibility relation, whose underlying poset is depicted on the left, induces the asynchronous graph depicted on the right: b c a {a, b, c} c b {a, b} {a, c} b c {a} a 1.4 The cube and the coherence properties We would like to characterize the class of asynchronous graphs generated by an event structure. The two models of computation are intrinsically different: computations are described as sequences of transitions modulo permutation in asynchronous graphs, whereas they are described as sets of events ordered by a causality order in event structures. As we will see, the situation is the same in game semantics when we associate an asynchronous game to a formula, which can be seen as an event structure with additional structure ; or when we try and extract the causality structure from a strategy defined as an asynchronous subgraph of the asynchronous game. This explains why we are so much interested in clarifying the relationship between asynchronous graphs and event structures, starting from the cube property for asynchronous graphs, which plays a fundamental role in the characterization. The cube property expresses a fundamental causality principle in the diagrammatic language of asynchronous transition systems [Bed88, Shi85, WN95]. It is related to stability in the sense of Berry [Ber79]. It was first noticed by Nielsen, Plotkin and Winskel in [NPW81], then reappeared in [PSS90] and [GLM92, Mel98, Mel05b] and was studied thoroughly by Kuske in his PhD thesis; see [DK02] for a survey. The characterization is based on the observation that every asynchronous graph of configurations generated by an event structure satisfies the following properties. Definition 1 (Cube property). An asynchronous graph G satisfies the cube property when a hexagonal diagram in G induced by two paths m n o : x y and p q r : x y, which are coinitial and cofinal, is filled by 2-dimensional tiles as pictured in the left-hand side of the diagram below, if and only it is filled by 2-dimensional 11

12 tiles as pictured in the right-hand side of the diagram: r y 1 y o q x 2 p x 3 x m y 2 x 1 n q y 1 r y x 2 p x o y 3 m y 2 x 1 n Definition 2 (Coherence property). An asynchronous graph G satisfies the coherence property when a hexagonal diagram in G is filled by 2-dimensional tiles as pictured in the left-hand side of the diagram below, if and only it is filled by 2-dimensional tiles as pictured in the right-hand side of the diagram: x 2 y 1 n x o x 3 y 3 m x 1 y 2 x 2 y 1 p x 3 y 3 r x 1 y q y 2 Definition 3 (Rooted graph). A position of an asynchronous graph G is called a root when there exists a path x to every position x of the graph. An asynchronous graph is rooted when it has a root. Definition 4 (Simply connected graph). An asynchronous graph G is simply connected when every pair s, t : x y of coinitial and cofinal paths are equivalent modulo homotopy. Conversely, we establish that the three properties are strong enough to characterize the asynchronous graphs G generated by an event structure. The idea is to reconstruct the asynchronous graph G from the following notion of events: Asynchronous events. Given a transition m : x y in an asynchronous graph G, we write [m] for the equivalence class of m by the residuation relation: this set [m] is called the asynchronous event (or the move) associated to the transition, and conversely we say that m is an instance of [m]. Event structure associated to an asynchronous graph. The event structure E(G) associated to an asynchronous graph G is defined as follows: its events are the asynchronous events of the asynchronous graph, e 1 < G e 2 when for every transition m 2 : x 2 y 2 in the event e 2, there exists a transition m 1 : x 1 y 1 and a path y 1 x 2, e 1 # G e 2 when every transition m 1 : x 1 y 1 in the event e 1 and every transition m 2 : x 2 y 2 in the event e 2 have incompatible target positions y 1 and y 2, in the sense that there exists no position z such that s 1 : y 1 z and s 2 : y 2 z for some paths s 1 and s 2. 12

