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18
Full Abstraction for PCF
 Information and Computation
, 1996
"... 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 "historyfree" strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certai ..."
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Cited by 192 (14 self)
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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 "historyfree" 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 orderextensional fully abstract model of PCF by quotienting the intensional model by the intrinsic preorder. This is the first syntaxindependent 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.
Algorithmic Game Semantics
 In Schichtenberg and Steinbruggen [16
, 2001
"... 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 syntaxindependen ..."
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Cited by 48 (3 self)
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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 syntaxindependent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with nonfunctional features such as control operators and locallyscoped 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 computerassisted verification and program analysis. Some promising steps have already been taken in this
Uniform rules and dialogue games for fuzzy logics
 LPAR 2004, volume 3452 of Lecture Notes in Computer Science
, 2004
"... Abstract. We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental tnorm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequentsofrelations. Such a f ..."
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Cited by 17 (8 self)
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Abstract. We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental tnorm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequentsofrelations. 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 CoNP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics. 1
Games in the Semantics of Programming Languages
 Dept. of Philosophy, University of Amsterdam
, 1997
"... ion for PCF Motivated by the full completeness results, it became of compelling interest to reexamine perhaps the bestknown "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 highero ..."
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Cited by 8 (1 self)
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ion for PCF Motivated by the full completeness results, it became of compelling interest to reexamine perhaps the bestknown "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 higherorder functional programming language; modulo issues of the parameterpassing strategies, it forms a fragment of any programming language with higherorder procedures (which includes any reasonably expressive objectoriented language). The aspect of the Full Abstraction problem I personally found most interesting was: to construct a syntaxindependent 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...
From intuitionistic logic to GödelDummett logic via parallel dialogue games
 IN PROCEEDINGS OF THE 33RD IEEE INTERNATIONAL SYMPOSIUM ON MULTIPLEVALUED LOGIC
, 2003
"... 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 GödelDummett logic G. Among other things, this provides a computational interpretation of Avron’s hypersequent calculus for G ..."
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Cited by 8 (4 self)
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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 GödelDummett logic G. Among other things, this provides a computational interpretation of Avron’s hypersequent calculus for G.
A dialogue game for intuitionistic fuzzy logic based on comparison of degrees of truth
 In Proceedings of InTech’03
, 2003
"... 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 strategie ..."
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Cited by 4 (2 self)
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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, countermodels 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 socalled complete dialogue trees can be viewed as tableau style proof search procedure.
Parallel Dialogue Games and Hypersequents for Intermediate Logics
 Proceedings of TABLEAUX 2003, Automated Reasoning with Analytic Tableaux and Related Methods
, 2003
"... 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 game ..."
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Cited by 3 (2 self)
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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
Giles’s Game and the Proof Theory of ̷Lukasiewicz Logic
"... Abstract. In the 1970s, Robin Giles introduced a game combining Lorenzenstyle dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in ̷Lukasiewicz logic. In this pa ..."
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Cited by 3 (2 self)
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Abstract. In the 1970s, Robin Giles introduced a game combining Lorenzenstyle dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in ̷Lukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work on the proof theory of ̷Lukasiewicz logic.
Combining supervaluation and degree based reasoning under vagueness
"... 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 makin ..."
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Cited by 2 (2 self)
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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, tnorm 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