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Computational foundations of basic recursive function theory
 Journal of Theoretical Computer Science
, 1993
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From Operational to Denotational Semantics
 In MFPS 1991
, 1989
"... In this paper it is shown how operational semantic methods may be naturally extended to encompass many of the concepts of denotational semantics. This work builds on the standard development of an operational semantics as an interpreter and operational equivalence. The key addition is an operational ..."
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Cited by 18 (6 self)
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In this paper it is shown how operational semantic methods may be naturally extended to encompass many of the concepts of denotational semantics. This work builds on the standard development of an operational semantics as an interpreter and operational equivalence. The key addition is an operational ordering on sets of terms. From properties of this ordering a closure construction directly yields a fully abstract continuous cpo model. Furthermore, it is not necessary to construct the cpo, for principles such as soundness of fixedpoint induction may be obtained by direct reasoning from this new ordering. The end result is that traditional denotational techniques may be applied in a purely operational setting in a natural fashion, a matter of practical importance for developing semantics of realistic programming languages. 1 Introduction This paper aims to accomplish a degree of unification between operational and denotational approaches to programming language semantics by recasting d...
Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ΒΈ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...
Extracting Recursive Programs in Type Theory
"... MartinLof's constructive type theory is a foundational theory of mathematics and programming. The key to using type theory as a logic is the formulas as types principle, whereby propositional assertions are directly expressed by types. Furthermore, using the extraction method programs can aut ..."
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MartinLof's constructive type theory is a foundational theory of mathematics and programming. The key to using type theory as a logic is the formulas as types principle, whereby propositional assertions are directly expressed by types. Furthermore, using the extraction method programs can automatically be extracted from proofs. One weakness of the use of the extraction method to date, however, is that it is impossible to extract arbitrary recursively defined programs from proofs, because all functions in type theory must be total. We show that under some extensions to type theory extraction of recursive programs is direct and useful. 1 Introduction We believe the motivations and directions of this work are best seen in the light of historical development, and will thus take a short historical digression into intuitionism, realizability, proofs as programs, and formulas as types. The most basic tenet of the philosophy of intuitionism, as set forth by Brouwer, is that a mathematical p...
Reflective Semantics of Constructive Type Theory
, 1991
"... It is wellknown that the proof theory of many sufficiently powerful logics may be represented internally by Godelization. Here we show that for Constructive Type Theory, it is furthermore possible to define a semantics of the types in the type theory itself, and that this procedure results in new r ..."
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It is wellknown that the proof theory of many sufficiently powerful logics may be represented internally by Godelization. Here we show that for Constructive Type Theory, it is furthermore possible to define a semantics of the types in the type theory itself, and that this procedure results in new reasoning principles for type theory. Paradoxes are avoided by stratifying the definition in layers. 1 Introduction Given a sufficiently powerful logical theory L such as Peano Arithmethic, it is wellknown that the proof theory of L may be expressed internally via Godelization. This is accomplished by defining a metafunction dAe that encodes formulas A as data, and a predicate Provable L (dAe) which is true just when formula A is provable. This gives L knowledge of its own proof theory, but it doesn't know that it knows it: the embedded proof theory could just as well be for some different logic. What is needed then are principles of selfknowledge that connect the provability predicate wit...