This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Non-local control flow is shown to fit into this framework as the violation of strong and weak ‘bracketing ’ conditions, related to linear behaviour. The language µPCF (Parigot’s λµ with constants and recursion) is adopted as a simple basis for higher-type, sequential computation with access to the flow of control. A simple operational semantics for both call-by-name and call-by-value evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of µPCF.

) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. 1 Introduction We report here on the current status of research on the Logic of Types and Computation at Carnegie Mellon University [SAB + ]. The general goal of this research program is to develop a logical fra...

We investigate the development of theories of types and computability via realizability.

In an impure functional language, there are programs whose behaviour is completely functional (in that they behave extensionally on inputs), but the functions they compute cannot be written in the purely functional fragment of the language. That is, the class of programs with functional behaviour is more expressive than the usual class of pure functional programs. In this paper we introduce this extended class of "functional" programs by means of examples in Standard ML, and explore what they might have to offer to programmers and language implementors. After reviewing some theoretical background, we present some examples of functions of the above kind, and discuss how they may be implemented. We then consider two possible programming applications for these functions: the implementation of a search algorithm, and an algorithm for exact real-number integration. We discuss the advantages and limitations of this style of programming relative to other approaches. We also consider the incr...

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. non-uniform.

The relationship between Hyland-Ong-style games and Berry-Curien sequential algorithms is investigated, with the object of describing semantic solutions to two problems | to characterise eectively the \minimal models" of the simply-typed -calculus and the fully abstract model of PCF with control operators | which are shown to be equivalent.

Abstract. We give a simple order-theoretic construction of a Cartesian closed category of sequential functions. It is based on bistable biorders, which are sets with a partial order — the extensional order — and a bistable coherence, which captures equivalence of program behaviour, up to permutation of top (error) and bottom (divergence). We show that monotone and bistable functions (which are required to preserve bistably bounded meets and joins) are strongly sequential, and use this fact to prove universality results for the bistable biorder semantics of the simply-typed lambda-calculus (with atomic constants), and an extension with arithmetic and recursion. We also construct a bistable model of SPCF, a higher-order functional programming language with non-local control. We use our universality result for the lambda-calculus to show that the semantics of SPCF is fully abstract. We then establish a direct correspondence between bistable functions and sequential algorithms by showing that sequential data structures give rise to bistable biorders, and that each bistable function between such biorders is computed by a sequential algorithm. 1.

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It is known that the strongly stable functions which arise in the semantics of PCF can be realized by sequential algorithms, which can be considered as deterministic strategies in games associated to PCF types. Studying the connection between strongly stable functions and sequential algorithms, two dual classes of hypercoherences naturally arise: the parallel and serial hypercoherences. The objects belonging to the intersection of these two classes are in bijective correspondence with the so-called "serial-parallel" graphs, that can essentially be considered as games. We show how to associate to any hypercoherence a parallel hypercoherence together with a projection onto the given hypercoherence and present some properties of this construction. Intuitively, it makes explicit the computational time of a hypercoherence.