# Completion procedures as semidecision procedures

## Abstract

In this paper we give a new abstract framework for the study of Knuth-Bendix type completion procedures, which are regarded as *semidecision procedures* for theorem proving.

First, we extend the classical proof orderings approach started in [6] in such a way that proofs of different theorems can also be compared. This is necessary for the application of proof orderings to theorem proving derivations. We use proof orderings to uniformly define all the fundamental concepts in terms of *proof reduction*.

A completion procedure is given by a set of *inference rules* and a *search plan*. The inference rules determine what can be derived from given data. The search plan chooses at each step of the derivation which inference rule to apply to which data. Each inference step either reduces the proof of a theorem or deletes a *redundant* sentence. Our definition of *redundancy* is based on the assumed proof ordering. We have shown in [16] that our definition subsume those given in [50, 13].

We prove that if the inference rules are *refutationally complete* and the search plan is *fair*, a completion procedure is a semidecision procedure for theorem proving. The key part of this result is the notion of *fairness*. Our definition of fairness is the first definition of fairness for completion procedures which addresses the theorem proving problem. It is new in three ways: it is *target oriented*, that is it keeps the theorem to be proved into consideration, it is explicitly stated as a property of the search plan and it is defined in terms of proof reduction, so that expansion inferences and contraction inferences are treated uniformly. According to this definition of fairness, it is not necessary to consider all critical pairs in a derivation for the derivation to be fair. This is because not all critical pairs are necessary to prove a given theorem. Considering all critical pairs is an unnecessary source of inefficiency in a theorem proving derivation.

We also show that the process of diproving inductive theorems by the so called *inductionless induction* method is a semidecision process. Finally, we present according to our framework, some equational completion procedures based on Unfailing Knuth-Bendix completion.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]S.Anantharaman and J.Mzali, Unfailing Completion modulo a set of equations, Technical Report, LRI, Université de Paris Sud, 1989.Google Scholar
- [2]S. Anantharaman, J. Hsiang and J. Mzali, SbReve2: A Term Rewriting Laboratory with (AC)-Unfailing Completion, in N. Dershowitz (ed.),
*Proceedings of the Third International Conference on Rewriting Techniques and Applications*, Chapel Hill, NC, USA, April 1989, Springer Verlag, Lecture Notes in Computer Science 355, 533–537, 1989.Google Scholar - [3]S. Anantharaman, J. Hsiang, Automated Proofs of the Mougang Identities in Alternative Rings,
*Journal of Automated Reasoning*, Vol. 6, No. 1, 76–109, 1990.Google Scholar - [4]S.Anantharaman, N.Andrianarivelo, Heuristical Criteria in Refutational Theorem Proving, in
*Proceedings of the Symposium on the Design and Implementation of Systems for Symbolic Computation*, 184–193, Capri, Italy, April 1990.Google Scholar - [5]S.Anantharaman, N.Andrianarivelo, M.P.Bonacina, J.Hsiang, SBR3: A Refutational Prover for Equational Theorems, to appear in
*Proceedings of the Second International Workshop on Conditional and Typed Rewriting Systems*, Montreal, Canada, June 1990.Google Scholar - [6]L.Bachmair, N.Dershowitz, J.Hsiang, Orderings for Equational Proofs, in
