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.

The Full Abstraction Problem for PCF [23, 20, 7, 11] is one of the longest-standing problems in the semantics of programming languages. There is quite widespread agreement that it is one of the most difficult; there is much less agreement as to what exactly the problem is, or more particularly as to the precise criteria for a solution. The usual formulation is that one wants a "semantic characterization" of the fully abstract model (by which we mean the inequationally fully abstract order-extensional model, which Milner proved to be uniquely specified up to isomorphism by these properties [20]). The problem is to understand what should be meant by a "semantic characterization". Our view is that the essential content of the problem, what makes it important, is that it calls for a semantic characterization of sequential, functional computation at hig...

. We present a general semantic universe of call-by-value computation based on elements of game semantics, and validate its appropriateness as a semantic universe by the full abstraction result for call-by-value PCF, a generic typed programming language with call-by-value evaluation. The key idea is to consider the distinction between call-by-name and call-by-value as that of the structure of information flow, which determines the basic form of games. In this way the call-by-name computation and call-by-value computation arise as two independent instances of sequential functional computation with distinct algebraic structures. We elucidate the type structures of the universe following the standard categorical framework developed in the context of domain theory. Mutual relationship between the presented category of games and the corresponding call-by-name universe is also clarified. 1. Introduction The call-by-value is a mode of calling procedures widely used in imperative and function...

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]. The presented computation model, forming the class of hereditarily sequential functionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. This approach is influenced by the work of Kleene [Kle78], Gandy [Gan67], Kahn and Plotkin [KP78], Berry and Curien [BC82, Cur86, Cur92], and Cartwright and Felleisen [CF92]. We characterize the computable elements in this model in two different ways: (a) by recursiveness requirements for the game, and (b) as definability with the schemata (S1)-- (S8), (S11), which is related to definability in PCF. It turns out that both definitio...

We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli.

One of the major challenges in denotational semantics is the construction of a fully abstract semantics for a higher-order sequential programming language. For the past fifteen years, research on this problem has focused on developing a semantics for PCF, an idealized functional programming language based on the typed λ-calculus. Unlike most practical languages, PCF has no facilities for observing and exploiting the evaluation order of arguments to procedures. Since we believe that these facilities play a crucial role in sequential computation, this paper focuses on a sequential extension of PCF, called SPCF, that includes two classes of control operators: a possibly empty set of error generators and a collection of catch and throw constructs. For each set of error generators, the paper presents a fully abstract semantics for SPCF. If the set of error generators is empty, the semantics interprets all procedures—including catch and throw—as Berry-Curien sequential algorithms. If the language contains error generators, procedures denote manifestly sequential functions. The manifestly sequential functions form a Scott domain that is isomorphic to a domain of decision trees, which is the natural

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

ion Robert Cartwright Matthias Felleisen Department of Computer Science Rice University Houston, TX 77251-1892 Abstract One of the major challenges in denotational semantics is the construction of fully abstract models for sequential programming languages. For the past fifteen years, research on this problem has focused on developing models for PCF, an idealized functional programming language based on the typed lambda calculus. Unlike most practical languages, PCF has no facilities for observing and exploiting the evaluation order of arguments in procedures. Since we believe that such facilities are crucial for understanding the nature of sequential computation, this paper focuses on a sequential extension of PCF (called SPCF) that includes two classes of control operators: error generators and escape handlers. These new control operators enable us to construct a fully abstract model for SPCF that interprets higher types as sets of error-sensitive functions instead of continuous...

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Sieber has described a model of PCF consisting of continuous functions that are invariant under certain (finitary) logical relations, and shown that it is fully abstract for closed terms of up to third-order types. We show that one may achieve full abstraction at all types using a form of "Kripke logical relations" introduced by Jung and Tiuryn to characterize -definability. To appear in Information and Computation. (Accepted, October 1994) Supported by NSF grant CCR-92110829. 1 Introduction The nature of sequential functional computation has fascinated computer scientists ever since Scott remarked on a curious incompleteness phenomenon when he introduced LCF (Logic for Computable Functions) and its continuous function model in 1969 (Scott, 1993). Scott noted that although the functionals definable by terms in PCF---the term language of LCF---admitted a sequential evaluation strategy, there were functions in the model that seemed to require a parallel evaluation strategy. "Sequen...