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Hereditarily Sequential Functionals
 In Proceedings of the Symposium on Logical Foundations of Computer Science: Logic at St. Petersburg, Lecture notes in Computer Science
, 1994
"... In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full ..."
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Cited by 59 (0 self)
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In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full abstraction problem for PCF, raised in [Plo77]. The presented computation model, forming the class of hereditarily sequential functionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. This approach is influenced by the work of Kleene [Kle78], Gandy [Gan67], Kahn and Plotkin [KP78], Berry and Curien [BC82, Cur86, Cur92], and Cartwright and Felleisen [CF92]. We characterize the computable elements in this model in two different ways: (a) by recursiveness requirements for the game, and (b) as definability with the schemata (S1) (S8), (S11), which is related to definability in PCF. It turns out that both definitio...
Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
Correspondence between Operational and Denotational Semantics
 Handbook of Logic in Computer Science
, 1995
"... This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational ..."
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This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of orderextensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simplytyped lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dIdomains and stable functions Homepa...
What Is an Algorithm?
, 2000
"... Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (o ..."
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Cited by 23 (3 self)
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Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The ChurchTuring Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...
Computational Foundations of Basic Recursive Function Theory
 Theoretical Computer Science
, 1988
"... The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on complet ..."
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The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the ...
Density and Choice for Total Continuous Functionals
 About and Around Georg Kreisel
, 1996
"... this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are rel ..."
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Cited by 8 (3 self)
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this paper is to give complete proofs of the density theorem and the choice principle for total continuous functionals in the natural and concrete context of the partial continuous functionals [Ers77], essentially by specializing more general treatments in the literature. The proofs obtained are relatively short and hopefully perspicious, and may contribute to redirect attention to the fundamental questions Kreisel originally was interested in. Obviously this work owes much to other sources. In particular I have made use of work by Scott [Sco82] (whose notion of an information system is taken as a basis to introduce domains), Roscoe [Ros87], Larsen and Winskel [LW84] and Berger [Ber93]. The paper is organized as follows. Section 1 treats information systems, and in section 2 it is shown that the partial orders defined by them are exactly the (Scott) domains with countable basis. Section 3 gives a characterization of the continuous functions between domains, in terms of approximable mappings. In section 4 cartesian products and function spaces of domains and information systems are introduced. In section 5 the partial and total continuous functionals are defined. Section 6 finally contains the proofs of the two theorems above; it will be clear that the same proofs also yield effective versions of these theorems.
1996], Computation on abstract data types. The extensional approach, with an application to streams
 Annals of Pure and Applied Logic
"... In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definiti ..."
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In this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functionals of type level ≤ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type. 1
On sequential functionals of type 3
 Math. Structures Comput. Sci
, 2006
"... We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1 ..."
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We show that the extensional ordering of the sequential functionals of pure type 3, e.g. as defined via game semantics [2, 4], is not cpoenriched. This shows that this model does not equal Milner’s [9] fully abstract model for P CF. 1
The Unfolding of NonFinitist Arithmetic
, 2000
"... The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA ..."
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Cited by 5 (3 self)
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The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is prooftheoretically equivalent to predicative analysis.
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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Cited by 4 (2 self)
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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.