Abstract
We provide the extended ground-reducibility test which is essential for induction with term-rewriting systems based on [Küc89]: Given a term, determine at which sets of positions it is ground-reducible by which subsets of rules. The core of our method is a new parallel cover algorithm based on recursive decomposition. From this we obtain a separation algorithm which determines constructors and defined function symbols in a term-algebra presented by a rewrite system. We then reduce our main problem of extended ground-reducibility to separation and cover. Furthermore, using the knowledge of algebra separation, we refine the bounds of [JK86] for the size of ground reduction test-sets. Both our cover algorithm and our extended ground-reducibility test are engineered to be adaptive to actual problem structure, i.e., to allow for lower than the worst case bounds for test-sets on well conditioned problems, including well conditioned subproblems of difficult cases.
A substantial part of this work was done while the first author was supported by a Fulbright scholarship and both authors were at the Department for Computer and Information Sciences, University of Delaware, Newark, Delaware.
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References
Leo Bachmair and Nachum Dershowitz. Critical pair criteria for completion. Journ. Symbolic Computation, 6(1):1–18, August 1988.
Robert S. Boyer and J Strother Moore. A Computational Logic. Academic Press, Orlando, Florida, 1979.
Reinhard Bündgen. Design, implementation, and application of an extended ground-reducibility test. Master's thesis, Computer and Information Sciences, University of Delaware, Newark, DE 19716, 1987.
Reinhard Bündgen. Design, implementation, and application of an extended ground-reducibility test. Technical Report 88-05, Computer and Information Sciences, University of Delaware, Newark, DE 19716, December 1987.
B. F. Caviness, editor. Eurocal'85, volume 204 of LNCS. Springer-Verlag, 1985. (European Conference on Computer Algebra, Linz, Austria, April 1985).
Laurent Fribourg. A strong restriction of the inductive completion procedure. In Proc. 13th ICALP, volume 226 of LNCS, Rennes, France, 1986. Springer-Verlag. (To appear in J. Symbolic Computation.).
Jean H. Gallier. Logic for Computer Science. Harper & Row, New York, 1986.
Gérard Huet and Jean-Marie Hullot. Proofs by induction in equational theories with constructors. J. Computer and System Sciences, 25:239–266, 1982.
Gérard Huet and Derek C. Oppen. Equations and rewrite rules: A survey. In Ron Book, editor, Formal Languages: Perspectives and Open Problems, pages 349–405. Academic Press, 1980.
Jean-Pierre Jouannaud and Emmanuel Kounalis. Proofs by induction in equational theories without constructors. Rapport de Recherche 295, Laboratoire de Recherche en Informatique, Université Paris 11, Orsay, France, September 1986. (To appear in Information and Computation, 1989.).
Deepak Kapur and David R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125–157, February 1987.
Deepak Kapur, Paliath Narendran, and Hantao Zhang. Proof by induction using test sets. In Jörg H. Siekmann, editor, 8th International Conference on Automated Deduction, volume 230 of LNCS, pages 99–117. Springer-Verlag, 1986.
Deepak Kapur, Paliath Narendran, and Hantao Zhang. On sufficient-completeness and related properties of term rewriting systems. Acta Informatica, 24(4):395–415, 1987.
Emmanuel Kounalis. Completeness in data type specifications. In Caviness [Cav85], pages 348–362.
Wolfgang Küchlin. A confluence criterion based on the generalised Knuth-Bendix algorithm. In Caviness [Cav85], pages 390–399.
Wolfgang Küchlin. Equational Completion by Proof Transformation. PhD thesis, Swiss Federal Institute of Technology (ETH), CH-8092 Zürich, Switzerland, June 1986.
Wolfgang Küchlin. Inductive completion by ground proof transformation. In H. Aït-Kaci and M. Nivat, editors, Rewriting Techniques, volume 2 of Resolution of Equations in Algebraic Structures, chapter 7. Academic Press, 1989.
Tobias Nipkow and G. Weikum. A decidability result about sufficient completeness of axiomatically specified abstract data types. In Sixth GI Conference on Theoretical Computer Science, volume 145 of LNCS, pages 257–268, 1982.
David Plaisted. Semantic confluence and completion methods. Information and Control, 65:182–215, 1985.
Jean-Jacques Thiel. Stop loosing sleep over incomplete data type specifications. In Proc. 11th PoPL, Salt Lake City, Utah, 1984. ACM.
Hantao Zhang, Deepak Kapur, and Mukkai S. Krishnamoorthy. A mechanizable induction principle for equational specifications. In E. Lusk and R. Overbeek, editors, 9th International Conference on Automated Deduction, volume 310 of LNCS, pages 162–181. Springer-Verlag, 1988.
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Bündgen, R., Küchlin, W. (1989). Computing ground reducibility and inductively complete positions. In: Dershowitz, N. (eds) Rewriting Techniques and Applications. RTA 1989. Lecture Notes in Computer Science, vol 355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51081-8_100
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DOI: https://doi.org/10.1007/3-540-51081-8_100
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