Dynamically-typed computations for order-sorted equational presentations
Identifieur interne : 00D664 ( Main/Merge ); précédent : 00D663; suivant : 00D665Dynamically-typed computations for order-sorted equational presentations
Auteurs : Claus Hintermeier [France] ; Hélène Kirchner [France] ; Claude Kirchner [France]Source :
Abstract
Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt the G-algebra framework, where equational, membership and existence formulas can be expressed, and that provides a complete deduction calculus which incorporates the interaction between all these formulas. To practically deal with this calculus, we introduce an operational semantics for G-algebra using rewrite systems over so-called decorated terms, that have assertions concerning the sort membership of any subterm in its head node. Decorated rewrite rules perform equational replacement, decoration rewrite rules enrich the decorations and record sort information. Therefore we use the semantic sort principle, i.e. equal terms belong to equal sorts, rather than the syntactic sort principle that does not use the equational part of a presentation. In order to have a complete and decidable unification on decorated terms, we restrict to sort inheriting theories. The sort inheritance property is undecidable in general but we provide a test to check it on a given presentation. The test provides information on how to extend the presentation in a model conservative way, in order to obtain sort inheritance. then a completion procedure on decorated terms is designed to compute all interactions between equational and membership formulas. When the completion terminates, the resulting set of rewrite rules provides a way to decide equational theorems of the form (t = t') and typing theorems of the form (t : A).
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cedex</addrLine><country key="FR"></country></address></desc></org></tutelle><tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName><orgName type="acronym">INPL</orgName><date type="end">2011-12-31</date><desc><address><country key="FR"></country></address></desc></org></tutelle><tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName><desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle></tutelles></hal:affiliation><country>France</country><placeName><settlement type="city">Nancy</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Université Nancy 2</orgName><orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName><placeName><settlement type="city">Nancy</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Institut national polytechnique de Lorraine</orgName><orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName></affiliation></author><author><name sortKey="Kirchner, Claude" sort="Kirchner, Claude" uniqKey="Kirchner C" first="Claude" last="Kirchner">Claude Kirchner</name><affiliation wicri:level="1"><hal:affiliation type="researchteam" xml:id="struct-2360" status="OLD"><idno type="RNSR">199221357D</idno><orgName>Constraints, automatic deduction and software properties proofs</orgName><orgName type="acronym">PROTHEO</orgName><desc><address><country key="FR"></country></address><ref type="url">http://www.inria.fr/equipes/protheo</ref></desc><listRelation><relation active="#struct-160" type="direct"></relation><relation name="UMR7503" active="#struct-441569" type="indirect"></relation><relation active="#struct-300009" type="indirect"></relation><relation active="#struct-300291" type="indirect"></relation><relation active="#struct-300292" type="indirect"></relation><relation active="#struct-300293" type="indirect"></relation><relation active="#struct-2496" type="direct"></relation></listRelation><tutelles><tutelle active="#struct-160" type="direct"><org type="laboratory" xml:id="struct-160" status="OLD"><orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName><orgName type="acronym">LORIA</orgName><desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine><country key="FR"></country></address><ref 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Recherche en Informatique et en Automatique</orgName><orgName type="acronym">Inria</orgName><desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/en/</ref></desc></org></tutelle><tutelle active="#struct-300291" type="indirect"><org type="institution" xml:id="struct-300291" status="OLD"><orgName>Université Henri Poincaré - Nancy 1</orgName><orgName type="acronym">UHP</orgName><date type="end">2011-12-31</date><desc><address><addrLine>24-30 rue Lionnois, BP 60120, 54 003 NANCY cedex, France</addrLine><country key="FR"></country></address></desc></org></tutelle><tutelle active="#struct-300292" type="indirect"><org type="institution" xml:id="struct-300292" status="OLD"><orgName>Université Nancy 2</orgName><date type="end">2011-12-31</date><desc><address><addrLine>91 avenue de la Libération, BP 454, 54001 Nancy cedex</addrLine><country key="FR"></country></address></desc></org></tutelle><tutelle active="#struct-300293" type="indirect"><org type="institution" xml:id="struct-300293" status="OLD"><orgName>Institut National Polytechnique de Lorraine</orgName><orgName type="acronym">INPL</orgName><date type="end">2011-12-31</date><desc><address><country key="FR"></country></address></desc></org></tutelle><tutelle active="#struct-2496" type="direct"><org type="laboratory" xml:id="struct-2496" status="OLD"><orgName>INRIA Lorraine</orgName><desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine><country key="FR"></country></address><ref type="url">http://www.inria.fr/centre-de-recherche-inria/nancy-grand-est</ref></desc><listRelation><relation active="#struct-300009" type="direct"></relation></listRelation></org></tutelle></tutelles></hal:affiliation><country>France</country><placeName><settlement type="city">Nancy</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Université Nancy 2</orgName><orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName><placeName><settlement type="city">Nancy</settlement><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region></placeName><orgName type="university">Institut national polytechnique de Lorraine</orgName><orgName type="institution" wicri:auto="newGroup">Université de Lorraine</orgName></affiliation></author></analytic></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt the G-algebra framework, where equational, membership and existence formulas can be expressed, and that provides a complete deduction calculus which incorporates the interaction between all these formulas. To practically deal with this calculus, we introduce an operational semantics for G-algebra using rewrite systems over so-called decorated terms, that have assertions concerning the sort membership of any subterm in its head node. Decorated rewrite rules perform equational replacement, decoration rewrite rules enrich the decorations and record sort information. Therefore we use the semantic sort principle, i.e. equal terms belong to equal sorts, rather than the syntactic sort principle that does not use the equational part of a presentation. In order to have a complete and decidable unification on decorated terms, we restrict to sort inheriting theories. The sort inheritance property is undecidable in general but we provide a test to check it on a given presentation. The test provides information on how to extend the presentation in a model conservative way, in order to obtain sort inheritance. then a completion procedure on decorated terms is designed to compute all interactions between equational and membership formulas. When the completion terminates, the resulting set of rewrite rules provides a way to decide equational theorems of the form (t = t') and typing theorems of the form (t : A).</div></front></TEI></record>Pour manipuler ce document sous Unix (Dilib)
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