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AC-KBO revisited*


Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper, we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.

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The research described in this paper is supported by the Austrian Science Fund (FWF) international project I963, the bilateral programs of the Japan Society for the Promotion of Science and the KAKENHI Grant No. 25730004.

This is an extended version of a paper presented at the Twelfth International Symposium on Functional and Logic Programming (FLOPS 2014), invited as a rapid publication in TPLP. The authors acknowledge the assistance of the conference chairs Michael Codish and Eijiro Sumii.

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Alarcón, B., Lucas, S. and Meseguer, J. 2010. A dependency pair framework for AC-termination. In Proc. 8th International Workshop on Rewriting Logic and its Applications (WRLA 2010), Lecture Notes in Computer Science, vol. 6381. Springer Berlin Heidelberg, 3551.
Arts, T. and Giesl, J. 2000. Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 12, 133–178.
Bachmair, L. and Plaisted, D. A. 1985. Termination orderings for associative-commutative rewriting systems. Journal of Symbolic Computation 1, 329349.
Ben, Cherifa, A and Lescanne, P. 1987. Termination of rewriting systems by polynomial interpretations and its implementation. Science of Computer Programming 9, 2, 137159.
Codish, M., Giesl, J., Schneider-Kamp, P. and Thiemann, R. 2012. SAT solving for termination proofs with recursive path orders and dependency pairs. Journal of Automated Reasoning 49, 1, 5393.
Dershowitz, N. 1982. Orderings for term-rewriting systems. Theoretical Computer Science 17, 3, 279301.
Giesl, J. and Kapur, D. 2001. Dependency pairs for equational rewriting. In Proc. 12th International Conference on Rewriting Techniques and Applications (RTA 2001), Lecture Notes in Computer Science, vol. 2051. Springer Berlin Heidelberg, 93108.
Knuth, D. and Bendix, P. 1970. Simple word problems in universal algebras. In Computational Problems in Abstract Algebra, Leech, J., Ed. Pergamon Press, New York, 263297.
Korovin, K. and Voronkov, A. 2003a. An AC-compatible Knuth-Bendix order. In Proc. 19th International Conference on Automated Deduction (CADE 2003), Lecture Notes in Artificial Intelligence, vol. 2741. Springer Berlin Heidelberg, 4759.
Korovin, K. and Voronkov, A. 2003b. Orienting rewrite rules with the Knuth-Bendix order. Information and Computation 183, 2, 165186.
Krishnamoorthy, M. and Narendran, P. 1985. On recursive path ordering. Theoretical Computer Science 40, 323328.
Kusakari, K. 2000. AC-termination and dependency pairs of term rewriting systems. Ph.D. thesis, JAIST, Nomi, Japan.
Kusakari, K. and Toyama, Y. 2001. On proving AC-termination by AC-dependency pairs. IEICE Transactions on Information and Systems E84-D, 5, 439447.
Lankford, D. 1979. On proving term rewrite systems are noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA.
Löchner, B. 2006. Things to know when implementing KBO. Journal of Automated Reasoning 36, 4, 289310.
Ludwig, M. and Waldmann, U. 2007. An extension of the Knuth-Bendix ordering with LPO-like properties. In Proc. 14th International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR 2007), Lecture Notes in Artificial Intelligence, vol. 4790. Springer Berlin Heidelberg, 348362.
Marché, C. and Urbain, X. 2004. Modular and incremental proofs of AC-termination. Journal of Symbolic Computation 38, 1, 873897.
Middeldorp, A. and Zantema, H. 1997. Simple termination of rewrite systems. Theoretical Computer Science 175, 1, 127158.
Rubio, A. 2002. A fully syntactic AC-RPO. Information and Computation 178, 2, 515533.
Schrijver, A. 1986. Theory of Linear and Integer Programming. Wiley, West Sussex, England.
Steinbach, J. 1990. AC-termination of rewrite systems: A modified Knuth-Bendix ordering. In Proc. 2nd International Conference on Algebraic and Logic Programming (ALP 1990), Lecture Notes in Computer Science, vol. 463. Springer Berlin Heidelberg, 372386.
Thiemann, R., Allais, G. and Nagele, J. 2012. On the formalization of termination techniques based on multiset orderings. In Proc. 23rd International Conference on Rewriting Techniques and Applications (RTA 2012), Leibniz International Proceedings in Informatics, vol. 15. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 339354.
Winkler, S. 2013. Termination tools in automated reasoning. Ph.D. thesis, UIBK, Innsbruck, Austria.
Winkler, S., Zankl, H. and Middeldorp, A. 2012. Ordinals and Knuth-Bendix orders. In Proc. 18th International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR-18), LNCS Advanced Research in Computing and Software Science, vol. 7180. Springer Berlin Heidelberg, 420434.
Yamada, A., Winkler, S., Hirokawa, N. and Middeldorp, A. 2014. AC-KBO revisited. In Proc. 12th International Symposium on Functional and Logic Programming (FLOPS 2014), Lecture Notes in Computer Science, vol. 8475. Springer International Publishing, 319335.
Zankl, H., Hirokawa, N. and Middeldorp, A. 2009. KBO orientability. Journal of Automated Reasoning 43, 2, 173201.
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Theory and Practice of Logic Programming
  • ISSN: 1471-0684
  • EISSN: 1475-3081
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