Abstract
Resolution-based calculi are among the most widely used calculi for theorem proving in first-order logic. Numerous refinements of resolution are nowadays available, such as e.g. basic superposition, a calculus highly optimized for theorem proving with equality. However, even such an advanced calculus does not restrict inferences enough to obtain decision procedures for complex logics, such as \(\mathcal{SHIQ}\). In this paper, we present a new decomposition inference rule, which can be combined with any resolution-based calculus compatible with the standard notion of redundancy. We combine decomposition with basic superposition to obtain three new decision procedures: (i) for the description logic \(\mathcal{SHIQ}\), (ii) for the description logic \(\mathcal{ALCHIQ}b\), and (iii) for answering conjunctive queries over \(\mathcal{SHIQ}\) knowledge bases. The first two procedures are worst-case optimal and, based on the vast experience in building efficient theorem provers, we expect them to be suitable for practical usage.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Bachmair, L., Ganzinger, H.: Resolution Theorem Proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 19–99. Elsevier, Amsterdam (2001)
Bachmair, L., Ganzinger, H., Lynch, C., Snyder, W.: Basic Paramodulation. Information and Computation 121(2), 172–192 (1995)
Calvanese, D., De Giacomo, G., Lenzerini, M.: On the Decidability of Query Containment under Constraints. In: Proc. PODS 1998, pp. 149–158. ACM Press, New York (1998)
Chandra, A.K., Merlin, P.M.: Optimal implementation of conjunctive queries in relational data bases. In: Proc. STOC 1977, pp. 77–90. ACM Press, New York (1977)
de Nivelle, H.: Splitting through new proposition symbols. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 172–185. Springer, Heidelberg (2001)
Dershowitz, N., Plaisted, D.A.: Rewriting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 535–610. Elsevier, Amsterdam (2001)
Fermüller, C., Leitsch, A., Hustadt, U., Tammet, T.: Resolution Decision Procedures. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1791–1849. Elsevier Science, Amsterdam (2001)
Horrocks, I., Sattler, U., Tobies, S.: Practical Reasoning for Very Expressive Description Logics. Logic Journal of the IGPL 8(3), 239–263 (2000)
Hustadt, U., Motik, B., Sattler, U.: Reasoning for Description Logics around SHIQ in a Resolution Framework. Technical Report 3-8-04/04, FZI, Germany (2004), http://www.fzi.de/wim/publikationen.php?id=1172
Hustadt, U., Motik, B., Sattler, U.: Reducing SHIQ− Description Logic to Disjunctive Datalog Programs. In: Proc. KR 2004, pp. 152–162. AAAI Press, Menlo Park (2004)
Joyner Jr., W.H.: Resolution Strategies as Decision Procedures. J. ACM 23(3), 398–417 (1976)
Lutz, C., Sattler, U.: The Complexity of Reasoning with Boolean Modal Logics. In: Advances in Modal Logics, vol. 3, CSLI Publications, Stanford (2001)
Nieuwenhuis, R., Rubio, A.: Theorem Proving with Ordering and Equality Constrained Clauses. J. Logic and Computation 19(4), 312–351 (1995)
Nieuwenhuis, R., Rubio, A.: Paramodulation-Based Theorem Proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 371–443. Elsevier Science, Amsterdam (2001)
Plaisted, D.A., Greenbaum, S.: A Structure-preserving Clause Form Transformation. J. Symbolic Logic and Computation 2(3), 293–304 (1986)
Riazanov, A., Voronkov, A.: Splitting Without Backtracking. In: Proc. IJCAI 2001, pp. 611–617. Morgan Kaufmann, San Francisco (2001)
Schmidt, R.A., Hustadt, U.: A Resolution Decision Procedure for Fluted Logic. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 433–448. Springer, Heidelberg (2000)
Tessaris, S.: Questions and answers: reasoning and querying in Description Logic. PhD thesis, University of Manchester, UK (2001)
Tobies, S.: Complexity Results and Practical Algorithms for Logics in Knowledge Representation. PhD thesis, RWTH Aachen, Germany (2001)
Weidenbach, C.: Combining Superposition, Sorts and Splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1965–2013. Elsevier Science, Amsterdam (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hustadt, U., Motik, B., Sattler, U. (2005). A Decomposition Rule for Decision Procedures by Resolution-Based Calculi. In: Baader, F., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2005. Lecture Notes in Computer Science(), vol 3452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32275-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-32275-7_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25236-8
Online ISBN: 978-3-540-32275-7
eBook Packages: Computer ScienceComputer Science (R0)