Complete Sets of Transformations for General E-Unification

This paper is concerned with E-unification in arbitrary equational theories. We extend the method of transformations on systems of terms, developed by Martelli-Montanari for standard unification, to Eunification by giving two sets of transformations, BT and T, which are proved to be sound and complete in the sense that a complete set of E-unifiers for any equational theory E can be enumerated by either of these sets. The set T is an improvement of BT, in that many E-unifiers produced by BT will be weeded out by T. In addition, we show that a generalization of surreduction (also called narrowing) combined with the computation of critical pairs is complete. A new representation of equational proofs as certain kinds of trees is used to prove the completeness of the set BT in a rather direct fashion that parallels the completeness of the transformations in the case of (standard) unification. The completeness of T and the generalization of surreduction is proved by a method inspired by the concept of unfailing completion, using an abstract (and simpler) notion of the completion of a set of equations. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-89-12. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/779 COMPLETE SETS OF TRANSFORMATIONS FOR GENERAL E-UNIFICATION Jean H. Gallier and Wayne Snyder MS-CIS-89-12 LOGIC & COMPUTATION 03 Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania Philadelphia, PA 19104 (Revised February 1989) To appear In special Issue of "Theoretical Computer Science" (1988) Acknowledgements: This research was supported in part by NSF grants DCR 86-07156, MCS-8219196-CER, IR184-10413-A02 and U.S. Army grants DAA29-84-K-0061, DAA29-84-9-0027. COMPLETE SETS OF TRANSFORMATIONS FOR GENERAL E-UNIFICATION Jean H. Gallier and Wayne Snyder Department of Computer and Information Science University of Pennsylvania Philadelphia, Pa 19104