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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.
Parallel computable higher type functionals (Extended Abstract)
 In Proceedings of IEEE 34th Annual Symposium on Foundations of Computer Science, Nov 35
, 1993
"... ) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different way ..."
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) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursiontheoretic, prooftheoretic and machinetheoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifierfree, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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Dialogue Games and Innocent Strategies: An Approach to (Intensional) Full Abstraction for PCF
, 1993
"... ion for PCF Preliminary Announcement Martin Hyland and Luke Ong 26th July 1993 This note is (intended to be) released in conjunction with a preliminary announcement of Abramsky, Jagadeesan and Malacaria entitled Games and Full Abstraction of PCF. Like Abramsky et al. (but independently), we have ..."
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ion for PCF Preliminary Announcement Martin Hyland and Luke Ong 26th July 1993 This note is (intended to be) released in conjunction with a preliminary announcement of Abramsky, Jagadeesan and Malacaria entitled Games and Full Abstraction of PCF. Like Abramsky et al. (but independently), we have found an intensionally fully abstract model for pcf [Plo77]. Our model is a Cartesian closed category of Scott domains all of whose compact elements are definable in pcf. Using Stoughton's Theorem [Sto88], the model can be extensionally collapsed by means of a continuous homomorphism to the least fixpoint, orderextensional, fully abstract model which is shown to be unique by Milner [Mil77]. It is unclear at this stage how our model relates to that of Abramsky et al. Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players. This approach is very concrete and in nature goes back to Kleene [Kle78] and Gandy in o...
Applications of the KleeneKreisel Density Theorem to Theoretical Computer Science
, 2006
"... The KleeneKreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbe ..."
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The KleeneKreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbers as inputs and finally survey some recent applications of this generalization. 1
Computing with functionals  computability theory or computer science
 Bulletin of Symbolic Logic
, 2006
"... We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject. ..."
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We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.
The Analysis of Programming Structure
 ACM SIGACT News
, 1997
"... This paper has explored three examples of good semantical analyses of programming structures. The three examples share two characteristics: the semantic models are abstract enough to be applicable in many situations, and the models lead to proofs of noncomputability. Other examples of programming s ..."
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This paper has explored three examples of good semantical analyses of programming structures. The three examples share two characteristics: the semantic models are abstract enough to be applicable in many situations, and the models lead to proofs of noncomputability. Other examples of programming structures have been omitted from this short essay: foundations for objectoriented languages, descriptions of languages with local variables, and the theory of database query languages. Each of these examples have corresponding semantical theories that enjoy the two characteristics above. The richness of programming structure suggests a corollary: it is folly to look for one universal model to explain all programming structures. Of course, as a theoretical subject, semantics benefits from the reduction of many concepts to a primitive, common level. Nevertheless, reduction must often be resisted. We have seen how computability theory loses all kinds of relevant distinctions. Another example is the naive semantics of PCF based on dcpos: the model is not abstract enough,
Complexity and Intensionality in a Type1 Framework for Computable Analysis
 Computer Science Logic: 19th International Workshop, CSL 2005, 14th Annual Conference of the EACSL
"... Abstract. Implementations of real number computations have largely been unusable in practice because of their very bad performance, especially in comparison to floating point arithmetic implemented in hardware. This performance problem is to a very large extent due to the type2 nature of the comput ..."
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Abstract. Implementations of real number computations have largely been unusable in practice because of their very bad performance, especially in comparison to floating point arithmetic implemented in hardware. This performance problem is to a very large extent due to the type2 nature of the computable analysis frameworks usually employed. This problem can be overcome by employing a type1 approach. This paper presents such an approach and deals with properties of it that have not been well studied before, namely the introduction of complexity measures for type1 representations of real functions and ways to define intensional functions, i.e. functions that may return different real numbers for the same real argument given in different representations. 1