Many-valued Horn Logic is Hard

Fuzzy Description Logics (FDLs) have been introduced to reason about vague or imprecise knowledge in application domains. In recent years, reasoning in many FDLs based on infinitely many values has been proved to be undecidable [3,15] and systematic studies have been undertaken on this topic [8]. On the other hand, every finite-valued FDL that has been studied in the recent literature has not only been proved to be decidable, but even to belong to the same complexity class as the corresponding crisp DL [6,7,11,12]. A question that naturally arises is whether the finite-valued fuzzy framework is not more complex (w.r.t. computational complexity) than the crisp-valued formalism in general. A common opinion is that everything that can be expressed in finite-valued FDLs can be reduced to the corresponding crisp DLs without any serious loss of efficiency. Indeed, although some known translations of finite-valued FDLs into crisp DLs are exponential [5], more efficient reasoning can be achieved through direct algorithms. The fact that a significant difference in computational complexity between the crisp and the finite-valued case has not yet been found is mainly due to the high expressivity of the languages studied so far. Indeed, these languages already contain significant sources of nondeterminism in the crisp case. Our idea is that the proof of a possible difference in the complexity between both formalisms has to be searched in languages that allow for a lower expressivity. In such languages, the sources of nondeterminism inherent in the classical framework have not yet shown up, while the same languages may be already affected by the inherent nondeterminism of the basic logical connectives in the finite-valued framework. For this purpose, we want to take advantage of the “revival” that simple DL