Finite Lattices Do Not Make Reasoning in ALCI Harder

We consider the fuzzy description logic $${\mathcal {ALCOI}}$$i¾?with semantics based on a finite residuated De Morgan lattice. We show that reasoning in this logic is ExpTime-complete w.r.t. general TBoxes. In the sublogics $${\mathcal {ALCI}}$$ and $${\mathcal {ALCO}}$$, it is PSpace-complete w.r.t. acyclic TBoxes. This matches the known complexity bounds for reasoning in classical description logics between $${\mathcal ALC} $$ and $${\mathcal {ALCOI}}$$.

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