Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-03T11:38:18.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of unification modulo ACUI

Published online by Cambridge University Press:  11 November 2019

Franz Baader ,
Pavlos Marantidis Open the ORCID record for Pavlos Marantidis [Opens in a new window] ,
Antoine Mottet  and
Alexander Okhotin
Franz Baader
Affiliation:
Theoretical Computer Science, Technische Universität Dresden, Germany
Pavlos Marantidis*
Affiliation:
Theoretical Computer Science, Technische Universität Dresden, Germany
Antoine Mottet
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Alexander Okhotin
Affiliation:
St. Petersburg State University, St. Petersburg, Russia
*
*Corresponding author. Email: pavlos.marantidis@tu-dresden.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.

Keywords

Unification theoryACUIapproximate unificationground identities
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Special Issue 6: Special Issue: Unification , June 2020 , pp. 597 - 626
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge University Press 2019

Footnotes

†

Supported by DFG Graduiertenkolleg 1763 (QuantLA).

‡

Partially supported by DFG Graduiertenkolleg 1763 (QuantLA).

References

Baader, F. (1996). Using automata theory for characterizing the semantics of terminological cycles. Ann. of Mathematics and Artificial Intelligence 18, 175–219.CrossRefGoogle Scholar
Baader, F., Borgwardt, S. and Morawska, B. (2012). Extending unification in ℰℒ towards general TBoxes. In: Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR 2012), AAAI Press/The MIT Press, 568–572.Google Scholar
Baader, F., Fernández Gil, O. and Marantidis, P. (2018). Matching in the description logic {↕0 with respect to general TBoxes. In: Barthe, G., Sutcliffe, G. and Veanes, M. (eds.) Proceedings of the 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR 2018), vol. 57. EPiC Series in Computing, EasyChair, 76–94.Google Scholar
Baader, F., Küsters, R. and Molitor, R. (1999). Computing least common subsumers in description logics with existential restrictions. In: Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI 1999), 96–101.Google Scholar
Baader, F., Marantidis, P. and Okhotin, A. (2016). Approximate unification in the description logic ℱℒ0. In: Michael, L. and Kakas, A. C. (eds.) Proceedings of the 15th European Conference on Logics in Artificial Intelligence (JELIA 2016), vol. 10021. Lecture Notes in Artificial Intelligence. Springer, 49–63.Google Scholar
Baader, F. and Morawska, B. (2009). Unification in the description logic ℰℒ. In: Treinen, R. (ed.) Proceedings of the 20th International Conference on Rewriting Techniques and Applications (RTA 2009), vol. 5595. Lecture Notes in Computer Science. Springer, 350–364.Google Scholar
Baader, F. and Narendran, P. (2001). Unification of concept terms in description logics. Journal of Symbolic Computation 31 (3), 277–305.CrossRefGoogle Scholar
Baader, F. and Nipkow, T. (1998). Term Rewriting and All That, Cambridge University Press.CrossRefGoogle Scholar
Baader, F. and Schulz, K. U. (1993). General A- and AX-unification via optimized combination procedures. In: Abdulrab, H. and Pécuchet, J.-P. (eds.) Word Equations and Related Topics, vol. 677. Lecture Notes in Computer Science. Springer, 23–42. ISBN: 978-3-540-47636-8CrossRefGoogle Scholar
Baader, F. and Schulz, K. U. (1996). Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation 21, 211–243.CrossRefGoogle Scholar
Baader, F. and Siekmann, J. H. (1994). Unification theory. In: Gabbay, D. M., Hogger, C. J. and Robinson, J. A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, Oxford, UK, Oxford University Press, 41–125.Google Scholar
Baader, F. and Snyder, W. (2001). Unification theory. In: Robinson, J. A. and Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I. Elsevier Science Publishers, 447–533.Google Scholar
Dovier, A., Pontelli, E. and Rossi, G. (2006). Set unification. TPLP 6 (6), 645–701. doi:10.1017/S1471068406002730.Google Scholar
Dowling, W. F. and Gallier, J. (1984). Linear-time algorithms for testing the satisfiability of propositional horn formulae. Journal of Logic Programming 1 (3), 267–284.CrossRefGoogle Scholar
Garey, M. R. and Johnson, D. S. (1990). Computers and Intractability; A Guide to the Theory of NP-Completeness. New York, NY, USA, W. H. Freeman & Co. ISBN: 0716710455.Google Scholar
Iranzo, P. J. and Rubio-Manzano, C. (2015). Proximity-based unification theory. Fuzzy Sets and Systems, 262, 21–43. doi:10.1016/j.fss.2014.07.006.CrossRefGoogle Scholar
Jaumard, B. and Simeone, B. (1987). On the complexity of the maximum satisfiability problem for Horn formulas. Information Processing Letters 26 (1), 1–4.CrossRefGoogle Scholar
Kapur, D. and Narendran, P. (1992). Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning 9 (2), 261–288. ISSN: 1573-0670. doi:10.1007/BF00245463.CrossRefGoogle Scholar
Kapur, D. and Narendran, P. (1986). NP-completeness of the set unification and matching problems. In: Siekmann, J. H. (ed.) Proceedings of the 8th International Conference on Automated Deduction, vol. 230. Lecture Notes in Computer Science. Oxford, UK, Springer, 489–495.CrossRefGoogle Scholar
Lehmann, K. and Turhan, A.-Y. (2012). A framework for semantic-based similarity measures for ℰℒℋ-concepts. In: Proceedings of the 13th European Conference on Logics in Artificial Intelligence (JELIA’2012), vol. 7519. Lecture Notes in Computer Science. Springer, 307–319.Google Scholar
Marché, C. (1996). Normalized rewriting: An alternative to rewriting modulo a set of equations. Journal of Symbolic Computation 21 (3), 253–288. ISSN: 0747-7171. https://doi.org/10.1006/jsco.1996.0011.CrossRefGoogle Scholar
Marques-Silva, J., Ignatiev, A. and Morgado, A. (2017). Horn maximum satisfiability: Reductions, algorithms and applications. In: Oliveira, E. et al. (eds.) Progress in Artificial Intelligence, Cham, Springer International Publishing, 681–694.CrossRefGoogle Scholar
Narendran, P. and Rusinowitch, M. (1996). Any ground associative-commutative theory has a finite canonical system. Journal of Automated Reasoning 17 (1), 131–143. doi: 10.1007/BF00247671.CrossRefGoogle Scholar
Pan, J. Z., Ren, Y. and Zhao, Y. (2016). Tractable approximate deduction for OWL. Artificial Intelligence 235, 95–155.CrossRefGoogle Scholar
Siekmann, J. H. (1989). Unification theory: A survey. Journal of Symbolic Computation 7 (3–4), 207–274.CrossRefGoogle Scholar