## Notre Dame Journal of Formal Logic

### Deciding Unifiability and Computing Local Unifiers in the Description Logic $\mathcal{E\!L}$ without Top Constructor

#### Abstract

Unification in description logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive description logic $\mathcal{E\!L}$ is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using $\mathcal{E\!L}$. On the other hand, unification in $\mathcal{E\!L}$ has been shown to be NP-complete and, thus, of considerably lower complexity than unification in other description logics of similarly restricted expressive power.

However, $\mathcal{E\!L}$ allows the use of the top concept ($\top$), which represents the whole interpretation domain, whereas the large medical ontology SNOMED CT makes no use of this feature. Surprisingly, removing the top concept from $\mathcal{E\!L}$ makes the unification problem considerably harder. More precisely, we will show that unification in $\mathcal{E\!L}$ without the top concept is PSPACE-complete. In addition to the decision problem, we also consider the problem of actually computing $\mathcal{E\!L}^{{-}\top}\!$-unifiers.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 4 (2016), 443-476.

Dates
Accepted: 21 August 2013
First available in Project Euclid: 19 July 2016

https://projecteuclid.org/euclid.ndjfl/1468952202

Digital Object Identifier
doi:10.1215/00294527-3555507

Mathematical Reviews number (MathSciNet)
MR3565533

Zentralblatt MATH identifier
1358.68272

Keywords
unification description logics

#### Citation

Baader, Franz; Binh, Nguyen Thanh; Borgwardt, Stefan; Morawska, Barbara. Deciding Unifiability and Computing Local Unifiers in the Description Logic $\mathcal{E\!L}$ without Top Constructor. Notre Dame J. Formal Logic 57 (2016), no. 4, 443--476. doi:10.1215/00294527-3555507. https://projecteuclid.org/euclid.ndjfl/1468952202

