# Reducing the complexity of the Knuth-Bendix completion algorithm: A "unification" of different approaches

## Abstract

The Knuth-Bendix completion procedure for rewrite rule systems is of wide applicability in symbolic and algebraic computation. Attempts to reduce the complexity of this completion algorithm are reported in the literature. Already in their seminal 1967 paper D.E. Knuth and P.B. Bendix have suggested to keep all the rules interreduced during the execution of the algorithm. G. Huet has presented a version of the completion algorithm in which every rewrite rule is kept in reduced form with respect to all the other rules in the system. Borrowing an idea of Buchberger's for the completion of bases of polynomial ideals the author has proposed in 1983 a criterion for detecting "unnecessary" critical pairs. If a critical pair is recognized as unnecessary then one need not apply the costly process of computing normal forms to it. It has been unclear whether these approaches can be combined. We demonstrate that it is possible to keep all the rewrite rules interreduced and still use a criterion for eliminating unnecessary critical pairs.

## Keywords

Normal Form Induction Hypothesis Theorem Prove Polynomial Ideal Critical Pair## Preview

Unable to display preview. Download preview PDF.

## References

- [Bu65]B. Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, doctoral dissertation, Univ. Innsbruck, 1965.Google Scholar
- [Bu70]B. Buchberger: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4/3, 374–383, 1970.CrossRefGoogle Scholar
- [Bu79]B. Buchberger: A Criterion for Detecting Unnecessary Reductions in the Construction of Gröbner Bases, Proc. EUROSAM 79, Marseille, 1979, Lecture Notes in Computer Science 72, 3–21, Berlin-Heidelberg-New York, Springer-Verlag, 1979.Google Scholar
- [Bu85]B. Buchberger: Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, in: Recent Trends in Multidimensional Systems Theory, N.K. Bose, ed., D. Reidel Publishing Comp., to appear in 1985. Available also as CAMP-Report Nr. 83-29.0, Inst. of Math., Univ. Linz, 1983.Google Scholar
- [BC83]B. Buchberger, G.E. Collins, R. Loos: Computer Algebra — Symbolic and Algebraic Computation, 2nd ed., Springer-Verlag, 1983.Google Scholar
- [De83]N. Dershowitz: Applications of the Knuth-Bendix Completion Procedure, Laboratory Operations, The Aerospace Corp., El Segundo, Calif. 90245, Aerospace Rep. ATR-83(8478)-2, 1983.Google Scholar
- [Hs82]J. Hsiang: Topics in Automated Theorem Proving and Program Generation, Ph.D. thesis, Dept. of Comp. Sci., Univ. of Illinois, Urbana, Ill., 1982.Google Scholar
- [Hu80]G. Huet: Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems, JACM 27/4, 797–821, 1980.CrossRefGoogle Scholar
- [Hu81]G. Huet: A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm, J. Computer and System Sciences 23, 11–21, 1981.CrossRefGoogle Scholar
- [HH80]G. Huet, J.M. Hullot: Proofs by Induction in Equational Theories with Constructors, 21st IEEE Symp. on Foundations of Comp. Sci., 96–107, 1980.Google Scholar
- [KB67]D.E. Knuth, P.B. Bendix: Simple Word Problems in Universal Algebra, Proc. of the Conf. on Computational Problems in Abstract Algebra, Oxford, 1967, J. Leech (ed.), Pergamon Press, 1970.Google Scholar
- [Mu80]D.R. Musser: On Proving Inductive Properties of Abstract Data Types, 7th ACM Symp. on Principles of Progr. Languages, 154–162, 1980.Google Scholar
- [Ne42]M.H.A. Newman: On Theories with a Combinatorial Definition of "Equivalence", Ann. Math. 43/2, 223–243, 1942.Google Scholar
- [Wi83]F. Winkler: A Criterion for Eliminating Unnecessary Reductions in the Knuth-Bendix Algorithm, Tech. Rep. CAMP-83-14.1, Inst. f. Math., J. Kepler Univ., Linz, 1983.Google Scholar
- [Wi84]F. Winkler: The Church-Rosser Property in Computer Algebra and Special Theorem Proving: An Investigation of Critical-Pair/Completion Algorithms, doctoral dissertation, Inst. f. Math., J. Kepler Univ., Linz, 1984.Google Scholar
- [Wi85]F. Winkler: A Note on Improving the Complexity of the Knuth-Bendix Algorithm, Tech. Rep. No. 85-04, Dept. of Comp. and Inf. Sci., Univ. of Delaware, 1985.Google Scholar
- [WB83]F. Winkler, B. Buchberger: A Criterion for Eliminating Unnecessary Reductions in the Knuth-Bendix Algorithm, Proc. Colloquium on Algebra, Combinatorics and Logic in Computer Science, Györ, Hungary, 1983, Colloquia Mathematica Societatis J. Bolyai, J. Bolyai Math. Soc. and North-Holland, 1985.Google Scholar