*Proceedings of the First Annual IEEE Symposium on Logic in Computer Science*, 346–357, Cambridge, MA, June 1986.Google Scholar - [7]L. Bachmair, N. Dershowitz, Completion for rewriting modulo a congruence, in P.Lescanne (ed.),
*Proceedings of the Second International Conference on Rewriting Techniques and Applications*, Bordeaux, France, May 1987, Springer Verlag, Lecture Notes in Computer Science 256, 192–203, 1987.Google Scholar - [8]L.Bachmair, N.Dershowitz, Inference Rules for Rewrite-Based First-Order Theorem Proving, in
*Proceedings of the Second Annual Symposium on Logic in Computer Science*, Ithaca, New York, June 1987.Google Scholar - [9]L. Bachmair, Proofs Methods for Equational Theories, Ph.D. thesis, Department of Computer Science, University of Illinois, Urbana, IL.,1987.Google Scholar
- [10]L.Bachmair, Proof by consistency in equational theories, in
*Proceedings of the Third Annual IEEE Symposium on Logic in Computer Science*, 228–233, Edinburgh, Scotland, July 1988.Google Scholar - [11]L. Bachmair, N. Dershowitz and D.A. Plaisted, Completion without failure, in H. Ait-Kaci, M. Nivat (eds.),
*Resolution of Equations in Algebraic Structures*, Vol. II: Rewriting Techniques, 1–30, Academic Press, New York, 1989.Google Scholar - [12]L.Bachmair, H.Ganzinger, On Restrictions of Ordered Paramodulation with Simplification, in
*Proceedings of the Tenth International Conference on Automated Deduction*, Kaiserslautern, Germany, July 1990.Google Scholar - [13]L.Bachmair, H.Ganzinger, Completion of First-Order Clauses with Equality by Strict Superposition, to appear in M.Okada, S.Kaplan (eds.),
*Proceedings of the Second International Workshop on Conditional and Typed Rewriting Systems*, Montreal, Canada, June 1990.Google Scholar - [14]M.P. Bonacina, G. Sanna, K Blab: An Equational Theorem Prover for the Macintosh, in N.Dershowitz (ed.),
*Proceedings of the Third International Conference on Rewriting Techniques and Applications*, Chapel Hill, NC, USA, April 1989, Springer Verlag, Lecture Notes in Computer Science 355, 548–550, 1989.Google Scholar - [15]M.P.Bonacina, J.Hsiang, Operational and Denotational Semantics of Rewrite Programs, to appear in
*Proceedings of the North American Conference on Logic Programming*, Austin, TX, October 1990.Google Scholar - [16]M.P.Bonacina, J.Hsiang, On fairness of completion-based theorem proving strategies, Technical report, Department of Computer Science, SUNY at Stony Brook.Google Scholar
- [17]M.P.Bonacina, J.Hsiang, The Knuth-Bendix-Huet theorem and its extensions, in preparation.Google Scholar
- [18]C.L. Chang, R.C. Lee,
*Symbolic Logic and Mechanical Theorem Proving*, Academic Press, New York, 1973.Google Scholar - [19]N. Dershowitz, Z. Manna, Proving termination with multisets orderings,
*Communications of the ACM*, Vol. 22, No. 8, 465–476, August 1979.Google Scholar - [20]N.Dershowitz, N.A.Josephson, Logic Programming by Completion, in
*Proceedings of the Second International Conference on Logic Programming*, 313–320, Uppsala, Sweden, 1984.Google Scholar - [21]N. Dershowitz, Computing with Rewrite Systems,
*Information and Control*, Vol. 65, 122–157, 1985.Google Scholar - [22]N.Dershowitz, D.A.Plaisted, Logic Programming Cum Applicative Programming, in
*Proceedings of the IEEE Symposium on Logic Programming*, 54–66, Boston, MA, 1985.Google Scholar - [23]N. Dershowitz, Termination of Rewriting,
*Journal of Symbolic Computation*, Vol. 3, No. 1 & 2, 69–116, February/April 1987.Google Scholar - [24]N.Dershowitz, Completion and its Applications, in
*Proceedings of Conference on Resolution of Equations in Algebraic Structures*, Lakeway, Texas, May 1987.Google Scholar - [25]N.Dershowitz, J.-P.Jouannaud, Rewrite Systems, Technical Report 478, LRI, Université de Paris Sud, April 1989 and Chapter 15 of Volume B of
