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Uniform proofs as a foundation for logic programming
 ANNALS OF PURE AND APPLIED LOGIC
, 1991
"... A prooftheoretic characterization of logical languages that form suitable bases for Prologlike programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its ..."
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Cited by 374 (108 self)
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A prooftheoretic characterization of logical languages that form suitable bases for Prologlike programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cutfree sequent proofs called uniform proofs. A uniform proof is one that can be found by a goaldirected search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that firstorder and higherorder Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that firstorder and higherorder versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to firstorder Horn clauses is briefly discussed.
Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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An Overview of λProlog
 In Fifth International Logic Programming Conference
, 1988
"... Abstract: λProlog is a logic programming language that extends Prolog by incorporating notions of higherorder functions, λterms, higherorder unification, polymorphic types, and mechanisms for building modules and secure abstract data types. These new features are provided in a principled fashion ..."
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Cited by 99 (34 self)
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Abstract: λProlog is a logic programming language that extends Prolog by incorporating notions of higherorder functions, λterms, higherorder unification, polymorphic types, and mechanisms for building modules and secure abstract data types. These new features are provided in a principled fashion by extending the classical firstorder theory of Horn clauses to the intuitionistic higherorder theory of hereditary Harrop formulas. The justification for considering this extension a satisfactory logic programming language is provided through the prooftheoretic notion of a uniform proof. The correspondence between each extension to Prolog and the new features in the stronger logical theory is discussed. Also discussed are various aspects of an experimental implementation of λProlog. Appears in the Fifth International Conference Symposium on Logic Programming, 15 – 19 August 1988, Seattle, Washington. This is a slightly corrected version of
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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Cited by 67 (0 self)
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
From operational semantics to abstract machines
 Mathematical Structures in Computer Science
, 1992
"... We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediatelevel specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctnesspreserving t ..."
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Cited by 59 (6 self)
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We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediatelevel specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctnesspreserving transformations to operational semantics until the resulting specifications have the form of abstract machines. Though not automatable in general, this approach to constructing machine implementations can be mechanized, providing machineverified correctness proofs. As examples we present the transformation of specifications for both callbyname and callbyvalue evaluation of the untyped λcalculus into abstract machines that implement such evaluation strategies. We also present extensions to the callbyvalue machine for a language containing constructs for recursion, conditionals, concrete data types, and builtin functions. In all cases, the correctness of the derived abstract machines follows from the (generally transparent) correctness of the initial operational semantic specification and the correctness of the transformations applied. 1.
Natural Deduction for Intuitionistic Linear Logic
, 1993
"... The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential !) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL + , is described in this paper. I ..."
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Cited by 34 (0 self)
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The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential !) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL + , is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL + , instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I + , which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph "Natural Deduction". It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL + ; in particular one can formulate a notion of strong validity (as in Prawitz's "Ideas and Results in Proof Theory") permitting a proof of strong normalization. T...
On an interpretation of second order quantification in first order intuitionistic propositional logic
 JOURNAL OF SYMBOLIC LOGIC, 57: 33 { 52
, 1992
"... We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a ..."
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Cited by 23 (0 self)
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We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula there exists a formula Ap (e ectively computable from), containing only variables not equal to p which occur in, and such that for all formulas not involving p, ` ! Ap if and only if ` !. Consequently quanti cation over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on rst order propositions. An immediate corollary is the strengthening of the usual Interpolation Theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC² can be constructed whose algebra of truthvalues is equal to any given Heyting algebra.
A Logic Program for Transforming Sequent Proofs to Natural Deduction Proofs
, 1991
"... In this paper, we show that an intuitionistic logic with secondorder function quantification, called hh 2 here, can serve as a metalanguage to directly and naturally specify both sequent calculi and natural deduction inference systems for firstorder logic. For the intuitionistic subset of first ..."
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Cited by 16 (3 self)
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In this paper, we show that an intuitionistic logic with secondorder function quantification, called hh 2 here, can serve as a metalanguage to directly and naturally specify both sequent calculi and natural deduction inference systems for firstorder logic. For the intuitionistic subset of firstorder logic, we present a set of hh 2 formulas which simultaneously specifies both kinds of inference systems and provides a direct and concise account of the correspondence between cutfree sequential proofs and normal natural deduction proofs. The logic of hh 2 can be implemented using such logic programming techniques as providing operational interpretations to the connectives and implementing unification on terms. With respect to such an interpreter, our specification provides a description of how to convert a proof in one system to a proof in the other. The operation of converting a sequent proof to a natural deduction proof is functional in the sense that there is always one na...
Encoding Modal Logics in Logical Frameworks
 Studia Logica
, 1997
"... We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce severa ..."
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Cited by 14 (8 self)
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We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed calculus metalanguage of the Logical Frameworks. These formalizations yield readily proofeditors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and NaturalDeduction proof systems for Modal Logics, Logical Frameworks, Typed calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...