An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certain simple extension of PCF. We then introduce an intrinsic preorder on strategies, and show that it satisfies some remarkable properties, such that the intrinsic preorder on function types coincides with the pointwise preorder. We then obtain an order-extensional fully abstract model of PCF by quotienting the intensional model by the intrinsic preorder. This is the first syntax-independent description of the fully abstract model for PCF. (Hyland and Ong have obtained very similar results by a somewhat different route, independently and at the same time.) We then consider the effective version of our model, and prove a Universality Theorem: every element of the effective extensional model is definable in PCF. Equivalently, every recursive strategy is definable up to observational equivalence.

Introduction SAMSON ABRAMSKY (samson@comlab.ox.ac.uk) Oxford University Computing Laboratory 1. Introduction Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntax-independent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with non-functional features such as control operators and locally-scoped references [4, 21, 5, 19, 2, 22, 17, 11]. A substantial survey of the state of the art of Game Semantics circa 1997 was given in a previous Marktoberdorf volume [6]. Our aim in this tutorial presentation is to give a first indication of how Game Semantics can be developed in a new, algorithmic direction, with a view to applications in computer-assisted verification and program analysis. Some promising steps have already been taken in this

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Abstract. We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental t-norm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequents-of-relations. Such a framework can be interpreted via a particular class of dialogue games combined with bets, where the rules reflect possible moves in the game. The problem of determining the validity of atomic relational hypersequents is shown to be polynomial for each logic, allowing us to develop Co-NP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics. 1

ion for PCF Motivated by the full completeness results, it became of compelling interest to re-examine perhaps the best-known "open problem" in the semantics of programming languages, namely the "Full Abstraction problem for PCF", using the new tools provided by game semantics. 2 PCF is a higher-order functional programming language; modulo issues of the parameterpassing strategies, it forms a fragment of any programming language with higher-order procedures (which includes any reasonably expressive object-oriented language). The aspect of the Full Abstraction problem I personally found most interesting was: to construct a syntax-independent model in which every element is the denotation of some program (note the analogy with full completeness, whose definition had in turn been motivated in part by this aspect of full abstraction). This is not how the problem was originally formulated, but by "general abstract nonsense", given such a model one can always quotient it to get a fully ab...

Building on a version of Lorenzen’s dialogue foundation for intuitionistic logic, we show that a suitable game of communicating parallel dialogues is sound and complete for Gödel-Dummett logic G. Among other things, this provides a computational interpretation of Avron’s hypersequent calculus for G.

Abstract: A dialogue game for fuzzy logic, based on the comparison of truth degrees, is presented. It is shown that the game is adequate for G △ ∞, i.e., intuitionistic fuzzy logic enriched by the projection operator △. Any given countermodel to a formula can be used to construct a winning strategies for one of the players, called Opponent. Conversely, counter-models can be extracted from each winning strategy for Opponent. Winning strategies for the other player, Proponent, correspond to proofs of validity. The systematic construction of so-called complete dialogue trees can be viewed as tableau style proof search procedure.

Abstract. A parallel version of Lorenzen’s dialogue theoretic foundation for intuitionistic logic is shown to be adequate for a number of important intermediate logics. The soundness and completeness proofs proceed by relating hypersequent derivations to winning strategies for parallel dialogue games. This also provides a computational interpretation of hypersequents. 1

Abstract. Two popular approaches to formalize adequate reasoning with vague propositions are usually deemed incompatible: On the one hand, there is supervaluation with respect to precisification spaces, which consist in collections of classical interpretations that represent admissible ways of making vague atomic statements precise. On the other hand, t-norm based fuzzy logics model truth functional reasoning, where reals in the unit interval [0,1] are interpreted as degrees of truth. We show that both types of reasoning can be combined within a single logic SŁ, that extends both: Łukasiewicz logic Ł and (classical) S5, where the modality corresponds to ‘... is true in all complete precisifications’. Our main result consists in a game theoretic interpretation of SŁ, building on ideas already introduced by Robin Giles in the 1970s to obtain a characterization of Ł in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. In our case the experiments are replaced by random evaluations with respect to a given probability distribution over permissible precisifications. 1

. Game theory can be a basis for theorem proving. In Dialogue Logic the validity of a formula F is examined in two person, perfect information games. After a brief description of Dialogue Logic and introductory examples we discuss some implementation issues of Colosseum, our theorem prover implementing Dialogue Games. We describe the search space one has to conquer and propose search tactics to direct the search. Finally, we compare Colosseum's performance with that of two tableaux provers, namely ft and ileantab. 1 Dialogue Logic Game theory can be a basis for theorem proving. In Dialogue Logic, originally developed by Lorenzen et al. [6], the validity of some given formula F is examined in two person, perfect information games. There are two players moving alternatively, both having complete information of the current situation of the game. The player who claims being able to justify F is called the proponent, his adversary the opponent. The initial state, the dialogue setting, is ...