On the (un)decidability of fuzzy description logics under Łukasiewicz 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 image with GCIs does not have the finite model property under Łukasiewicz 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 Łukasiewicz Logic. We additionally provide a decision algorithm for acyclic image knowledge bases under Łukasiewicz 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 image knowledge bases under Łukasiewicz Logic, none of them exhibit formal proofs of their correctness and completeness, which is the additional contribution here.