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

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 (Neyman-Pearson) 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.

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I present a semantics for the language of first order additive-multiplicative 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 (resource-using, problem-solving) 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...

. We discuss the extent to which game semantics is implicit in the basic concepts of linear logic. Introduction The purpose of this paper is to show that a version of game semantics for linear logic is implicit in the logic itself and the basic intuitions underlying the logic. Like the talk at CSL'93 on which it is based, the body of this paper is intended to be accessible to people with little or no previous knowledge of linear logic or game semantics. Comments that do presuppose such prior knowledge have been relegated to a series of notes at the end of the paper. Propositions as Types The relevance of various constructive propositional logics, including linear logic, to computation and particularly to type theory is largely based on the propositionsas -types paradigm, also often called the Curry-Howard isomorphism [8, 9, 13]. In its simplest form, this paradigm involves a correspondence between the constructive logic of implication and simple typed combinatory logic. Constructive...

Abstract. Theorem proving, or algorithmic proof-search, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proof-search as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important examples of control, namely backtracking and uniform proof. 1 Introduction to reductive logic and proof-search Theorem proving, or algorithmic proof-search, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proof-search as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important

While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in many, diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around recent developments in the theory of sequent calculus proofs. We outline how new tools and methodologies can be developed around a boarder approach to computational logic. Computational logic, unity of logic, proof theory 1. SOFTWARE AND HARDWARE CORRECTNESS IS CRITICALLY IMPORTANT Computer systems are everywhere in our societies and their integration with all parts of our lives is constantly increasing. There are a host of computer systems—such as those in cars, airplanes, missiles, hospital equipment—where correctness of software is

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The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.