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Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... 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 ..."
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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.
Bistructures, Bidomains and Linear Logic
 in Proc. 21st ICALP
, 1997
"... Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of func ..."
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Cited by 14 (3 self)
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Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has an associated coKleisli category which is closely related to that of Berry's bidomains (both have equivalent nontrivial full subcartesian closed categories).
Timeless Games
 Computer Science Logic: 11th International Workshop Proceedings, volume 1414 of Lecture Notes in Computer Science. EACSL
, 1998
"... . Two models of classical linear logic are set up. First our recent version of AJM games model which will be our source model. Then the target model, polarized pointed relations, a variant of the plain relational model which is constructed in two steps: first the model of pointed relations, then the ..."
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Cited by 14 (1 self)
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. Two models of classical linear logic are set up. First our recent version of AJM games model which will be our source model. Then the target model, polarized pointed relations, a variant of the plain relational model which is constructed in two steps: first the model of pointed relations, then the additional polarization structure which yields a proper duality. Then the natural timeforgetting map is shown to generate a lax functor from the source to the target. Finally a further refinement of the target model using bipolarities is sketched, giving a closer link with the games model for the interpretation of syntax. Thus a bridge is constructed that goes from a dynamic model to a static model of evaluation. 1 Introduction The basic mathematical reflex was to model types, programs, evaluation with sets, functions, composition. As years went by, denotational semantics pictured the syntactic triple with increasingly sophisticated tools: lattices, posets, concrete data structures, coher...
A Fully Abstract Model for Sequential Computation
, 1998
"... In 1977, G. Plotkin pointed out the problem of finding a fully abstract model for the sequential programming language PCF [16], which had been originally developed by D. Scott [19]. This question turned out to be one of the most enduring problems of semantics. A very nice description of the differen ..."
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In 1977, G. Plotkin pointed out the problem of finding a fully abstract model for the sequential programming language PCF [16], which had been originally developed by D. Scott [19]. This question turned out to be one of the most enduring problems of semantics. A very nice description of the different approaches
Realizability Models for Sequential Computation
, 1998
"... We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of seq ..."
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We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...