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On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
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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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
Partial Combinatory Algebras of Functions
, 2009
"... We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative st ..."
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We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is, that every realizability topos is a quotient of a realizability topos on a total combinatory algebra. AMS Subject Classification (2000): 03B40,68N18
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...
Resourcebounded Continuity and Sequentiality for Typetwo Functionals
"... Devices]: Complexity Measures and Classesrelations among complexity classes; F.1.1 [Computation by Abstract Devices]: Models of Computationrelations between models; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logiccomputability theory General Terms: Theory, Algorithms Ad ..."
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Devices]: Complexity Measures and Classesrelations among complexity classes; F.1.1 [Computation by Abstract Devices]: Models of Computationrelations between models; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logiccomputability theory General Terms: Theory, Algorithms Additional Key Words and Phrases: Higherorder complexity, sequential computation, decision trees 1.
Part II Local Realizability Toposes and a Modal Logic for
"... 5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define: ..."
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5.1 Definition and Examples 5.1.1 Definition and Definability Results A tripos is a weak tripos with disjunction which has a (weak) generic object. Explicitly we define:
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...
Towards a Logic of Sequential Computation: A synthetic account of Sequential Domain Theory
"... Synthetic Domain Theory (SDT) was originally suggested by Dana Scott to obtain a uniform and logicbased account of domain theory. In SDT the domain structure is intrinsic to a chosen class of sets with “good” properties. SDT is uniform in the sense that it applies to different models of the basic a ..."
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Synthetic Domain Theory (SDT) was originally suggested by Dana Scott to obtain a uniform and logicbased account of domain theory. In SDT the domain structure is intrinsic to a chosen class of sets with “good” properties. SDT is uniform in the sense that it applies to different models of the basic axioms thus giving rise to different kinds of domains. E.g. in [2] one finds a sheaf model for a kind of “stable ” SDT and in [6, 3] realizability models for a kind of “strongly stable ” domain theory in the sense of [1]. Axiomatising these different kinds of models would give rise to different kinds of “flavours ” of SDT as suggested by Martin Hyland. The pca used in [6, 3] is derived from the universal object U = [N → N] of the category SA of countably based sequential algorithms (where N is the concrete data structure of natural numbers). As shown in [4, 5] the type U = [N → N] is also universal in the wellpointed category OSA of observably sequential algorithms which contains SA as a lluf subccc. In this paper we consider the realizability model for SDT arising from U in OSA and discuss possible axiomatizations of sequential SDT inspired by this model. In particular, we exploit the fact that there is a type O such that O → O contains a dominance Σ corresponding to the one considered in the realizability model over U in SA.