13 This leads to the following proposition. Proposition 5. Every rooted and simply connected asynchronous graph G satisfying the cube property and the coherence property coincides with the asynchronous graph of configurations generated by the event structure E(G). From this follows immediately that in a rooted and simply connected asynchronous graph G satisfying the cube property and the coherence property, one has: Corollary 6. The set of paths homotopy equivalent to a given path s : x y is the linearization of the partial order G on the events appearing in the transition path s. It should be mentioned that the corollary does not need the coherence property to hold. The cube property is typically satisfied by every asynchronous transition system and every transition system with independence in the sense of [WN95, SNW96]. The correspondence between homotopy classes and sets of linearizations of a partial order adapts to our setting a standard result on pomsets and asynchronous transition systems with dynamic independence due to Bracho, Droste and Kuske [BDK97]. The proposition is best illustrated by an example. Consider the following asynchronous graph in which every two maximal paths are homotopic: x 2 d b g y h x 4 x 5 e f x 3 c x 1 a The asynchronous graph G is rooted, simply connected, and satisfies the cube property as well as the coherence property. In that case, the proposition states that the asynchronous graph G is generated by an event structure E(G) on the asynchronous events of the asynchronous graph. It is not difficult to see that the asynchronous events of the graph (12) are e 1 = {a}, e 2 = {b, e, h}, e 3 = {c, d} and e 4 = {f, g}. Now, the partial order on these events is defined by stating that e i e j whenever every element of e j appears after an element of e i in a maximal path from the root. The induced partial order is thus e 4 (12) e 2 e 3 e 1 and Proposition 5 states that the asynchronous graph (12) is (isomorphic to) the asynchronous graph of configurations generated by this partial order. Let us explain briefly how the proof of Proposition 5 is established. First of all, the hypothesis that the asynchronous game G = (V, E, s, t, ) is simply connected enables to define a partial order on the set of positions, defined by reachability: x y precisely when there exists a path x y. Define then [x, y] as the interval between two positions x and y such that x y, equipped with the reachability order : [x, y] = { z V x z y }. 13

14 The proof of Proposition 5 reduces then to showing that in every simply connected asynchronous graph satisfying the cube property: Lemma 7. The interval [x, y] is a finite distributive lattice, and for every pair of positions x, y satisfying x y. The statement of Proposition 5 follows then by applying Birkhoff s representation theorem [Bir33] and a careful reconstruction of the paths using the coherence property, see the second author s PhD thesis [Mim08] for details. 2 Asynchronous games 2.1 Asynchronous games An asynchronous game G = (G,, λ) consists of an asynchronous graph G = (V, E, s, t) (which is simply connected and satisfies the cube property) together with a distinguished initial position V and a function λ : E {O, P } which to every transition associates a polarity: either O for Opponent or P for Proponent. A transition is supposed to have the same polarity as its residuals, polarity is therefore well-defined on moves. We also suppose that every position x is reachable from the initial position, i.e. that there exists a path x. A morphism of asynchronous games is a morphism between the underlying asynchronous graphs which respects the initial position and polarities. An embedding ι : G H of a game G into a game H is a monomorphism ι between them. Given a game G, we write G for the game G with polarities inverted, we also write ˆG for the game obtained from G by adding a new position, which becomes the initial position, and a new Opponent transition n : G. The dual operation is denoted G = (ˆ(G )). Given two games G and H, we define their asynchronous product G H as the game whose underlying asynchronous graph is the asynchronous tensor product G H of the underlying asynchronous graphs of G and H, whose polarity function is the one induced by those of G and H and whose initial position is ( G, H ). We also define their sum G + H as the disjoint union of the two games in which the two initial positions are identified. 2.2 Ingenuous strategies Properties characterizing definable strategies were studied in the case of alternating strategies (where Opponent and Proponent moves should strictly alternate in plays) in [Mel05a] and generalized to the non-alternating setting that we are considering here in [MM07, Mim08]. We recall here the basic properties of definable strategies. An interesting feature of these properties is that they allow one to reason about strategies in a purely local and diagrammatic fashion. It is easy to show that every definable family σ χ of strategies is uniform and moreover the strategies σ constituting it satisfy the following properties. Positionality. and u : x y, A strategy σ is positional when for every three paths s, t : x s u σ and s t and t σ implies t u σ This property essentially means that a strategy is a subgraph of the game: a strategy σ induces a subgraph G σ of the game which consists of all the positions and transitions contained in at least one play in σ, conversely every play in this subgraph belongs 14

