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HiLog: A foundation for higherorder logic programming
 JOURNAL OF LOGIC PROGRAMMING
, 1993
"... We describe a novel logic, called HiLog, and show that it provides a more suitable basis for logic programming than does traditional predicate logic. HiLog has a higherorder syntax and allows arbitrary terms to appear in places where predicates, functions and atomic formulas occur in predicate calc ..."
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Cited by 213 (40 self)
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We describe a novel logic, called HiLog, and show that it provides a more suitable basis for logic programming than does traditional predicate logic. HiLog has a higherorder syntax and allows arbitrary terms to appear in places where predicates, functions and atomic formulas occur in predicate calculus. But its semantics is firstorder and admits a sound and complete proof procedure. Applications of HiLog are discussed, including DCG grammars, higherorder and modular logic programming, and deductive databases.
Inheritance As Implicit Coercion
 Information and Computation
, 1991
"... . We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. ..."
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Cited by 116 (3 self)
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. We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can typecheck in more than one way. Since interpretations follow the type...
Specifying and Implementing Theorem Provers in a HigherOrder Logic Programming Language
, 1989
"... We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped λterms, replacing ..."
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Cited by 46 (7 self)
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We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped λterms, replacing firstorder unification with higherorder unification, and allowing implication and universal quantification in queries and the bodies of clauses. Inference rules for a variety of proof systems can be naturally specified in this language. The higherorder features of the language contribute to a concise specification of provisos concerning variable occurrences in formulas and the discharge of assumptions present in many proof systems. In addition, abstraction in metaterms allows the construction of terms representing object level proofs which capture the notions of abstractions found in many proof systems. The operational interpretations of the connectives of the language provide a set of basic search operations which describe goaldirected search for proofs. To emphasize the generality of the metalanguage, we compare it to another general specification language: the Logical Framework (LF). We describe a translation which compiles a specification of a logic in LF to a set of formulas of our metalanguage, and
Le Fun: Logic, equations, and Functions
 In Proc. 4th IEEE Internat. Symposium on Logic Programming
, 1987
"... Abstract † We introduce a new paradigm for the integration of functional and logic programming. Unlike most current research, our approach is not based on extending unification to generalpurpose equation solving. Rather, we propose a computation delaying mechanism called residuation. This allows a ..."
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Cited by 44 (1 self)
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Abstract † We introduce a new paradigm for the integration of functional and logic programming. Unlike most current research, our approach is not based on extending unification to generalpurpose equation solving. Rather, we propose a computation delaying mechanism called residuation. This allows a clear distinction between functional evaluation and logical deduction. The former is based on the λcalculus, and the latter on Horn clause resolution. In clear contrast with equationsolving approaches, our model supports higherorder function evaluation and efficient compilation of both functional and logic programming expressions, without being plagued by nondeterministic termrewriting. In addition, residuation lends itself naturally to process synchronization and constrained search. Besides unification (equations), other residuations may be any grounddecidable goal, such as mutual exclusion (inequations), and comparisons (inequalities). We describe an implementation of the residuation paradigm as a prototype language called Le Fun—Logic, equations, and Functions.
KripkeStyle Models for Typed Lambda Calculus
 Annals of Pure and Applied Logic
, 1996
"... The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke ..."
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Cited by 44 (3 self)
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The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately \semantic." However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, wellpointedness, or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations ...
A Proof of the ChurchRosser Theorem and its Representation in a Logical Framework
, 1992
"... We give a detailed, informal proof of the ChurchRosser property for the untyped lambdacalculus and show its representation in LF. The proof is due to Tait and MartinLöf and is based on the notion of parallel reduction. The representation employs higherorder abstract syntax and the judgmentsast ..."
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Cited by 36 (8 self)
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We give a detailed, informal proof of the ChurchRosser property for the untyped lambdacalculus and show its representation in LF. The proof is due to Tait and MartinLöf and is based on the notion of parallel reduction. The representation employs higherorder abstract syntax and the judgmentsastypes principle and takes advantage of term reconstruction as it is provided in the Elf implementation of LF. Proofs of metatheorems are represented as higherlevel judgments which relate sequences of reductions and conversions.
Topological Incompleteness and Order Incompleteness of the Lambda Calculus
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2001
"... A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In th ..."
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Cited by 23 (15 self)
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A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by BastoneroGouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is nontrivial and metrizable.
On The Algebraic Models Of Lambda Calculus
 Theoretical Computer Science
, 1997
"... . The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory ..."
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Cited by 20 (11 self)
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. The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a firstorder algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus...
A Continuum of Theories of Lambda Calculus Without Semantics
 16TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2001), IEEE COMPUTER
, 2001
"... In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we ..."
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Cited by 16 (11 self)
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In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by BastoneroGouy [6, 7] and by Berline [10], that the strongly stable semantics is incomplete. 1
Some Results on the Full Abstraction Problem for Restricted Lambda Calculi
 In Proc. of the 18 th International Symposium on Mathematical Foundations of Computer Science, MFCS'93, Springer LNCS
, 1993
"... ion Problem for Restricted Lambda Calculi Furio Honsell and Marina Lenisa Dipartimento di Matematica e Informatica, Universit`a di Udine, via Zanon,6  Italy Email:fhonsell,lenisag@udmi5400.cineca.it dedicated to Corrado Bohm on the occasion of his 70 th birthday Abstract. Issues in the mathemat ..."
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Cited by 14 (1 self)
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ion Problem for Restricted Lambda Calculi Furio Honsell and Marina Lenisa Dipartimento di Matematica e Informatica, Universit`a di Udine, via Zanon,6  Italy Email:fhonsell,lenisag@udmi5400.cineca.it dedicated to Corrado Bohm on the occasion of his 70 th birthday Abstract. Issues in the mathematical semantics of two restrictions of the calculus, i.e. Icalculus and V calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scottcontinuous functions. A correct, albeit nonfully abstract, model for the SECD evaluation of the latter is defined using Girard's coherence spaces and stable functions. These results are used to illustrate the interest of the analysis of the fine structure of mathematical models of programming languages. 1 Introduction D. Scott, in the late sixties, discovered a truly mathematical semantics for  calculus using continuous lattices. His D1 model was the first example of ...