Abstract
We introduce an extension to Description Logics that allows us to use prototypes to define concepts. To accomplish this, we introduce the notion of prototype distance functions (pdfs), which assign to each element of an interpretation a distance value. Based on this, we define a new concept constructor of the form \(P_{\sim n}(d)\) for \({\sim }\in \{<,\le ,>,\ge \}\), which is interpreted as the set of all elements with a distance \({}\sim n\) according to the pdf d. We show how weighted alternating parity tree automata (wapta) over the non-negative integers can be used to define pdfs, and how this allows us to use both concepts and pointed interpretations as prototypes. Finally, we investigate the complexity of reasoning in \(\mathcal {ALCP} (\text {wapta})\), which extends the Description Logic \(\mathcal {ALC}\) with the constructors \(P_{\sim n}(d)\) for pdfs defined using wapta.
A. Ecke—Supported by DFG in the Research Training Group QuantLA (GRK 1763).
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Notes
- 1.
Recall that a pointed interpretation is an interpretation together with an element of the interpretation domain.
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Baader, F., Ecke, A. (2016). Reasoning with Prototypes in the Description Logic \({\mathcal {ALC}}\) Using Weighted Tree Automata. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_5
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