# An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras

## Abstract

A new approach based on computing the Gröbner basis of polynomial ideals is developed for solving word problems and unification problems for finitely presented commutative algebras. This approach is simpler and more efficient than the approaches based on generalizations of the Knuth-Bendix completion procedure to handle associative and commutative operators. It is shown that (i) the word problem over a finitely presented commutative ring with unity is equivalent to the polynomial equivalence problem modulo a polynomial ideal over the integers, (ii) the unification problem for linear forms is decidable for finitely presented commutative rings with unity, (iii) the word problem and unification problem for finitely presented boolean polynomial rings are co-NP-complete and co-NP-hard respectively, and (iv) the set of all unifiers of two forms over a finitely presented abelian group can be computed in polynomial time. Examples and results of algorithms based on the Gröbner basis computation are also reported.

## Key Words

Word Problem Unification Problem Finitely Presented Algebras Commutative Algebras Gröbner Basis Polynomial Ideals Term Rewriting Knuth-Bendix Completion Procedure## Preview

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## 7. References

- [1]Angarsson, S., Kandri-Rody, A., Kapur, D., Narendran, P., and Saunders, B.D., “The Complexity of Testing whether a Polynomial Ideal is Nontrivial,” Proceedings of the
*Third MACSYMA User's Conference*, Schenectady, NY, July, 1984, pp. 452–458.Google Scholar - [2]Ballantyne, A.M., and Lankford, D.S., “New Decision Algorithms for Finitely Presented Commutative Semigroups,”
*Computer and Mathematics with Applications*Vol. 7 pp. 159–165, 1981.Google Scholar - [3]Buchberger, B.,
*An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal*(in German), Ph.D. Thesis, Univ. of Innsbruck, Austria, Math., Inst., 1965.Google Scholar - [4]Buchberger, B., “A Critical-Pair/Completion Algorithm in Reduction Rings,”
*Proc. Logic and Machines: Decision Problems and Complexity*, (eds. by E. Borger, G. Hasenjaeger, D. Rodding), Spring LNCS 171, pp 137–161, 1984.Google Scholar - [5]Buchberger, B., and Loos, R., “Algebraic Simplification,” in
*Computer Algebra: Symbolic and Algebraic Computation*(Eds. B. Buchberger, G.E. Collins and R. Loos), Computing Suppl. 4 (1982), Springer Verlag, New York, pp. 11–43.Google Scholar - [6]Chou, T.J., and Collins, G.E., “Algorithms for the Solutions of Systems of Linear Diophantine Equations,”
*Siam J. Comput.*11 (1982), pp. 687–708.Google Scholar - [7]Garey, M.R. and Johnson, D.S.,
*Computers and Intractability*, W.H. Freeman, 1979.Google Scholar - [8]Hsiang, J.,
*Topics in Theorem Proving and Program Synthesis*, Ph.D. Thesis, University of Illinois, Urbana-Champagne, July 1983.Google Scholar - [9]Huet, G., “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems,”
*JACM*, Vol. 27, No. 4, October 1980, pp. 797–821. on Automata, Languages, and Programming (1983), Spain.Google Scholar - [10]Kandri-Rody, A., and Kapur, D., “Computing the Gröbner Basis of a Polynomial Ideal over Integers,” Proceedings of the
*Third MACSYMA Users' Conference*, Schenectady, NY, July 1984, pp. 436–451.Google Scholar - [11]Kandri-Rody, A., and Kapur, D., “Algorithms for Computing the Gröbner Bases of Polynomial Ideals over Various Euclidean Rings,” Proceedings of the
*EUROSAM, '84*, Cambridge, England, July 1984.Google Scholar - [12]Kandri-Rody, A., and Kapur, D., "An Algorithm for Computing the Gröbner Basis of a Polynomial Ideal over a Euclidean Ring,” General Electric Corporate Research and Development Report 84CRD045, April, 1984; Revised, December, 1984.Google Scholar
- [13]Kannan, R., and Bachem, A., “Prolynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix,”
*Siam J. Comput.*8 (1979), pp. 499–507.Google Scholar - [14]Kapur, D., and Narendran, P., “An Equational Approach to Theorem Proving in First-Order Predicate Calculus,” to appear in the
*Proc. of the IJCAI-85*, Los Angeles, August, 1985.Google Scholar - [15]Knuth, D.E. and Bendix, P.B., “Simple Word Problems in Universal Algebras,” in
*Computational Problems in Abstract Algebras*. (Ed. J. Leech), Pergamon Press, 1970, pp. 263–297.Google Scholar - [16]Lankford, D.S., and Ballantyne, A.M., “Decision Procedures for Simple Equational Theories with Commutative-Associative Axioms: Complete Sets of Commutative-Associative Reductions,” Automatic Theorem Proving Project, Dept. of Math. and Computer Science, University of Texas, Austin, TX 78712, Report ATP-39, August 1977.Google Scholar
- [17]Lankford, D.S., Butler, G., and Brady, B., “Abelian Group Unification Algorithms for Elementary Terms,”
*Proceedings of a NSF Workshop on the Rewrite Rule Laboratory*, September 6–9, 1983, General Electric Report, April, 1984.Google Scholar - [18]Lankford, D.S., Butler, G., and Ballantyne, A.M., “A Progress Report on New Decision Algorithms for Finitely Presented Abelian Groups,”
*7th Conference on Automated Deduction*, Springer Verlag LNCS 170 (ed. R.E. Shostak), NAPA Valley, Calif., May, 1984, pp. 128–141.Google Scholar - [19]Lankford, D.S, and Butler, G., “On the Foundations of Applied Equational Logic,”
*Talk Given at General Electric Research and Development Center*, Schenectady, NY, Feb., 1984.Google Scholar - [20]Lauer, M., “Canonical Representatives for Residue Classes of a Polynomial Ideal,”
*SYMSAC*1976, pp. 339–345.Google Scholar - [21]Le Chenadec, P., “Canonical Forms in Finitely Presented Algebras,”
*7th International Conf. on Automated Deduction*, Springer Verlag LNCS 170 (ed. R.E. Shostak), Napa Valley, Calif, May, 1984, pp. 142–165.Google Scholar - [22]Matijasevitch, J., “Enumerable Sets are Diophantine,”
*Dokl. Akad. Nauk. SSSR*191 (1970), pp. 279–282.Google Scholar - [23]Peterson, G.L., and Stickel, M.E., “Complete Sets of Reductions for Some Equational Theories,”
*JACM*28 (1981), pp. 233–264.Google Scholar - [24]Schaller, S.,
*Algorithmic Aspects of Polynomial Residue Class Rings*, Ph.D. Thesis, Computer Science Tech. Report 370, University of Wisconsin, Madison, 1979.Google Scholar - [25]Stickel, M.E., “A Unification Algorithm for Associative-Commutative Functions,”
*JACM*28 (1980), pp. 423–434.Google Scholar - [26]Szmielew, W., “Elementary Properties of Abelian Groups,”
*Fund. Math.*XLI (1954), pp. 203–271.Google Scholar - [27]van der Waerden, B.L.,
*Modern Algebra*, Vols. I and II, Fredrick Ungar Publishing Co., New York, 1966.Google Scholar - [28]Zacharias, G.,
*Generalized Grobner Bases in Commutative Polynomial Rings*, Bachelor Thesis, Lab. for Computer Science, MIT, 1978.Google Scholar