#### References

• [1] Baader, F., “Terminological cycles in a description logic with existential restrictions,” pp. 325–30 in Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI’03), edited by G. Gottlob and T. Walsh, Morgan Kaufmann, San Francisco, Calif., 2003.
• [2] Baader, F., N. T. Binh, S. Borgwardt, and B. Morawska, “Computing local unifiers in the description logic $\mathcal{EL}$ without the top concept,” pp. 2–8 in Proceedings of the 25th International Workshop on Unification (UNIF’11), edited by F. Baader, B. Morawska, and J. Otop, 2011.
• [3] Baader, F., N. T. Binh, S. Borgwardt, and B. Morawska, “Unification in the description logic $\mathcal{EL}$ without the top concept,” pp. 70–84 in Automated Deduction—CADE-23, edited by N. Bjørner and V. Sofronie-Stokkermans, vol. 6803 of Lecture Notes in Computer Science, Springer, Heidelberg, 2011.
• [4] Baader, F., S. Borgwardt, and B. Morawska, “A goal-oriented algorithm for unification in $\mathcal{ELH}_{R^{+}}$ w.r.t. cycle-restricted ontologies,” pp. 493–504 in AI 2012: Advances in Artificial Intelligence, edited by M. Thielscher and D. Zhang, vol. 7691 of Lecture Notes in Computer Science, Springer, Heidelberg, 2012.
• [5] Baader, F., S. Borgwardt, and B. Morawska, “SAT- encoding of unification in $\mathcal{ELH}_{R^{+}}$ w.r.t. cycle-restricted ontologies,” pp. 30–44 in Automated Reasoning, edited by B. Gramlich, D. Miller, and U. Sattler, vol. 7364 of Lecture Notes in Computer Science, Springer, Heidelberg, 2012.
• [6] Baader, F., S. Borgwardt, and B. Morawska, “Extending unification in $\mathcal{EL}$ towards general TBoxes,” pp. 568–72 in Proceedings of the 13th International Conference on Principles of Knowledge Representation and Reasoning (KR’12), edited by G. Brewka, T. Eiter, and S. A. McIlraith, AAAI Press, Palo Alto, Calif., 2012.
• [7] Baader, F., S. Brandt, and C. Lutz, “Pushing the $\mathcal{EL}$ envelope,” pp. 364–69 in Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI’05), edited by L. P. Kaelbling and A. Saffiotti, Morgan Kaufmann, San Francisco, Calif., 2005.
• [8] Baader, F., D. Calvanese, D. McGuinness, D. Nardi, and P. F. Patel-Schneider, eds., The Description Logic Handbook: Theory, Implementation, and Applications, Cambridge University Press, Cambridge, 2003.
• [9] Baader, F., and S. Ghilardi, “Unification in modal and description logics,” Logic Journal of the IGPL, vol. 19 (2011), pp. 705–30.
• [10] Baader, F., and B. Morawska, “Unification in the description logic $\mathcal{EL}$,” pp. 350–64 in Rewriting Techniques and Applications, edited by R. Treinen, vol. 5595 of Lecture Notes in Computer Science, Springer, Berlin, 2009.
• [11] Baader, F., and B. Morawska, “SAT encoding of unification in $\mathcal{EL}$,” pp. 97–111 in Logic for Programming, Artificial Intelligence, and Reasoning, edited by C. G. Fermüller and A. Voronkov, vol. 6397of Lecture Notes in Computer Science, Springer, Berlin, 2010.
• [12] Baader, F., and B. Morawska, “Unification in the description logic $\mathcal{EL}$,” Logical Methods in Computer Science, vol. 6 (2010), no. 17.
• [13] Baader, F., and P. Narendran, “Unification of concept terms in description logics,” Journal of Symbolic Computation, vol. 31 (2001), pp. 277–305.
• [14] Baader, F., and W. Snyder, “Unification theory,” pp. 445–532 in Handbook of Automated Reasoning, edited by J. A. Robinson and A. Voronkov, MIT Press, Cambridge, Mass., 2001.
• [15] Birget, J.-C., “State-complexity of finite-state devices, state compressibility and incompressibility,” Mathematical Systems Theory, vol. 26 (1993), pp. 237–69.
• [16] Blackburn, P., J. van Benthem, and F. Wolter, Handbook of Modal Logic, Elsevier, 2006.
• [17] Chandra, A. K., D. C. Kozen, and L. J. Stockmeyer, “Alternation,” Journal of the ACM, vol. 28 (1981), pp. 114–33.
• [18] Dijkstra, E. W., “A note on two problems in connexion with graphs,” Numerische Mathematik, vol. 1 (1959), pp. 269–71.
• [19] Garey, M. R., and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, Calif., 1979.
• [20] Horrocks, I., U. Sattler, and S. Tobies, “Practical reasoning for very expressive description logics,” Logic Journal of the IGPL, vol. 8 (2000), pp. 239–64.
• [21] Jiang, T., and B. Ravikumar, “A note on the space complexity of some decision problems for finite automata,” Information Processing Letters, vol. 40 (1991), pp. 25–31.
• [22] Kozen, D., “Lower bounds for natural proof systems,” pp. 254–66 in 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), IEEE Computer Science, Long Beach, Calif., 1977.
• [23] Nieuwenhuis, R., A. Oliveras, and C. Tinelli, “Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL (T),” Journal of the ACM, vol. 53 (2006), pp. 937–77.
• [24] Savitch, W. J., “Relationships between nondeterministic and deterministic tape complexities,” Journal of Computer and System Sciences, vol. 4 (1970), pp. 177–92.
• [25] Schild, K., “A correspondence theory for terminological logics: Preliminary report,” pp. 466–71 in Proceedings of the 12th International Joint Conference on Artificial Intelligence (IJCAI’91), Kaufmann, San Francisco, Calif., 1991.
• [26] Sofronie-Stokkermans, V., “Locality and subsumption testing in $\mathcal{EL}$ and some of its extensions,” pp. 315–39 in Advances in Modal Logic, vol. 7, edited by C. Areces and R. Goldblatt, College Publications, London, 2008.
• [27] Suntisrivaraporn, B., F. Baader, S. Schulz, and K. Spackman, “Replacing SEP-triplets in SNOMED CT using tractable description logic operators,” pp. 287–91 in Proceedings of the 11th Conference on Artificial Intelligence in Medicine (AIME’07), edited by R. Bellazzi, A. Abu-Hanna, and J. Hunter, vol. 4594 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
• [28] Wolter, F., and M. Zakharyaschev, “Undecidability of the unification and admissibility problems for modal and description logics,” ACM Transactions on Computational Logic, vol. 9 (2008), no. 25.