Equational Formulas in Order-Sorted Algebras

We propose a set of transformation rules for first order formulas whose atoms are either equations between terms or “sort constraints” t e s where s is a regular tree language (or a sort in the algebraic specification community). This set of rules is proved to be correct, terminating and complete. This shows in particular that the first order theory of any rational tree language is decidable, extending the results of [Mal71,CL89,Mah88]. We also show how to apply our results to automatic inductive proofs in equational theories.

[1]  Peter D. Mosses,et al.  Unified algebras and institutions , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[2]  Kenneth Kunen,et al.  Negation in Logic Programming , 1987, J. Log. Program..

[3]  Hubert Comon,et al.  Inductive Proofs by Specification Transformation , 1989 .

[4]  Gert Smolka,et al.  Logic Programming with Polymorphically Order-Sorted Types , 1988, ALP.

[5]  Jean-Pierre Jouannaud,et al.  Proofs by induction in equational theories without constructors , 1985, Bull. EATCS.

[6]  Michael J. Maher,et al.  Unification Revisited , 1988, Foundations of Deductive Databases and Logic Programming..

[7]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[8]  José Meseguer,et al.  Principles of OBJ2 , 1985, POPL.

[9]  José Meseguer,et al.  Operational Semantics of OBJ-3 (Extended Abstract) , 1988, ICALP.

[10]  Claude Kirchner,et al.  Order-sorted equational unification , 1988 .

[11]  Michael J. Maher.  Complete axiomatizations of the algebras of finite, rational and infinite trees , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[12]  Hubert Comon-Lundh,et al.  Equational Problems and Disunification , 1989, J. Symb. Comput..

[13]  Alain Colmerauer,et al.  Equations and Inequations on Finite and Infinite Trees , 1984, FGCS.

[14]  J. Goguen,et al.  Order-Sorted Equational Computation , 1989 .

[15]  Hubert Comon.  Unification et disunification : théorie et applications , 1988 .

[16]  J. Meseguer,et al.  Order-Sorted Algebra Solves the Constructor-Selector, Multiple Representation, and Coercion Problems , 1993, Inf. Comput..

[17]  Jean-Pierre Jouannaud,et al.  Operational Semantics for Order-Sorted Algebra , 1985, ICALP.

[18]  David A. Plaisted,et al.  Semantic Confluence Tests and Completion Methods , 1985, Inf. Control..