Abstract
Unification in the Description Logic (DL) \(\mathcal {FL}_0\) is known to be ExpTime-complete and of unification type zero. We investigate whether a lower complexity of the unification problem can be achieved by either syntactically restricting the role depth of concepts or semantically restricting the length of role paths in interpretations. We show that the answer to this question depends on whether the number formulating such a restriction is encoded in unary or binary: for unary coding, the complexity drops from ExpTime to PSpace. As an auxiliary result, we prove a PSpace-completeness result for a depth-restricted version of the intersection emptiness problem for deterministic root-to-frontier tree automata. Finally, we show that the unification type of \(\mathcal {FL}_0\) improves from type zero to unitary (finitary) for unification without (with) constants in the restricted setting.
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Notes
- 1.
For unary coding, the size of the input k is the number k, whereas for binary coding it is the size of its binary encoding, i.e., \(\log k\).
- 2.
Intuitively, \(\rho \) is the number of different role names occurring in the unification problem and each letter \(i, 1\le i\le \rho \), stands for a role name \(r_i\).
- 3.
- 4.
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Acknowledgements
Franz Baader was partially supported by DFG TRR 248 (cpec, grant 389792660), Oliver Fernández Gil by DFG in project number 335448072, and Maryam Rostamigiv by a DAAD Short-Term Grant, 2021 (57552336). The authors should like to thank Patrick Koopmann for determining the maximal role depth of concepts in ontologies from Bioportal 2017 and in SNOMED CT.
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Baader, F., Gil, O.F., Rostamigiv, M. (2021). Restricted Unification in the DL \(\mathcal {FL}_0\). In: Konev, B., Reger, G. (eds) Frontiers of Combining Systems. FroCoS 2021. Lecture Notes in Computer Science(), vol 12941. Springer, Cham. https://doi.org/10.1007/978-3-030-86205-3_5
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