On the (un)decidability of fuzzy description logics under Lukasiewicz t-norm

Recently there have been some unexpected results concerning Fuzzy Description Logics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike the classical case, the DL ALC with GCIs does not have the finite model property under ?ukasiewicz Logic or Product Logic, the proposed reasoning algorithms are neither correct nor complete and, specifically, knowledge base satisfiability is an undecidable problem for Product Logic. In this work, we show that knowledge base satisfiability is also an undecidable problem for ?ukasiewicz Logic. We additionally provide a decision algorithm for acyclic ALC knowledge bases under ?ukasiewicz Logic via a Mixed Integer Linear Programming (MILP) based procedure (note, however, that the decidability of this problem is already known). While similar MILP based algorithms have been proposed in the literature for acyclic ALC knowledge bases under ?ukasiewicz Logic, none of them exhibit formal proofs of their correctness and completeness, which is the additional contribution here.