Using formal tools in computer science to describe games is an interesting problem. We give games, exactly two person games, an axiomatical foundation based on the process algebra ACP (Algebra of Communicating Process). A fresh operator called opponent's alternative composition operator (OA) is introduced into ACP to describe game trees and game strategies, called GameACP. And its sound and complete axiomatical system is naturally established. To model the outcomes of games (the co-action of the player and the opponent), correspondingly in GameACP, the execution of GameACP processes, another operator called playing operator (PO) is extended into GameACP. We also establish a sound and complete axiomatical system for PO. Finally, we give the correctness theorem between the outcomes of games and the deductions of GameACP processes.
The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of ?-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 Mellies. It also requires an interactive reformulation of the usual acyclicity criterion of linear logic, as well as a directed variant, as a scheduling criterion. The manipulation of objects with state which changes over time is all-pervasive in computing. Perhaps the simplest example of such objects are the program variables of classical imperative languages. An important strand of work within the study of such languages, pioneered by John Reynolds, focusses on Idealized Algol, an elegant synthesis of imperative and functional features.   In game semantics, the higher-order value passing mechanisms of the λ-calculus are decomposed as sequences of atomic actions exchanged by a Player and its Opponent. Seen from this angle, game semantics is reminiscent of trace semantics in concurrency theory, where a process is identified to the sequences of requests it generates in the course of time. Asynchronous game semantics is an attempt to bridge the gap between the two subjects, and to see mainstream game semantics as a refined and interactive form of trace semantics. Asynchronous games are positional games played on Mazurkiewicz traces, which reformulate (and generalize) the familiar notion of arena game. The interleaving semantics of λ-terms, expressed as innocent strategies, may be analysed in this framework, in the perspective of true concurrency. The analysis reveals that innocent strategies are positional strategies regulated by forward and backward confluence properties. This captures, we believe, the essence of innocence. We conclude the article by defining a non-uniform variant of the λ-calculus, in which the game semantics of a λ-term is formulated directly as a trace semantics, performing the syntactic exploration or parsing of that λ-term. Connections between the sequentialitysconcurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. A games model of a programming language with higher-order store in the style of ML-references is introduced. The category used for the model is obtained by relaxing certain behavioural conditions on a category of games previously used to provide fully abstract models of pure functional languages. The model is shown to be fully abstract by means of factorization arguments which reduce the question of definability for the language with higher-order store to that for its purely functional fragment.  A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for Multiplicative-Additive Linear Logic is proved for this semantics.  We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy: strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al   In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full abstraction problem for PCF, raised in [Plo77].