FroCoS 2013: Frontiers of Combining Systems pp 87-102

# Uniform Interpolation of $$\mathcal{ALC}$$-Ontologies Using Fixpoints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8152)

## Abstract

We present a method to compute uniform interpolants with fixpoints for ontologies specified in the description logic $$\mathcal{ALC}$$. The aim of uniform interpolation is to reformulate an ontology such that it only uses a specified set of symbols, while preserving consequences that involve these symbols. It is known that in $$\mathcal{ALC}$$ uniform interpolants cannot always be finitely represented. Our method computes uniform interpolants for the target language $$\mathcal{ALC}\mu$$, which is $$\mathcal{ALC}$$ enriched with fixpoint operators, and always computes a finite representation. If the result does not involve fixpoint operators, it is the uniform interpolant in $$\mathcal{ALC}$$. The method focuses on eliminating concept symbols and combines resolution-based reasoning with an approach known from the area of second-order quantifier elimination to introduce fixpoint operators when needed. If fixpoint operators are not desired, it is possible to approximate the interpolant.

## Keywords

Modal Logic Description Logic Conjunctive Normal Form Domain Element Small Clause
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Ackermann, W.: Untersuchungen über das Eliminationsproblem der mathematischen Logik. Mathematische Annalen 110(1), 390–413 (1935)
2. 2.
Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Handbook of Automated Reasoning, pp. 19–99. Elsevier, MIT Press (2001)Google Scholar
3. 3.
Calvanese, D., Giacomo, G.D., Lenzerini, M.: Reasoning in expressive description logics with fixpoints based on automata on infinite trees. In: Proc. IJCAI 1999, pp. 84–89. Morgan Kaufmann (1999)Google Scholar
4. 4.
D’Agonstino, G., Hollenberg, M.: Uniform interpolation, automata and the modal μ-calculus. In: AiML, vol. 1, pp. 73–84. CSLI Pub. (1998)Google Scholar
5. 5.
D’Agostino, G., Lenzi, G.: On modal μ-calculus with explicit interpolants. J. Applied Logic 4(3), 256–278 (2006)
6. 6.
Gabbay, D., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. In: Proc. KR 1992, pp. 425–435. Morgan Kaufmann (1992)Google Scholar
7. 7.
Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second Order Quantifier Elimination: Foundations, Computational Aspects and Applications. College Publ. (2008)Google Scholar
8. 8.
Goranko, V., Hustadt, U., Schmidt, R.A., Vakarelov, D.: SCAN is complete for all Sahlqvist formulae. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 149–162. Springer, Heidelberg (2004)
9. 9.
Grau, B.C., Motik, B.: Reasoning over ontologies with hidden content: The import-by-query approach. J. Artificial Intelligence Research 45, 197–255 (2012)
10. 10.
Herzig, A., Mengin, J.: Uniform interpolation by resolution in modal logic. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 219–231. Springer, Heidelberg (2008)
11. 11.
Lutz, C., Piro, R., Wolter, F.: $$\mathcal{EL}$$-concepts go second-order: Greatest fixpoints and simulation quantifiers. In: Proc. DL 2010, pp. 43–54. CEUR-WS.org (2010)Google Scholar
12. 12.
Lutz, C., Wolter, F.: Foundations for uniform interpolation and forgetting in expressive description logics. In: Proc. IJCAI 2011, pp. 989–995. AAAI Press (2011)Google Scholar
13. 13.
Nikitina, N.: Forgetting in General EL Terminologies. In: Description Logics. Proc. DL 2011. CEUR-WS.org (2011)Google Scholar
14. 14.
Nonnengart, A., Szałas, A.: A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In: Logic at Work, pp. 307–328. Springer (1999)Google Scholar
15. 15.
Schmidt, R.A.: The Ackermann approach for modal logic, correspondence theory and second-order reduction. J. Appl. Logic 10(1), 52–74 (2012)
16. 16.
Simancik, F., Kazakov, Y., Horrocks, I.: Consequence-based reasoning beyond Horn ontologies. In: Proc. IJCAI 2011, pp. 1093–1098. AAAI Press (2011)Google Scholar
17. 17.
Szałas, A.: Second-order reasoning in description logics. J. Appl. Non-Classical Logics 16(3-4), 517–530 (2006)
18. 18.
Wang, Z., Wang, K., Topor, R., Zhang, X.: Tableau-based forgetting in $$\mathcal{ALC}$$ ontologies. In: Proc. ECAI 2010, pp. 47–52. IOS Press (2010)Google Scholar