We prove that validity and satisfiability of assertions in the Fuzzy Description Logic based on infinite-valued Product Logic with universal and existential quantifiers (which are non-interdefinable) is decidable when we only consider quasi-witnessed interpretations.
As a first step we will expose Hájek's results on decidability of assertion satisfiability in Fuzzy Description Logic based on Lukasiewicz Logic and explain why the same method does not apply to the product case.
Subsequently we will give an algorithm that reduces the problem of satisfiability of assertions in our Fuzzy Description Logic (restricted to quasi-witnessed interpretations) to a semantic consequence problem, with finite number of hypotheses, on infinite-valued propositional Product Logic.
Since in Fuzzy Description Logics the interesting semantics is the one having the unit real interval [0,1] as a domain, we will prove that, in the cases of validity and positive satisfiability, the restriction to quasi-witnessed models is not necessary and point out that whether the same result holds for 1-satisfiability is a problem that we leave open.