*Handbook of Theoretical Computer Science*, North-Holland, 1989.Google Scholar - [26]N.Dershowitz, J.-P.Jouannaud, Notations for Rewriting, Rapport de Recherche 478, LRI, Université de Paris Sud, January 1990.Google Scholar
- [27]N.Dershowitz, A Maximal-Literal Unit Strategy for Horn Clauses, to appear in M.Okada, S.Kaplan (eds.),
*Proceedings of the Second International Workshop on Conditional and Typed Rewriting Systems*, Montreal, Canada, June 1990.Google Scholar - [28]F.Fages, Associative-commutative unification, in R.Shostak (ed.),
*Proceedings of the Seventh International Conference on Automated Deduction*, Napa Valley, CA, USA, 1984, Springer Verlag, Lecture Notes in Computer Science 170, 1984.Google Scholar - [29]L.Fribourg, A Strong Restriction to the Inductive Completion Procedure, in
*Proceedins of the Thirteenth International Conference on Automata Languages and Programming*, Rennes, France, July 1986, Springer Verlag, Lecture Notes in Computer Science 226, 1986.Google Scholar - [30]J.A.Goguen, How to prove algebraic inductive hypotheses without induction, in W.Bibel and R.Kowalski (eds.),
*Proceedings of the Fifth International Conference on Automated Deduction*, 356–373, Les Arcs, France, 1980, Springer Verlag, Lecture Notes in Computer Science 87, 1980.Google Scholar - [31]J.Hsiang, N.Dershowitz, Rewrite Methods for Clausal and Nonclausal Theorem Proving, in
*Proceedings of the Tenth International Conference on Automata, Languages and Programming*, Barcelona, Spain, July 1983, Springer Verlag, Lecture Notes in Computer Science 154, 1983.Google Scholar - [32]J. Hsiang, Refutational Theorem Proving Using Term Rewriting Systems,
*Artificial Intelligence*, Vol. 25, 255–300, 1985.Google Scholar - [33]J. Hsiang, M. Rusinowitch, A New Method for Establishing Refutational Completeness in Theorem Proving, in J.Siekmann (ed.),
*Proceedings of the Eighth Conference on Automated Deduction*, Oxford, England, July 1986, Springer Verlag, Lecture Notes in Computer Science 230, 141–152, 1986.Google Scholar - [34]J. Hsiang, Rewrite Method for Theorem Proving in First Order Theories with Equality,
*Journal of Symbolic Computation*, Vol. 3, 133–151, 1987.Google Scholar - [35]J. Hsiang, M. Rusinowitch, On word problems in equational theories, in Th.Ottman (ed.),
*Proceedings of the Fourteenth International Conference on Automata, Languages and Programming*, Karlsruhe, West Germany, July 1987, Springer Verlag, Lecture Notes in Computer Science 267, 54–71, 1987.Google Scholar - [36]J.Hsiang, M.Rusinowitch and K. Sakai, Complete Inference Rules for the Cancellation Laws, in
*Proceedings of the Tenth International Joint Conference on Artificial Intelligence*, Milano, Italy, August 1987, 990–992, 1987.Google Scholar - [37]J.Hsiang, M.Rusinowitch, Proving Refutational Completeness of Theorem Proving Strategies: the Transfinite Semantic Tree Method, to appear in
*Journal of the ACM*, 1990.Google Scholar - [38]G. Huet, Confluent reductions: abstract properties and applications to term rewriting systems,
*Journal of the ACM*, Vol. 27, 797–821, 1980.Google Scholar - [39]G. Huet, A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm,
*Journal of Computer and System Sciences*, Vol. 23, 11–21, 1981.Google Scholar - [40]G. Huet, J.M. Hullot, Proofs by Induction in Equational Theories with Constructors, Journal of Computer and System Sciences, Vol. 25, 239–266, 1982.Google Scholar
- [41]J.-P. Jouannaud, C. Kirchner, Completion of a set of rules modulo a set of equations,
*SIAM Journal of Computing*, Vol. 15, 1155–1194, November 1986.Google Scholar - [42]J.-P.Jouannaud, E.Kounalis, Proofs by induction in equational theories without constructors, in
