Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T16:23:08.289Z Has data issue: false hasContentIssue false
  • English
  • Français

AC-KBO revisited*†

Published online by Cambridge University Press:  08 June 2015

AKIHISA YAMADA ,
SARAH WINKLER ,
NAO HIROKAWA  and
AART MIDDELDORP
AKIHISA YAMADA
Affiliation:
Research Institute for Secure Systems, AIST, Amagasaki, Japan
SARAH WINKLER
Affiliation:
Institute of Computer Science, University of Innsbruck, Innsbruck, Austria
NAO HIROKAWA
Affiliation:
School of Information Science, JAIST, Nomi, Japan
AART MIDDELDORP
Affiliation:
Institute of Computer Science, University of Innsbruck, Innsbruck, Austria (e-mail: aart.middeldorp@uibk.ac.at)
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Term rewritingterminationassociative-commutative theoryKnuth-Bendix order
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 16 , Issue 2 , March 2016 , pp. 163 - 188
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

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.

References

Alarcón, B., Lucas, S. and Meseguer, J. 2010. A dependency pair framework for A ∨ C-termination. In Proc. 8th International Workshop on Rewriting Logic and its Applications (WRLA 2010), Lecture Notes in Computer Science, vol. 6381. Springer Berlin Heidelberg, 35–51.CrossRefGoogle Scholar
Arts, T. and Giesl, J. 2000. Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 1–2, 133–178.CrossRefGoogle Scholar
Bachmair, L. and Plaisted, D. A. 1985. Termination orderings for associative-commutative rewriting systems. Journal of Symbolic Computation 1, 329–349.Google Scholar
Ben, Cherifa, A and Lescanne, P. 1987. Termination of rewriting systems by polynomial interpretations and its implementation. Science of Computer Programming 9, 2, 137–159.Google Scholar
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, 53–93.CrossRefGoogle Scholar
Dershowitz, N. 1982. Orderings for term-rewriting systems. Theoretical Computer Science 17, 3, 279–301.Google Scholar
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, 93–108.Google Scholar
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, 263–297.Google Scholar
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, 47–59.Google Scholar
Korovin, K. and Voronkov, A. 2003b. Orienting rewrite rules with the Knuth-Bendix order. Information and Computation 183, 2, 165–186.CrossRefGoogle Scholar
Krishnamoorthy, M. and Narendran, P. 1985. On recursive path ordering. Theoretical Computer Science 40, 323–328.Google Scholar
Kusakari, K. 2000. AC-termination and dependency pairs of term rewriting systems. Ph.D. thesis, JAIST, Nomi, Japan.Google Scholar
Kusakari, K. and Toyama, Y. 2001. On proving AC-termination by AC-dependency pairs. IEICE Transactions on Information and Systems E84-D, 5, 439–447.Google Scholar
Lankford, D. 1979. On proving term rewrite systems are noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA.Google Scholar
Löchner, B. 2006. Things to know when implementing KBO. Journal of Automated Reasoning 36, 4, 289–310.CrossRefGoogle Scholar
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, 348–362.Google Scholar
Marché, C. and Urbain, X. 2004. Modular and incremental proofs of AC-termination. Journal of Symbolic Computation 38, 1, 873–897.Google Scholar
Middeldorp, A. and Zantema, H. 1997. Simple termination of rewrite systems. Theoretical Computer Science 175, 1, 127–158.CrossRefGoogle Scholar
Rubio, A. 2002. A fully syntactic AC-RPO. Information and Computation 178, 2, 515–533.CrossRefGoogle Scholar
Schrijver, A. 1986. Theory of Linear and Integer Programming. Wiley, West Sussex, England.Google Scholar
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, 372–386.Google Scholar
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, 339–354.Google Scholar
Winkler, S. 2013. Termination tools in automated reasoning. Ph.D. thesis, UIBK, Innsbruck, Austria.Google Scholar
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, 420–434.Google Scholar
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, 319–335.Google Scholar
Zankl, H., Hirokawa, N. and Middeldorp, A. 2009. KBO orientability. Journal of Automated Reasoning 43, 2, 173–201.Google Scholar
Supplementary material: PDF

Yamada supplementary material

Appendix

Download Yamada supplementary material(PDF)
PDF 238.3 KB