Solving Divergence in Knuth-Bendix Completion by Enriching Signatures

Abstract The Knuth-Bendix completion algorithm is a procedure which generates confluent and terminating sets of rewrite rules. The algorithm has many applications: the resulting rules can be used as a decision procedure for equality or, in the case of program synthesis, as a program. We present an effective algorithm to solve some cases of divergence in the Knuth-Bendix completion algorithm, starting from a grammar characterising the infinite rule set. We replace an infinite set of rewrite rules by a finite complete set by enriching the original (order-sorted) signature with new sorts and new operator arities, while remaining within a conservative extension of the original system and within the original term set. The complexity of the new rewriting system is no worse than that of the original system. We characterise the class of examples for which this approach is applicable and give some sufficient conditions for the algorithm to succeed.

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

[2]  Steffen Lange,et al.  Inductive Completion for Transformation of Equational Specifications , 1990, ADT.

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

[4]  A. J. J. Dick,et al.  Order-Sorted Term Rewriting , 1991, Comput. J..

[5]  E. Mark Gold,et al.  Language Identification in the Limit , 1967, Inf. Control..

[6]  Takashi Yokomori.  Learning Context-Free Languages Efficiently - A Report on Recent Results in Japan , 1989, AII.

[7]  Hélène Kirchner,et al.  Meta-rule Synthesis from Crossed Rewrite Systems , 1990, CTRS.

[8]  Nachum Dershowitz,et al.  Synthesis by Completion , 1985, IJCAI.

[9]  Manfred Schmidt-Schauß,et al.  Computational Aspects of an Order-Sorted Logic with Term Declarations , 1989, Lecture Notes in Computer Science.

[10]  Mark A Ardis.  Data Abstraction Transformations. , 1980 .

[11]  Gregory Kucherov.  On Relationship Between Term Rewriting Systems and Regular Tree Languages , 1991, RTA.

[12]  Hélène Kirchner,et al.  Schematization of Infinite Sets of Rewrite Rules Generated by Divergent Completion Processes , 1989, Theor. Comput. Sci..

[13]  Claude Kirchner,et al.  Implementations of Term Rewriting Systems , 1991, Comput. J..

[14]  Carl H. Smith,et al.  Inductive Inference: Theory and Methods , 1983, CSUR.

[15]  Hubert Comon.  Completion of Rewrite Systems with Membership Constraints , 1992 .

[16]  M. Schmidt-Schauβ.  Computational Aspects of an Order-Sorted Logic with Term Declarations , 1989 .

[17]  Carron Kirkwood,et al.  Some Experiments Using Term Rewriting Techniques for Concurrency , 1990, FORTE.

[18]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[19]  Yoshihito Toyama.  Membership conditional term rewriting systems , 1989 .

[20]  Miki Hermann.  Chain Properties of Rule Closures , 1989, STACS.

[21]  Nachum Dershowitz,et al.  Inductive Synthesis of Equational Programs , 1990, AAAI.

[22]  Muffy Calder,et al.  Inductive Inference for Solving Divergence in Knuth-Bendix Completion , 1989, AII.

[23]  Andrea Sattler-Klein,et al.  Divergence Phenomena during Completion , 1991, RTA.

[24]  Stef fen Lange,et al.  Towards a Set of Inference Rules for Solving Divergence in Knuth-Bendix Completion , 1989, AII.

[25]  Hong Chen,et al.  On Finite Representations of Infinite Sequences of Terms , 1990, CTRS.