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Etude Des Spйcifications Modulaires: Constructions De Colimites Finies, Diagrammes, Isomorphismes

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Etude Des Spйcifications Modulaires: Constructions De Colimites Finies, Diagrammes, Isomorphismes Etude des spécifications modulaires : constructions de colimites finies, diagrammes, isomorphismes Catherine Oriat To cite this version: Catherine Oriat. Etude des spécifications modulaires : constructions de colimites finies, diagrammes, isomorphismes. Autre [cs.OH]. Institut National Polytechnique de Grenoble - INPG, 1996. Français. tel-00005007 HAL Id: tel-00005007 https://tel.archives-ouvertes.fr/tel-00005007 Submitted on 23 Feb 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ई THESE prउesentउee par Catherine ORIAT p our obtenir le grade de DOCTEUR de lINSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE Arretउe ministउeriel du mars ङ Spउecialitउe Informatique उ Etude des spउecications mo dulaires constructions de colimites nies diagrammes isomorphismes Thईese soutenue le janvier Comp osition du jury Prउesident M JeanPierre VERJUS Rapp orteurs M Michel BIDOIT M Christian LAIR Examinateurs M Michel SINTZOFF M Didier BERT Lab oratoire Logiciels Systईemes et Rउeseaux LSRIMAG Remerciements Je remercie tout dab ord mon directeur de thईese M Didier Bert p our son soutien et p our la conance quil ma accordउee dans la prउeparation de cette thईese Je remercie M JeanPierre Verjus qui a acceptउe de prउesider ce jury Je remercie M Michel Bidoit p our lintउeret quil a p ortउe ईa mon travail et p our avoir acceptउe detre rapp orteur de ma thईese Je remercie M Christian Lair p our les amउeliorations quil ma suggउerउees et p our les erreurs quil ma signalउees avec b eaucoup dindulgence et de gentillesse Je suis particuliईerement honorउee quun catउegoricien de sa compउetence ait acceptउe de faire partie de mon jury en qualitउe de rapp orteur Je remercie M Michel Sintzo p our mavoir fait lhonneur de particip er ईa ce jury Ce travail doit b eaucoup ईa M JeanClaude Reynaud Je le remercie dune part p our le temps quil a consacrउe ईa suivre mon travail et dautre part p our toutes les suggestions quil ma faites Si par hasard ce travail contient quelques idउees intउeressantes cest probablement lui qui me les a souउees mais en paraphrasant Mac Lane je suis उevidemment resp onsable des erreurs qui p ourraient rester Je remercie enn les membres de lउequipe SCOP auxquels jasso cie les o ccupants prउesents ou passउes du batiment D p our leur accueil et leur soutien tout au long de ce travail Le dउepartement:::उep ousait lopinion traditionnelle qui voulait que la recherche fut une o ccupation soli taire et उerउemitique une mise ईalउepreuve du caractईere plus que de lउerudition quun abus de contact humain p ouvait vicier David Lo dge La chute du British Museum Rउesumउe La comp osition de spउecications mo dulaires p eut etre mo dउelisउee dans le for malisme des catउegories par des colimites de diagrammes La somme amalgamउee p ermet en particulier dassembler deux spउecications en prउecisant les parties com munes Notre travail p oursuit cette idउee classique selon trois axes Dun p ointde vue syntaxique nous dउenissons un langage p our reprउesenter les spउecications mo dulaires construites ईa partir dune catउegorie de spउecications et de morphismes de spउecications de base Ce langage est caractउerisउe formellement par une catउegorie de termes niment co complईete Dun p oint de vue sउemantique nous prop osons dasso cier ईa tout terme un dia gramme Cette interprउetation p ermet de faire abstraction de certains choix eectuउes lors de la construction de la spउecication mo dulaire Pour cela nous dउenissons une catउegorie de diagrammes concrईete cestadire dont les ईeches p euventetre mani pulउees eectivement En considउerant le quotient par une certaine congruence nous obtenons une complउetion de la catउegorie de base par colimites nies Nous