*Proceedings of the First Annual IEEE Symposium on Logic in Computer Science*, 358–366, Cambridge, MA, June 1986.Google Scholar - [43]J.-P.Jouannaud, E.Kounalis, Automatic proofs by induction in equational theories without constructors,
*Information and Computation*, 1989.Google Scholar - [44]J.-P.Jouannaud, C.Kirchner, Solving Equations in Abstract Algebras: A Rule-Based Survey of Unification, Rapport de Recherche, LRI, Université de Paris Sud, November 1989.Google Scholar
- [45]S. Kamin, J.-J. Lévy, Two generalizations of the recursive path ordering, Unpublished note, Department of Computer Science, University of Illinois, Urbana, Illinois, February 1980.Google Scholar
- [46]D.Kapur and P.Narendran, An equational approach to theorem proving in first order predicate calculus, in
*Proceedings of the Ninth International Joint Conference on Artificial Intelligence*, 1146–1153, Los Angeles, CA, August 1985.Google Scholar - [47]D. Kapur and D.R. Musser, Proof by consistency,
*Artificial Intelligence*, Vol. 31, No. 2, 125–157, February 1987.Google Scholar - [48]D. Kapur, P. Narendran and H. Zhang, Proof by induction using test sets, in J.Siekmann (ed.),
*Proceedings of the Eighth Conference on Automated Deduction*, Oxford, England, July 1986, Springer Verlag, Lecture Notes in Computer Science 230, 99–117, 1986.Google Scholar - [49]D.E. Knuth, P. Bendix, Simple Word Problems in Universal Algebras, in J. Leech (ed.),
*Proceedings of the Conference on Computational Problems in Abstract Algebras*, Oxford, England, 1967, Pergamon Press, Oxford, 263–298, 1970.Google Scholar - [50]E.Kounalis, M.Rusinowitch, On Word Problems in Horn Theories, in E.Lusk, R.Overbeek (eds.),
*Proceedings of the Ninth International Conference on Automated Deduction*, 527–537, Argonne, Illinois, May 1988, Springer Verlag, Lecture Notes in Computer Science 310, 1988.Google Scholar - [51]D.S. Lankford, Canonical inference, Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, May 1975.Google Scholar
- [52]D.S. Lankford, A simple explanation of inductionless induction, Technical report MTP-14, Mathematics Department, Louisiana Technical University, Ruston, LA, 1981.Google Scholar
- [53]D.Musser, On proving inductive properties of abstract data types, in
*Proceedings of the Seventh ACM Symposium on Principles of Programming Languages*, 154–162, Las Vegas, Nevada, 1980.Google Scholar - [54]G.E. Peterson, M.E. Stickel, Complete sets of reductions for some equational theories,
*Journal of the ACM*, Vol. 28, No. 2, 233–264, 1981.Google Scholar - [55]G.E. Peterson, A Technique for Establishing Completeness Results in Theorem proving with Equality,
*SIAM Journal of Computing*, Vol. 12, No. 1, 82–100, 1983.Google Scholar - [56]D.A. Plaisted, Semantic confluence tests and completion methods,
*Information and Control*, Vol. 65, 182–215, 1985.Google Scholar - [57]M.Rusinowitch, Theorem-proving with resolution and superposition: an extension of Knuth and Bendix procedure as a complete set of inference rules, Thèse d'Etat, Université de Nancy, 1987.Google Scholar
- [58]M.E.Stickel, Unification Algorithms for Artificial Intelligence Languages, Ph.D. thesis, Carnegie Mellon University 1976.Google Scholar
- [59]M.E. Stickel, A unification algorithm for associative-commutative functions,
*Journal of the ACM*, Vol. 28, No. 3, 423–434, 1981.Google Scholar - [60]H.Zhang, D.Kapur, First Order Theorem Proving Using Conditional Rewrite RUles, in E.Lusk, R.Overbeek (eds.),
*Proceedings of the Ninth International Conference on Automated Deduction*, 1–20, Argonne, Illinois, May 1988, Springer Verlag, Lecture Notes in Computer Science 310, 1988.Google Scholar