Unification in many-sorted equational theories
Unification in many-sorted equational theories is of special interest in automated deduction. It combines the work done on unification in unsorted equational theories and unification in many-sorted signatures without equational theories. The sort structure considered in this paper may be an arbitrary finite partially ordered set, the equational theory may be arbitrary. Combining them, however, requires some natural restrictions, one of which is that equal terms have equal sorts.
Given a unification algorithm A for an unsorted equational theory the corresponding unification algorithm for the many-sorted equational theory is obtained by simply postprocessing the substitutions generated by A with a uniform algorithm to be presented in this paper.
The results obtained concern the decidability of unification as well as the existence and the cardinality of complete and minimal sets of unifiers. The most useful results are obtained for a combination of finitary matching theories with a polymorphic signature.
Key wordsUnification Many-Sorted Logics Heterogeneous Algebras Equational Theories
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- Bu86b.Buettner,W., "Unification in the datastructure set", (to appear in this Proc. of 8th CADE)Google Scholar
- CD83.Cunningham, R.J., Dick, A.J.J., "Rewrite Systems on a Lattice of Types", Rep. No. DOC 83/7, Imperial College, London SW7 (1983)Google Scholar
- Co83.Cohn, A.G. "Improving the Expressiveness of Many-sorted Logic.", AAAI-83, Washington (1983)Google Scholar
- Fa83.Fages, F. "Formes canonique dans les algèbres booleènnes et application à la dèmonstration automatique en logique de premier ordre." Thèse du 3eme cycle, Paris, (1983)Google Scholar
- GM84.Goguen, J.A., Meseguer, J., "Equality, Types, Modules and Generics for Logic Programming", Journal of Logic Programming, (1984)Google Scholar
- GM85.Goguen, J.A., Meseguer, J., "Order Sorted Algebra I. Partial and Overloaded Operators, Errors and Inheritance.", SRI Report (1985)Google Scholar
- Gr79.Grätzer, G., "Universal algebra", Springer Verlag, (1979)Google Scholar
- Hay71.Hayes, P., "A Logic of Actions.", Machine Intelligence 6, Metamathematics Unit, University of Edinburgh (1971)Google Scholar
- He83.Herold, A., "Some basic notions of first order unification theory.", Internal report 15/83, Univ. Karlsruhe, (1983)Google Scholar
- HS85.Herold, A.,Siekmann, J., "Unification in abelian semigroups", technical report, Universität Kaiserslautern, Memo SEKI-85-III, (1985)Google Scholar
- Hen72.Henschen, L.J., "N-Sorted Logic for Automated Theorem Proving in Higher-Order Logic.", Proc. ACM Conference Boston, (1972)Google Scholar
- Hu76.Huet, G., "Resolution d'equations dans des languages d'ordere 1,2,..., ω;";, These d'Etat, Univ. de Paris, VII, (1976)Google Scholar
- HO80.Huet, G., Oppen, D.C., "Equations and Rewrite Rules", SRI Technical Report CSL-111, (1980)Google Scholar
- Hl80.Hullot, J.M., "Canonical forms and unification," Proc of the 5 th workshop on automated deduction, Springer Lecture Notes, vol. 87, pp. 318–334, (1980)Google Scholar
- KM84.Karl Mark G. Raph, "The Markgraf Karl Refutation Procedure", Technical report, Univ. Kaiserslautern, Memo SEKI-84-03, (1984)Google Scholar
- Lo78.Loveland, D., "Automated Theorem Proving", North Holland, (1980)Google Scholar
- LS76.Livesey,M., Siekmann, J., "Unification of sets and multisets," Technical report, Unive Kaiserslautern, Memo-SEKI-76-01, (1976)Google Scholar
- Ob62.Oberschelp, A., "Untersuchungen zur mehrsortigen Quantorenlogik.", Mathematische Annalen 145 (1962)Google Scholar
- Pl72.Plotkin, G., "Building in equational theories.", Machine Intelligence, vol. 7, 1972Google Scholar
- Ro65.Robinson, J.A., "A Machine-Oriented Logic Based on the Resolution Principle." JACM 12 (1965)Google Scholar
- Sch85a.Schmidt-Schauss, M., "A many-sorted calculus with polymorphic functions based on resolution and paramodulation.", Proc 9 th IJCAI, Los Angeles,(1985)Google Scholar
- Sch85b.Schmidt-Schauss, M., "Unification in a many-sorted calculus with declarations.", Proc. of GWAI, (1985)Google Scholar
- Sch86.Schmidt-Schauss, M., "Unification in many-sorted equational theories.", Internal report. Institut für Informatik, Kaiserslautern (forthcoming)Google Scholar
- Si84.Siekmann, J. H., "Universal Unification", Proc. of the 7 th CADE, Napa California, (1984)Google Scholar
- Sz82.Szabo, P., "Theory of First order Unification.", (in German, thesis) Univ. Karlsruhe, 1982Google Scholar
- Wa83.Walther, C., "A Many-Sorted Calculus Based on Resolution and Paramodulation.", Proc. of the 8 th I JCAI, Karlsruhe, (1983)Google Scholar
- Wa84.Walther, C., "Unification in Many-Sorted Theories.", Proc. of the 6 th ECAI, PISA, (1984)Google Scholar