Abstract
Mechanized systems for equational inference often produce many terms that are permutations of one another. We propose to gain efficiency by dealing with such sets of terms in a uniform manner, by the use of efficient general algorithms on permutation groups. We show how permutation groups arise naturally in equational inference problems, and study some of their properties. We also study some general algorithms for processing permutations and permutation groups, and consider their application to equational reasoning and term-rewriting systems. Finally, we show how these techniques can be incorproated into resolution theorem-proving strategies.
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Alexander, G. and Plaisted, D.: Proving equality theorems with hyper-linking, in Proceedings of the 11th International Conference on Automated Deduction, 1992, pp. 706-710. System abstract.
Bachmair, L., Dershowitz, N. and Plaisted, D.: Completion without failure, in H. Aït-Kaci and M. Nivat (eds), Resolution of Equations in Algebraic Structures 2: Rewriting Techniques, Academic Press, New York, 1989, pp. 1-30.
Bachmair, L. and Ganzinger, H.: On restrictions of ordered paramodulation with simplification, in M. Stickel (ed.), Proceedings of the 10th International Conference on Automated Deduction, Springer-Verlag, New York, 1990, pp. 427-441.
Bachmair, L. and Ganzinger, H.: Equational reasoning in saturation-based theorem proving, in W. Bibel and P. H. Schmitt (eds), Automated Deduction-A Basis for Applications, Vol. 1, Kluwer Academic Publishers, Boston, 1998, pp. 353-397.
Christian, J.: High Performance Permutative Completion, Ph.D. Thesis, University of Texas at Austin, 1989.
Chang, C. and Lee, R.: Symbolic Logic and Mechanical Theorem Proving, Academic Press, New York, 1973.
Dershowitz, N.: Termination of rewriting, J. Symbolic Comput. 3 (1987), 69-116.
Dershowitz, N. and Jouannaud, J.-P.: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, North-Holland, Amsterdam, 1990.
Domenjoud, E.: Number of minimal unifiers of the equation αx 1 + · · · + αx p = AC ß y1 + · · · + ßy q, J. Automated Reasoning 8 (1992), 39-44.
Furst, M., Hopcroft, J. and Luks, E.: Polynomial-time algorithms for permutation groups, in Proceedings of the 21st IEEE Symposium on Foundations of Computer Science, Syracuse, New York, 1980, pp. 36-41.
Fitting, M.: First-Order Logic and Automated Theorem Proving, Springer-Verlag, New York, 1990.
Frobenius, I.: Effiziente Behandlung von Äquivalenzklassen von Termen, Technical Report Projektarbeit, Universität Kaiserslautern, Kaiserslautern, Germany, 1989.
Hsiang, J. and Rusinowitch, M.: Proving refutational completeness of theorem-proving strategies: The transfinite semantic tree method, J. Assoc. Comput. Mach. 38(3) (1991), 559-587.
Klop, J. W.: Term rewriting systems, in S. Abramsky, D. M. Gabbay and T. S. E. Maibaum (eds), Handbook of Logic in Computer Science, Vol. 2, Oxford University Press, Oxford, 1992, Ch. 1, pp. 1-117.
Kirchner, C., Lynch, C. and Scharff, C.: A fine-grained concurrent completion procedure, in H. Ganzinger (ed.), Proceedings of RTA'96, Lecture Notes in Comput. Sci. 1103, Springer-Verlag, 1996, pp. 3-17.
Kapur, D. and Narendran, P.: Double-exponential complexity of computing a complete set of AC-unifiers, in Proceedings 7th IEEE Symposium on Logic in Computer Science, Santa Cruz, California, 1992, pp. 11-21.
Lankford, D., Butler, G. and Ballantyne, A.: A progresss report on new decision algorithms for finitely presented Abelian groups, in Proceedings of the 7th International Conference on Automated Deduction, Lecture Notes in Comput. Sci. 170, Springer-Verlag, 1984, pp. 128-141.
Leitsch, A.: The Resolution Calculus, Springer-Verlag, Berlin, 1997. Texts in Theoretical Computer Science.
Loveland, D.: Automated Theorem Proving: A Logical Basis, North-Holland, New York, 1978.
McCune, W.: 33 basic test problems: A practical evaluation of some paramodulation strategies, in R. Veroff (ed.), Automated Reasoning and Its Applications: Essays in Honor of Larry Wos, MIT Press, Cambridge, MA, 1997.
McCune, W. W.: Solution of the Robbins problem, J. Automated Reasoning 19(3) (1997), 263-276.
Plaisted, D.: Equational reasoning and term rewriting systems, in D. Gabbay, C. Hogger, J. A. Robinson and J. Siekmann (eds), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Oxford University Press, 1993, pp. 273-364.
Raoult, J.-C. and Vuillemin, J.: Operational and semantic equivalence between recursive programs, J. Assoc. Comput. Mach. 27(4) (1980), 772-796.
Stickel, M. E.: A unification algorithm for associative-commutative functions, J. Assoc. Comput. Mach. 28 (1981), 423-434.
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Avenhaus, J., Plaisted, D.A. General Algorithms for Permutations in Equational Inference. Journal of Automated Reasoning 26, 223–268 (2001). https://doi.org/10.1023/A:1006439522342
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DOI: https://doi.org/10.1023/A:1006439522342