montrons que le calcul du diagramme asso ciउea un terme dउenit une उequivalence entre la catउe gorie des termes et la catउegorie des diagrammes ce qui prouve la correction de cette interprउetation Enn nous prop osons un algorithme p our dउecider si deux diagrammes sont iso morphes dans le cas particulier o ईu la catउegorie de base est nie et sans cycle Cela p ermet de dउetecter des isomorphismes de construction entre spउecications mo dulaires cestadire des isomorphismes qui ne dउependent pas des spउecications de base mais seulement de la maniईere dont cellesci sont assemblउees Abstract The comp osition of mo dular sp ecications can b e mo deled in a category theo retic framework by means of colimits of diagrams Pushouts in particular allowus to gather two sp ecications sharing a common part Our work extends this classic idea along three lines From a syntactic p oint of view we dene a language to represent mo dular sp eci cations built from a category of base sp ecications and base sp ecication morphisms This language is formally characterized by a nitely co complete category of terms From a semantic p oint of view we prop ose to asso ciate with each term a diagram This interpretation allows us to abstract some choices made while constructing a mo dular sp ecication We thus dene a concrete category of diagrams in which arrows can actually b e handled Considering the quotient by a certain congruence relation we get a completion of the base category with nite colimits We prove that this calculus denes an equivalence b etween the category of terms and the category of diagrams which shows the soundness of this interpretation At last we prop ose an algorithm to decide whether two diagrams are isomorphic when the base category is nite and cycle free This allows us to detect construc tion isomorphisms b etween mo dular sp ecications ie isomorphisms which do not dep end on the base sp ecications but only on their combination Table des matiईeres Rउesumउe Introduction Spउecications algउebriques Thउeorie des catउegories Mo dularitउe Notre travail Comparaisons avec dautres travaux Plan de ce mउemoire Motivation Prउeliminaires Spउecications Comp osition de spउecications en LPG Sommes amalgamउees Somme amalgamउee de deux ensembles Somme amalgamउee de deux signatures Somme amalgamउee de deux spउecications Syntaxe abstraite p our les constructions mo dulaires Somme disjointe de deux spउecications Spउecications mo dulaires danneaux Isomorphisme de construction Cadre gउenउeral Institutions Terminologie et hypothईeses Graphes catउegories et diagrammes Graphes Catउegories foncteurs ::: Catउegories Foncteurs Transformations naturelles Adjonctions Diagrammes Catउegorie des diagrammes DIAGRघC ङ Colimites ई TABLE DES MATIERES Exemples de colimites Adjonction Colim a I ङ C C Conservation de colimites Aplatissement Catउegorie diagrघC ङ La catउegorie diagrघC ङ est niment co complईete La catउegorie diagrघC ङ est une complउetion de C par colimites nies Conclusion Syntaxe catउegorie des termes Prउecatउegories prउefoncteurs et prउecolimites Problईeme de circularitउe termes congruences Prउecatउegories C i Simplication des ईeches Extensions conservatrices Prउecatउegorie TERMEC ङ 0 Catउegorie TermeC ङ 0 Catउegorie libre LघC ङ 0 Conclusion Sउemantique des termes aux diagrammes Sउemantique Prउecatउegorie DIAGRघC ङ 0 Prउefoncteur D TERMEC ङ DIAGRघC ङ 0 0 Exemple des anneaux Diagramme asso ciउealaspउecication A 2 Diagramme asso ciउealaspउecication A 2 Diagramme asso ciउealaspउecication A 3 Diagramme asso ciउealaspउecication A 4 उ Equivalence entre les catउegories TermeघC ङ et diagrघC ङ 0 0 उ Equivalence entre catउegories उequivalence entre prउecatउegories उ Equivalence entre les prउecatउegories DIAGRघC ङ et TERMEC ङ 0 0 Correction du calcul de diagrammes Conclusion Isomorphismes entre diagrammes Transformations de diagrammes Substitution par ob jet isomorphe Suppression dun arc Ajout dun arc उetiquetउe
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