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An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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Cited by 17 (6 self)
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
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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Cited by 11 (5 self)
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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.
Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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Cited by 8 (7 self)
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
Synthetic domain theory and models of linear Abadi & Plotkin logic
, 2005
"... Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can r ..."
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Cited by 5 (4 self)
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Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPLstructure. This adds to the evidence that the notion of LAPLstructure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1
Inductive Construction of Repletion
 Appl. Categ. Structures
, 1997
"... Introduction In [2] Martin Hyland has proposed the notion of "Sreplete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains an ..."
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Cited by 2 (2 self)
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Introduction In [2] Martin Hyland has proposed the notion of "Sreplete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains and thus about meanings of functional programs where all functions between predomains are continuous and all endofunctions on domains have least fixpoints. Furthermore, SDT allows one to consider predomains as a full reflective subcategory of the ambient category of constructive sets. For this reason the ambient category is assumed to be a model of some sufficiently strong impredicative constructive type theory which will be used as the internal language for the ambient category of constructive sets in order to replace complicated external arguments by simpler proofs in the
Matching typed and untyped realizability (Extended abstract)
"... Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed ob ..."
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Cited by 1 (0 self)
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Realizability interpretations of logics are given by saying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might be drawn from an untyped universe of computation, such as a partial combinatory algebra, or they might be typed objects such as terms of a PCFstyle programming language. In some instances, one can show that a particular untyped realizability interpretation matches a particular typed one, in the sense that they give the same set of realizable formulae. In this case, we have a very good fit indeed between the typed language and the untyped realizability model—we refer to this condition as (constructive) logical full abstraction. We give some examples of this situation for a variety of extensions of PCF. Of particular interest are some models that are logically fully abstract for typed languages including nonfunctional features. Our results establish connections between what is computable in various programming languages, and what is true inside various realizability toposes. We consider some examples of logical formulae to illustrate these ideas, in particular their application to exact realnumber computability. The present article summarizes the material I presented at the Domains IV workshop, plus a few subsequent developments; it is really an extended abstract for a projected journal paper. No proofs are included in the present version. 0
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...
Domains in H
"... We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Int ..."
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We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domainlike sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a settheoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...
Notions of Computability for General Datatypes  Case For Support to accompany EPS(RP)
"... ion and information hiding III.1. Abstract types A more ambitious and speculative part of our programme will be the investigation of issues of computability for abstract types. In many modern programming languages, facilities for data abstraction are very important for the modular design of large p ..."
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ion and information hiding III.1. Abstract types A more ambitious and speculative part of our programme will be the investigation of issues of computability for abstract types. In many modern programming languages, facilities for data abstraction are very important for the modular design of large programs. The basic idea is that we can only interact with the data values through some prescribed interface. Indeed, the finite types are abstract types in a certain sense, since (typically) the only way to interact with a function is via application. Taking an extensional or "behavioural" view of datatypes, one is led to consider questions such as the following: ffl Which functions to and from the abstract type are computable? ffl When are two elements of the abstract type observationally equivalent? ffl When are two implementations of the same abstract type signature observationally indistinguishable ? Even for abstract types with firstorder signatures, these questions present a sign...