Abstract
We define and study crisp bisimulation and strong bisimilarity for fuzzy description logics (DLs) with involutive negation under the Gödel semantics. The considered logics are fuzzy extensions of the DL \(\mathcal {ALC}_{reg}\) with involutive negation and additional features among inverse roles, nominals, (qualified or unqualified) number restrictions, the universal role and local reflexivity of a role. We give results on invariance of concepts under crisp bisimulations as well as conditional invariance of TBoxes and ABoxes under strong bisimilarity in the mentioned fuzzy DLs. We also provide a theorem on the Hennessy-Milner property of crisp bisimulations in those logics. Furthermore, we also present our results on minimizing fuzzy interpretations by using strong bisimilarity.
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Notes
- 1.
In [12], \(\varPhi \)-bisimulation and \(\varPhi \)-bisimilarity are named \(\mathcal {L}_\varPhi \)-bisimulation and \(\mathcal {L}_\varPhi \)-bisimilarity, respectively. Here, we simplify the terms.
- 2.
Formally, the quotient fuzzy interpretation of \(\mathcal {I}\) w.r.t. the equivalence relation \({\mathop {\sim }\limits ^{.}}_{\varPhi ,\mathcal {I}}\) should be denoted by \(\mathcal {I}/_{{\mathop {\sim }\limits ^{.}}_{\varPhi ,\mathcal {I}}}\). We use \(\mathcal {I}/_{{\mathop {\sim }\limits ^{.}}_\varPhi }\) instead to simplify the notation.
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Nguyen, L.A., Nguyen, N.T. (2019). Bisimulations for Fuzzy Description Logics with Involutive Negation Under the Gödel Semantics. In: Nguyen, N., Chbeir, R., Exposito, E., Aniorté, P., Trawiński, B. (eds) Computational Collective Intelligence. ICCCI 2019. Lecture Notes in Computer Science(), vol 11683. Springer, Cham. https://doi.org/10.1007/978-3-030-28377-3_2
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