15 to the strategy when the strategy is positional. In fact, this graph G σ may be seen itself as an asynchronous graph by equipping it with the tiling relation induced by the underlying game: the fact that games and strategies are mathematical structures of comparable nature is conceptually appealing and technically useful in this setting. We sometimes say that a position x or a path s : x y is in a positional strategy σ to mean that it is in the induced graph G σ. Cube property. A positional strategy σ : G satisfies the cube property when the subgraph G σ it induces satisfies the cube property (Definition 1). Preservation of compatibility. A positional strategy σ : G preserves forward compatibility when every asynchronous tile of the shape (??) in the game G belongs to the subgraph G σ of the strategy σ when its two coinitial transitions m : x y 1 and n : x y 2 are transitions in the subgraph G σ. Diagrammatically, y 1 p σ m z q y 2 n σ x implies σ p y 1 σ m z q σ y 2 n σ x where the dotted edges indicate edges in G. Dually, it preserves backward compatibility when every asynchronous tile of the shape (??) in the game G belongs to the subgraph G σ of the strategy σ when its two cofinal transitions p : y 1 z and q : y 2 z are transitions in the subgraph G σ. Diagrammatically, σ p y 1 z q σ y 2 m x n implies σ p y 1 σ m z q σ y 2 n σ x These conditions ensure that the strategy is closed under residuals, and therefore the set of plays of the strategy σ reaching the same position x is regulated by a causality order on the moves occurring in these plays which refines the justification order on moves (in the sense of pointer games) specified by the asynchronous game. Determinism. If the graph G σ of the strategy contains a transition n : x y 2 and a Proponent transition m : x y 1 then there exists a residual of m along n, which is moreover contains in the strategy σ, this defining a tile of the form (??). Graphically, y 1 y 2 σ m n σ x implies σ p y 1 σ m z q σ y 2 n σ x Our concurrent notion of determinism is not exactly the same as the usual notion of determinism in sequential games: in particular, a strategy may play several Proponent moves from a given position, as long as it converges later, in other words deterministic strategies are closed under residual of paths after paths containing only Proponent moves. 15

16 Definition 8. A positional strategy which satisfies the three conditions above (cube property, preservation of compatibility, determinism) is called ingenuous. It is worth mentioning that the notion of ingenuous strategy may be alternatively formulated on strategies σ described as set of plays, rather than as an asynchronous subgraph G σ of the game, see the alternative definition in [MM07] and in the second author s PhD thesis [Mim08]. 2.3 Composition of ingenuous strategies Every pair of strategies σ : A B and τ : B C induces by interaction a strategy σ τ of the game A B C, defined as σ τ = { s A B C s A,B σ and s B,C τ } (we implicitly adjust the polarities of the moves). The composite σ; τ is the strategy of A C defined by hiding the moves in B from the interaction between σ and τ: σ; τ = { s A,C s σ τ } (13) We thus build a category G ing whose objects are games and whose morphisms from a game A to a game B are ingenuous strategies σ : A B. The composite of σ : A B and τ : B C is the strategy σ; τ : A C defined in (13). The identity id A on the game A is the copycat strategy. The fact that the composite of two ingenuous strategies is also ingenuous is not entirely obvious. It is based on a subtle confluence property, stated below. Lemma 9. Suppose that σ : A B and τ : B C are two ingenuous strategies and that u : (x, z) is a play in the composite strategy σ; τ. Suppose also that there exist two plays in σ and two plays in τ such that s 1 : (x, y 1 ) and s 2 : (x, y 2 ) t 1 : (y 1, z) and t 2 : (y 2, z) (s 1 ) A = (s 2 ) A, (s 1 ) B = (t 1 ) B, (s 2 ) B = (t 2 ) B, (t 1 ) C = (t 2 ) C then there exist a position y and two plays respectively in σ and in τ such that s : (x, y) and t : (y, z) s A = (s 1 ) A = (s 2 ) A, s B = t B, t C = (t 1 ) C = (t 2 ) C and both s 1 and s 2 (resp. t 1 and t 2 ) are prefixes of s (resp. of t) modulo homotopy. Diagrammatically, x y z s A x 0 y 1 (s 1) B y 2 y 0 (s 2) B z 0 t C 16

