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Complete sets of unifiers and matchers in equational theories

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CAAP'83 (CAAP 1983)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 159))

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Abstract

We propose an abstract framework to present unification and matching problems. We argue about the necessity of a somewhat complicated definition of basis of unifiers (resp. matchers). In particular we prove the non-existence of complete sets of minimal unifiers (resp. matchers) in some equational theories, even regular.

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Bibliography

  • Baxter L.D. The Complexity of Unification. Ph. D. Thesis, University of Waterloo. (1977)

    Google Scholar 

  • Baxter L.D. The Undecidability of the Third Order Dyadic Unification Problem. Information and Control 38, p170–178, (1978)

    Article  Google Scholar 

  • Colmerauer A. Un système de communication homme-machine en Français. Rapport préliminaire, Groupe de recherche en Intelligence Artificielle, U.E.R. de Luminy, Univ. d'Aix-Marseille. (octobre 1972)

    Google Scholar 

  • Colmerauer A. Prolog II, manuel de référence et modèle théorique. Rapport interne, Groupe d'Intelligence Artificielle, Univ. d'Aix-Marseille II. (Mars 1982)

    Google Scholar 

  • Fay M. First-order Unification in an Equational Theory. 4th Workshop on Automated Deduction, Austin, Texas, pp. 161–167. (Feb. 1979)

    Google Scholar 

  • Goldfarb W. The Undecidability of the Second-Order Unification Problem. Theoritical Computer Science 13, pp 225–230. North Holland Publishing Company. (1981)

    Google Scholar 

  • Gould W.E. A matching Procedure for Omega Order Logic. Scientific Report 1, AFCRL 66–781, contract AF19 (628)–3250. (1966)

    Google Scholar 

  • Guard J.R. Automated Logic for Semi-Automated Mathematics, Scientific Report 1, AFCRL 64, 411, Contract AF19 (628)–3250. (1964)

    Google Scholar 

  • Guard J.R., Oglesby F.C., Bennett J.H. and Settle L.G. Semi-automated Mathematics. JACM 16, pp. 49–62. (1969)

    Article  Google Scholar 

  • Herbrand J. Recherches sur la théorie de la démonstration. Thèse, U. de Paris, In: Ecrits logiques de Jacques Herbrand, PUF Paris 1968. (1930)

    Google Scholar 

  • Hsiang J. Topics in Automated Theorem Proving and Program Generation. Ph. D. Thesis, Univ. of Illinois at Urbana-Champaign. (Nov. 1982)

    Google Scholar 

  • Huet G. The Undecidability of Unification in Third Order Logic. Information and Control 22, pp 257–267. (1973)

    Article  Google Scholar 

  • Huet G. Résolution d'équations dans des langages d'ordre 1, 2, ... omega. Thèse d'Etat, Univ. de Paris VII. (1976)

    Google Scholar 

  • Huet G. and Oppen D. Equations and Rewrite Rules: a Survey. In Formal Languages: Perspectives and Open Problems, Ed. Book R., Academic Press. (1980)

    Google Scholar 

  • Hullot J.M. Compilation de Formes Canoniques dans les Théories Equationnelles. Thèse de 3ème cycle. U. de Paris Sud. (Nov. 1980)

    Google Scholar 

  • Kirchner C. and Kirchner H. Contribution à la résolution d'équations dans les algèbres libres et les variétés équationnelles d'algèbres. Thèse de 3ème cycle, Univ. de Nancy. (Mars 1982)

    Google Scholar 

  • Knuth D. and Bendix P. Simple Word Problems in Universal Algebras. In Computational Problems in Abstract Algebra, Ed. Leech J., Pergamon Press, pp. 263–297. (1970)

    Google Scholar 

  • Lankford D.S. A Unification Algorithm for Abelian Group Theory. Report MTP-1, Math. Dept., Louisiana Tech. U. (Jan. 1979)

    Google Scholar 

  • Livesey M. and Siekmann J. Unification of bags and sets. Internal Report 3|76, Institut fur Informatik I, U. Karlsruhe, (1976)

    Google Scholar 

  • Makanin G.S. The Problem of Solvability of Equations in a Free Semigroup. Akad. Nauk. SSSR, TOM pp. 233, 2. (1977)

    Google Scholar 

  • Martelli A. and Montanari U. An Efficient Unification Algorithm. ACM T.O.P.L.A.S., Vol. 4, No. 2, pp 258–282. (April 1982)

    Google Scholar 

  • Paterson M.S. and Wegman M.N. Linear Unification. J. of Computer and Systems Sciences 16, pp. 158–167. (1978)

    Article  Google Scholar 

  • Peterson G.E. and Stickel M.E. Complete Sets of Reduction for Equational Theories with Complete Unification Algorithms. JACM 28, 2 pp 233–264. (1981)

    Article  Google Scholar 

  • Plotkin G. Building-in Equational Theories. Machine Intelligence 7, pp. 73–90. (1972)

    Google Scholar 

  • Robinson G.A. and Wos L.T. Paramodulation and Theorem Proving in First-order Theories with Equality. Machine Intelligence 4, American Elsevier, pp. 135–150. (1969)

    Google Scholar 

  • Robinson J.A. A Machine-Oriented Logic Based on the Resolution Principle. JACM 12, pp. 32–41. (1965)

    Article  Google Scholar 

  • Robinson J.A. Computational Logic: the Unification Computation. Machine Intelligence 6, Eds B. Meltzer and D. Michie American Elsevier, New-York. (1971)

    Google Scholar 

  • Rulifson J.F., Derksen J.A. and Waldinger R.J. QA4: a Procedural Calculus for Intuitive Reasoning. Technical Note 73, Artificial Intelligence Center, Stanford Research Institute, Menlo Park, California. (November 1972)

    Google Scholar 

  • Siekmann J. Unification and Matching Problems. Ph. D. Thesis, Univ. Karlsruhe. (March 1978)

    Google Scholar 

  • Siekmann J. and Szabo P. Universal Unification in Regular Equational ACFM Theories. CADE 6th, New-York. (June 1982)

    Google Scholar 

  • Slagle J.R. Automated Theorem-Proving for Theories with Simplifiers, Commutativity and Associativity JACM 21, pp. 622–642. (1974)

    Article  Google Scholar 

  • Stickel M.E. A Complete Unification Algorithm for Associative-Commutative Functions. JACM 28, 3 pp. 423–434. (1981)

    Article  Google Scholar 

  • Stickel M.E. A Complete Unification Algorithm for Associative-Commutative Functions. 4th International Joint Conference on Artificial Intelligence, Tbilisi. (1975)

    Google Scholar 

  • Stickel M.E. Unification Algorithms for Artificial Intelligence Languages. Ph. D. Thesis, Carnegie-Mellon University. (1976)

    Google Scholar 

  • Venturini-Zilli M. Complexity of the Unification Algorithm for First-Order Expressions. Calcolo XII, Fasc. IV, p 361–372. (1975)

    Google Scholar 

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Giorgio Ausiello Marco Protasi

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© 1983 Springer-Verlag Berlin Heidelberg

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Fages, F., Huet, G. (1983). Complete sets of unifiers and matchers in equational theories. In: Ausiello, G., Protasi, M. (eds) CAAP'83. CAAP 1983. Lecture Notes in Computer Science, vol 159. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12727-5_12

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  • DOI: https://doi.org/10.1007/3-540-12727-5_12

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  • Print ISBN: 978-3-540-12727-7

  • Online ISBN: 978-3-540-38714-5

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