Holls: an Intentional First-Order Expression of Higher-Order Logic
Identifieur interne : 002810 ( Hal/Corpus ); précédent : 002809; suivant : 002811Holls: an Intentional First-Order Expression of Higher-Order Logic
Auteurs : Gilles Dowek ; Thérèse Hardin ; Claude KirchnerSource :
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Abstract
We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on $\lambda$-calculus, i.e. a proposition can be proved without the extensionality axioms in one theory if and only if it can in the other. We show that we can apply the \em Extended Narrowing and Resolution first-order proof-search method to this theory. We get this way a step by step simulation of higher-order resolution. Hence expressing higher-order logic as a first-order theory and applying a first-order proof search method is at least as efficient as a direct implementation. Furthermore, the well studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover as we stay in a first-order setting, extensions, such as equational higher-order resolution, are easier to handle.
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This presentation is intentionally equivalent to the usual presentation of higher-order logic based on $\lambda$-calculus, i.e. a proposition can be proved without the extensionality axioms in one theory if and only if it can in the other. We show that we can apply the \em Extended Narrowing and Resolution first-order proof-search method to this theory. We get this way a step by step simulation of higher-order resolution. Hence expressing higher-order logic as a first-order theory and applying a first-order proof search method is at least as efficient as a direct implementation. Furthermore, the well studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. 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Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="INRIA-ROCQ">INRIA Paris - Rocquencourt</idno><idno type="stamp" n="INRIA-RRRT">Rapports de recherche et Technique de l'Inria</idno><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno><idno type="stamp" n="INPL">Institut National Polytechnique de Lorraine</idno><idno type="stamp" n="LORIA2">Publications du LORIA</idno><idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno><idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno><idno type="stamp" n="LORIA-FM" p="LORIA">Méthodes formelles</idno><idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno><idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno><idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno></seriesStmt><notesStmt><note type="commentary">Projet COQ, Projet PARA, PROTHEO</note><note type="audience" n="0">Not set</note><note type="report" n="6">Research Report</note></notesStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">Holls: an Intentional First-Order Expression of Higher-Order Logic</title><author role="aut"><persName><forename type="first">Gilles</forename><surname>Dowek</surname></persName><idno type="halAuthorId">62174</idno><affiliation ref="#struct-34587"></affiliation></author><author role="aut"><persName><forename type="first">Thérèse</forename><surname>Hardin</surname></persName><idno type="halAuthorId">99580</idno></author><author role="aut"><persName><forename type="first">Claude</forename><surname>Kirchner</surname></persName><idno type="idHal">claudekirchner</idno><idno type="halAuthorId">66450</idno><idno type="ORCID">http://orcid.org/0000-0002-1686-1046</idno><affiliation ref="#struct-2360"></affiliation></author></analytic><monogr><idno type="reportNumber">RR-3556</idno><imprint><biblScope unit="pp">26</biblScope><date type="datePub">1998</date></imprint><authority type="institution">INRIA</authority></monogr></biblStruct></sourceDesc><profileDesc><langUsage><language ident="en">English</language></langUsage><textClass><keywords scheme="author"><term xml:lang="en">higher-order logic</term><term xml:lang="en">automated theorem proving</term><term xml:lang="en">deduction modulo</term><term xml:lang="en">rewriting</term><term xml:lang="en">skolemization</term></keywords><classCode scheme="halDomain" n="info.info-oh">Computer Science [cs]/Other [cs.OH]</classCode><classCode scheme="halTypology" n="REPORT">Reports</classCode></textClass><abstract xml:lang="en">We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on $\lambda$-calculus, i.e. a proposition can be proved without the extensionality axioms in one theory if and only if it can in the other. We show that we can apply the \em Extended Narrowing and Resolution first-order proof-search method to this theory. We get this way a step by step simulation of higher-order resolution. Hence expressing higher-order logic as a first-order theory and applying a first-order proof search method is at least as efficient as a direct implementation. Furthermore, the well studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover as we stay in a first-order setting, extensions, such as equational higher-order resolution, are easier to handle.</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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