17 Proof. The proof is done by induction on the length of the paths s 1, s 2, t 1 et t 2. It is trivial when one of these paths is empty (s 2 for example) since the position y = y 1 suits. Suppose that the path s 1 begins with a transition in A; we write this transition m : (x 0, y 0 ) (x 1, y 0 ) (the case where s 2 begins with a transition in A is similar). By hypothesis, we have (s 1 ) A = (s 2 ) A. The path s 2 is thus of the form s 2 = s 2 m s 2, where s 2 contains only transitions in the game B. By definition of the game A B, the positions (x 1, y 0 ) and (x, y 2 ) are compatible and therefore the path m (s 2 /m) : (x 0, y 0 ) (x, y 2 ) exists and is in the strategy σ since it preserves forward compatibility. We can therefore replace the path s 2 by the path m (s 2 /m) in our proof and suppose that the paths s 1 and s 2 both begin with the same transition m in A. The induction hypothesis can be applied to the paths s 1 /m : (x 1, y 0 ) (x, y 1 ), s 2 /m : (x 1, y 0 ) (x, y 2 ), t 1 et t 2, from which we can deduce that there exist two positions x and y and two paths s : (x 1, y 0 ) (x, y) and t : (y 0, z 0 ) (y, z) which satisfy the property. The paths m s and t enable us to conclude. The case where on of the paths t 1 or t 2 begins with a transition in C is similar. Otherwise, all the paths s 1, s 2, t 1 and t 2 begin with a transition in B. We write m : y 0 y 1 for the transition in B at the beginning of the paths s 1 and t 1, and n : y 0 y 2 the transition in B at the beginning of the paths s 2 and t 2. If the transitions m and n are the same, then we can conclude as previously, using the induction hypothesis. Otherwise, we will show that we can come down to this case. Suppose that m is a Proponent transition in B. Since the strategy σ is deterministic, the positions y 1 and y 2 are compatible, and the residual path m (s 2 /m) exists and is in the strategy σ since it preserves forward compatibility. We can therefore replace the path s 2 by m (s 2 /m) and t 2 by m (t 2 /m) in our proof and suppose that the four paths s 1, s 2, t 1 and t 2 begin with the same transition, from which we can conclude as previously. The case where n is a Proponent transition is similar. If both m and n are Opponent transitions, they are Proponent transitions in B C and the paths t 1 and t 2 begin with the same Proponent transition. We can therefore conclude by exchanging the roles of the strategies σ and τ in the proof above. This property states that given two interactions between the strategies σ and τ leading to the same play u in the composite strategy σ; τ, their union (wrt. the prefix order modulo homotopy) can be reached by interaction. From this property of maximal witness, which extends in our asynchronous setting the well-known zipping lemma of sequential game semantics, one can deduce that Proposition 10. The composite σ; τ : A C of two ingenuous strategies σ : A B and τ : B C is an ingenuous strategy. Proof. The fact that closure under prefix and positionality are preserved by composition is simple. The proofs that the other properties are preserved by composition are all similar, we only show here the preservation of determinism. Suppose that we are given a path u : (x, z), and two Proponent transitions m : (x, z) (x, z 1 ) and n : (x, z) (x, z 2 ) in the component C such that the plays u m and u n are played by the composite strategy σ; τ. The play u m (resp. u n) results from the interaction of two plays s 1 σ and t 1 m τ (resp. s 2 σ and t 2 n τ). Lemma 9 ensures that the two plays u m and u n result from the interaction of essentially the same pair of plays: s σ and t m τ for u m, s σ and t n τ for u n. Since the strategy τ is deterministic, the transitions m and n form a tile (??) and the plays t m p and t n q are in τ, from which we deduce that the paths m p and n q are also in the strategy σ; τ. 17

