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36
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 i ..."
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Cited by 243 (15 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.
A games semantics for linear logic
 Ann. Pure Appl. Logic
, 1992
"... We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard’s linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q ..."
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Cited by 152 (3 self)
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We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard’s linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier ‘almost ’ will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of gamesemantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard’s proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel’s Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.
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 68 (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
Propositional computability logic I
 ACM Transactions on Computational Logic
"... Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semanticsbased alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and ..."
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Cited by 61 (27 self)
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Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semanticsbased alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and “truth ” is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting’s intuitionistic calculus INT, whose language can be seen as a special fragment of that of CL. The constructivistic philosophy of INT, however, just like the resource philosophy of linear logic, has never really found an intuitively convincing and mathematically strict semantical justification. CL has good claims to provide such a justification and hence a materialization of Kolmogorov’s known thesis “INT = logic of problems”. The present paper contains a soundness proof for INT with respect to the CL semantics. It is expected to constitute part 1 of a twopiece series on the intuitionistic fragment of CL, with part 2 containing an anticipated completeness proof. 1
Representing Epistemic Uncertainty by means of Dialectical Argumentation
 Annals of Mathematics and AI
"... We articulate a dialectical argumentation framework for qualitative representation of epistemic uncertainty in scientific domains. The framework is grounded in specific philosophies of science and theories of rational mutual discourse. We study the formal properties of our framework and provide i ..."
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Cited by 37 (29 self)
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We articulate a dialectical argumentation framework for qualitative representation of epistemic uncertainty in scientific domains. The framework is grounded in specific philosophies of science and theories of rational mutual discourse. We study the formal properties of our framework and provide it with a game theoretic semantics. With this semantics, we examine the relationship between the snaphots of the debate in the framework and the long run position of the debate, and prove a result directly analogous to the standard (NeymanPearson) approach to statistical hypothesis testing. We believe this formalism for representating uncertainty has value in domains with only limited knowledge, where experimental evidence is ambiguous or conflicting, or where agreement between different stakeholders on the quantification of uncertainty is difficult to achieve. All three of these conditions are found in assessments of carcinogenic risk for new chemicals.
The logic of interactive Turing reduction
 Journal of Symbolic Logic
"... The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of ..."
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Cited by 25 (20 self)
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The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.
A Constructive Game Semantics for the Language of Linear Logic
, 1996
"... I present a semantics for the language of first order additivemultiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sente ..."
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Cited by 18 (13 self)
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I present a semantics for the language of first order additivemultiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sentences =games, sentences=resources or sentences=problems, where "truth" means existence of an effective winning (resourceusing, problemsolving) strategy. The paper introduces a decidable first order logic ET in the above language and gives a proof of its soundness and completeness (in the full language) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic. The semantics presented here is very similar to Blass's game semantics (A.Blass, "A game semantics for linear logic", APAL, 56). Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be ad...
The logic of tasks
 Annals of Pure and Applied Logic 117 (2002
"... The paper introduces a semantics for the language of classical first order logic supplemented with the additional operators ⊓ and ⊓. This semantics understands formulas as tasks. An agent (say, a machine or a robot), working as a slave for its master (say, the user or the environment), can carry out ..."
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Cited by 14 (11 self)
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The paper introduces a semantics for the language of classical first order logic supplemented with the additional operators ⊓ and ⊓. This semantics understands formulas as tasks. An agent (say, a machine or a robot), working as a slave for its master (say, the user or the environment), can carry out the task α⊓β if it can carry out any one of the two tasks α, β, depending on which of them was requested by the master; similarly, it can carry out ⊓xα(x) if it can carry out α(x) for any particular value for x selected by the master; an agent can carry out α→β if it can carry out β as long as it has, as a slave (resource), an agent who carries out α; finally, carrying out P, where P is an atomic formula, simply means making P true; in particular, ⊥ is a task that no agent can carry out. When restricted to the language of classical logic, the meaning of formulas is isomorphic to their classical meaning, which makes our semantics a conservative extension of classical semantics. This semantics can claim to be a formalization of the resource philosophy associated with linear logic, if resources are understood as agents carrying out tasks. The classical operators of our language correspond to the multiplicative operators of linear logic, while ⊓ and ⊓ correspond to the additive conjunction and universal quantifier, respectively. Our formalism may also have a potential to be used in AI as an alternative logic of planning and action. Its main appeal is that it is immune to the frame problem and the knowledge preconditions problem. The paper axiomatically defines a logic L in the above language and proves its soundness and completeness with respect to the task semantics in the following intuitive sense: L ⊢ α iff α can be carried out by an agent who has nothing but its intelligence (i.e. no physical resources or external sources of information) for carrying out tasks. This logic is shown to be semidecidable in the full language and decidable when the classical quantifier (but not ⊓) is forbidden in it. ∗ Supported by Summer Research Grant from Villanova University 1 1
The safe lambda calculus
 of Lecture Notes in Computer Science
, 2007
"... Abstract. Safety is a syntactic condition of higherorder grammars that constrains occurrences of variables in the production rules according to their typetheoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition ..."
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Abstract. Safety is a syntactic condition of higherorder grammars that constrains occurrences of variables in the production rules according to their typetheoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by transposing (and generalizing) the safety condition to the setting of the simplytyped lambda calculus. In contrast to the original definition of safety, our calculus does not constrain types (to be homogeneous). We show that in the safe lambda calculus, there is no need to rename bound variables when performing substitution, as variable capture is guaranteed not to happen. We also propose an adequate notion of βreduction that preserves safety. In the same vein as Schwichtenberg’s 1976 characterization of the simplytyped lambda calculus, we show that the numeric functions representable in the safe lambda calculus are exactly the multivariate polynomials; thus conditional is not definable. We also give a characterization of representable word functions. We then study the complexity of deciding betaeta equality of two safe simplytyped terms and show that this problem is PSPACEhard. Finally we give a gamesemantic analysis of safety: We show that safe terms are denoted by Pincrementally justified strategies. Consequently pointers in the game semantics of safe λterms are only necessary from order 4 onwards.