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Implementing Tactics and Tacticals in a HigherOrder Logic Programming Language
 Journal of Automated Reasoning
, 1993
"... We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement tactic style theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped terms ..."
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Cited by 68 (15 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 tactic style theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped terms, 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 inference 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 inference systems. Tactics and tacticals, which provide a framework for highlevel control over search for proofs, can be directly and naturally implemented in the extended language. This framework serves as a starting point for implementing theorem provers an...
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 52 (8 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
A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
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Cited by 36 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
A FineGrained Notation for Lambda Terms and Its Use in Intensional Operations
 Journal of Functional and Logic Programming
, 1996
"... We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to ..."
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Cited by 28 (9 self)
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We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to be performed on them and contains a mechanism for combining such substitutions so that they can be effected in a common structure traversal. The combination mechanism is a general one and consequently difficult to implement. We propose a simplification to it that retains its functionality in situations that occur commonly in fireduction. We then describe a system for annotating terms to determine if they can be affected by substitutions generated by external ficontractions. These annotations can lead to a conservation of space and time in implementations of reduction by permitting substitutions to be performed trivially in certain situations. The use of the resulting notation in the reduction...
Encoding Dependent Types in an Intuitionistic Logic
 LOGICAL FRAMEWORKS
, 1991
"... Various languages have been proposed as specification languages for representing a wide variety of logics. The development of typed calculi has been one approach toward this goal. The logical framework (LF), a calculus with dependent types is one example of such a language. A small subset of intui ..."
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Cited by 22 (6 self)
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Various languages have been proposed as specification languages for representing a wide variety of logics. The development of typed calculi has been one approach toward this goal. The logical framework (LF), a calculus with dependent types is one example of such a language. A small subset of intuitionistic logic with quantification over the simply typed calculus has also been proposed as a framework for specifying general logics. The logic of hereditary Harrop formulas with quantification at all nonpredicate types, denoted here as hh ! , is such a metalogic. In this paper, we show how to translate specifications in LF into hh ! specifications in a direct and natural way, so that correct typing in LF corresponds to intuitionistic provability in hh ! . In addition, we demonstrate a direct correspondence between proofs in these two systems. The logic of hh ! can be implemented using such logic programming techniques as providing operational interpretations to the connectives ...
Scoping Constructs In Logic Programming: Implementation Problems And Their Solution
, 1995
"... Machine (WAM). The provision of implications in goals results in the possibility of program clauses being added to the program for the purpose of solving specific subgoals. A naive scheme based on asserting and retracting program clauses does not suffice for implementing such additions for two reaso ..."
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Cited by 21 (9 self)
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Machine (WAM). The provision of implications in goals results in the possibility of program clauses being added to the program for the purpose of solving specific subgoals. A naive scheme based on asserting and retracting program clauses does not suffice for implementing such additions for two reasons. First, it is necessary to also support the resurrection of an earlier existing program in the face of backtracking. Second, the possibility for implication goals to be surrounded by quantifiers requires a consideration of the parameterization of program clauses by bindings for their free variables. Devices for supporting these additional requirements are described as also is the integration of these devices into the WAM. Further extensions to the machine are outlined for handling higherorder additions to the language. The ideas Work on this paper has been partially supported by NSF Grants CCR8905825 and CCR 9208465. Address correspondence to Gopalan Nadathur, Department of Compute...
A Logic Programming Approach to Implementing HigherOrder Term Rewriting
 Second International Workshop on Extensions to Logic Programming, volume 596 of Lecture Notes in Arti Intelligence
, 1992
"... Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higherorder term rewriting, which includes the usual notion of firstorder rewriting, can be naturally specified and implemented in a higherorder logic pr ..."
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Cited by 20 (4 self)
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Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higherorder term rewriting, which includes the usual notion of firstorder rewriting, can be naturally specified and implemented in a higherorder logic programming language. We adopt a notion of higherorder rewrite system which uses the simply typed calculus as the language for expressing rules, with a restriction on the occurrences of free variables on the left hand sides of rules so that matching of terms with rewrite templates is decidable. The logic programming language contains an implementation of the simplytyped lambda calculus including fij conversion and higherorder unification. In addition, universal quantification in queries and the bodies of clauses is permitted. For higherorder rewriting, we show how these operations implemented at the metalevel provide elegant mechanisms for the objectlevel operations of descending thro...
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 18 (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...
A Notation for Lambda Terms II: Refinements and Applications
, 1994
"... Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconver ..."
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Cited by 12 (2 self)
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Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconversion in the comparison of terms by using the `nameless' scheme of de Bruijn and also permits a delaying of substitutions by including a class of terms that encode other terms together with substitutions to be performed on them. The suspension notation contains a mechanism for `merging' substitutions so that they can be effected in a common structure traversal. The mechanism is cumbersome to implement in its full generality and a simplification to it is considered. In particular, the old merging operations are eliminated in favor of new ones that capture some of their functionality and that permit a simplified syntax for terms. The resulting notation is refined by the addition of annotations ...
Implementing a Notion of Modules in the Logic Programming Language Prolog
 In Evelina Lamma and Paola Mello, editors, Extensions of Logic Programming: Proceedings of the Third International Workshop
, 1993
"... Issues concerning the implementation of a notion of modules in the higherorder logic programming language Prolog are examined. A program in this language is a composite of type declarations and procedure definitions. The module construct that is considered permits large collections of such declarat ..."
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Cited by 10 (1 self)
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Issues concerning the implementation of a notion of modules in the higherorder logic programming language Prolog are examined. A program in this language is a composite of type declarations and procedure definitions. The module construct that is considered permits large collections of such declarations and definitions to be decomposed into smaller units. Mechanisms are provided for controlling the interaction of these units and for restricting the visibility of names used within any unit. The typical interaction between modules has both a static and a dynamic nature. The parsing of expressions in a module might require declarations in a module that it interacts with, and this information must be available during compilation. Procedure definitions within a module might utilize procedures presented in other modules and support must be provided for making the appropriate invocation during execution. Our concern here is largely with the dynamic aspects of module interaction. We describe a...