18 3 An asynchronous model of linear logic We consider formulas of the multiplicative and additive fragment of linear logic (MALL), which are generated by the grammar A ::= A ` A A A A & A A A X X where X is a variable (for brevity, we don t consider constants). The ` and & (resp. and ) connectives are sometimes called negative or asynchronous (resp. positive or synchronous). A position is a term generated by the following grammar x ::= x ` x x x & L x & R x L x R x The de Morgan dual A of a formula is defined as usual (A B) = A ` B for example and the dual of a position is defined similarly. Given a formula A, we write pos(a) for the set of valid positions of the formula which are defined inductively by pos(a) and if x pos(a) and y pos(b) then x ` y pos(a ` B), x y pos(a B), & L x pos(a & B), & R y pos(a & B), L x pos(a B) and R y pos(a B). A valid position x of a formula A thus corresponds of a prefix, or a partial exploration, of this formula, the position denoting a region of the formula that has not been explored. 3.1 Interpretation of formulas We write X for a list (X i ) 1 i n of variables. Suppose that A and B are formulas with X as free variables and χ is a fixed function, called a context of the formula, which to every variable X X associates a game G X. The game G χ A associated to A in the context χ is defined inductively as follows (we sometimes simply write G A instead of G χ A when the context is clear from the context). The interpretation of variables is given by the context: G χ X = χ(x). The game G A`B is obtained from the game G A G B by replacing every pair of positions (x, y) by x ` y, and adding a position and an Opponent transition ` : we therefore have G A`B = ˆ(GA G B ). The game G A&B is obtained from the disjoint union G A G B by replacing every position x of G A (resp. G B ) by & L x (resp. & R x), and adding a position and two Opponent transitions & L and & R : we therefore have G A&B = (ˆGA ) + (ˆG B ). The games associated to the other formulas are deduced by de Morgan duality: G A = G A. For example, if we interpret the variable X (resp. Y ) as the game with two positions and x (resp. and y) and one transition between them, the interpretation of the formula (X X ) & Y is & L (x x ) & L (x ) & L ( x ) & L ( ) & R y & L & R We have made explicit the positions of the games in order to underline the fact that they correspond to partial explorations of formulas, but the naming of a position won t play any role in the definition of asynchronous strategies (contrary for 18

19 example to ludics [Gir01] where the definition of games starts from logic). The interpretation of formulas can be extended to sequents S = A 1,..., A m B 1,..., B n by G χ S = ( ig χ A ) ( i G χ B i i ). 3.2 Interpretation of proofs A strategy σ on a game G is a non-empty prefix-closed set of plays, which are paths whose source is the initial position of the game. We write σ : G to indicate that σ is a strategy on a game G. To every proof π of a formula A, we associate a family σ χ, indexed by contexts χ, of strategies on the game G χ interpreting the sequent S in the context χ. This family of strategies describe the explorations of formulas that are allowed by the proof and are formally defined by induction of the proofs which are generated by the following sequent rules: X, X [ax] Γ, A, A [cut] Γ, Γ, A, B Γ, A ` B [`] Γ, A Γ, B [&] Γ, A & B Γ, A, B Γ,, A B [ ] Γ, A Γ, A B [ L] Γ, B Γ, A B [ R] Every axiom X, X is interpreted as the family of copycat strategies σ χ on the game χ(x), such a strategy σ being defined as the smallest strategy containing the empty play ε and such that for every play s : (x, x) in σ such that there exists an Opponent move m : x y or a Proponent move n : x y in χ(x), the plays s (x, m) (m, y) : (y, y) or s (n, x) (y, n) : (y, y) are also in σ. Suppose fixed an environment χ. A play in the interpretation of a proof whose first rule (from bottom up) is [`] s with π as premise is either empty or of the form ( ` ) (x`y) where s is a play in the interpretation of π. A play in the interpretation of a proof beginning with a [ ] rule with π 1 and π 2 s as premises is either empty or of the form ( ) (x y) where s is a play such that s Γ,A is a play in the interpretation of π 1 and s,b is a play in the interpretation of π 2. A play in the interpretation of a proof beginning with a [&] rule with π 1 and π 2 s as premises is either empty or of the form (& L ) (& L x) where s is a s play in the interpretation of π 1 or of the form (& R ) (& R x) where s is a play in the interpretation of π 2. A play in the interpretation of a proof beginning with a [ L ] with π as premise s is either empty or of the form ( L ) ( L x) where s is a play in the interpretation of π. A play in the interpretation of a proof beginning with a [ R ] with π as premise s is either empty or of the form ( R ) ( R x) where s is a play in the interpretation of π. Suppose that χ defines n variables, and χ is another context such that there exists a family (ι i ) of n embeddings ι i : χ(x i ) χ (X i ). In an obvious way, by a coproduct 19