Consistency reasoning in lattice-based fuzzy Description Logics

Abstract

Fuzzy Description Logics have been widely studied as a formalism for representing and reasoning with vague knowledge. One of the most basic reasoning tasks in (fuzzy) Description Logics is to decide whether an ontology representing a knowledge domain is consistent. Surprisingly, not much is known about the complexity of this problem for semantics based on complete De Morgan lattices. To cover this gap, in this paper we study the consistency problem for the fuzzy Description Logic L-SHI and its sublogics in detail.The contribution of the paper is twofold. On the one hand, we provide a tableaux-based algorithm for deciding consistency when the underlying lattice is finite. The algorithm generalizes the one developed for classical SHI. On the other hand, we identify decidable and undecidable classes of fuzzy Description Logics over infinite lattices. For all the decidable classes, we also provide tight complexity bounds.

The limits of decidability in fuzzy description logics with general concept inclusions

“…This leaves only the Gödel t-norm as a promising candidate for an implementation. As an alternative, one could use many-valued DLs that support only a finite set of truth values, arranged in a residuated lattice or a total order [31,33,34,40,41]. The complexity of reasoning in such logics is often the same as for the underlying classical DLs.…”

“…The reason is that several concept names need to be initialized to different values. However, this shows undecidability of local consistency [40,41], which is a decision problem between concept satisfiability and ontology consistency that asks for a model with an individual that has different degrees for several concepts at the same time. In the presence of fuzzy GCIs, at least the consistency of inequality assertions e 0 :C 1 ≥ p 1 , .…”

“…The idea to reduce consistency to consistency in classical DLs has been used for arbitrary finite chains of truth values with combination functions similar to ordinal sums of the Gödel and Łukasiewicz t-norms, and even for very expressive DLs such as SROIQ [33,34]; however, the reduction often increases the size of the input ontology by an exponential factor. In contrast, tight complexity bounds were shown for consistency and subsumption in fuzzy DLs below SHOI over an arbitrary finite lattice using a combination of automata-based constructions [40,56] and tableaux rules [41].…”

Conjunctive Query Answering in Finitely-Valued Fuzzy Description Logics

“…For example, if the concept Overutilized in O maps each CPU to the degree to which it is overutilized, then the concept Overutilized 0.6 in O C represents the set of CPUs that are overutilized to a degree of at least 0. 6.…”

A new view of relationship between atomic posets and complete (algebraic) lattices

“…Of the three research areas mentioned, order theory undoubtedly is the youngest. In recent years, as order and partial ordered set theory were widely applied in the combinatorics [1,9,13,37,43], fuzzy mathematics [7,32,40,42,44], computer science [2,39], and even in the social science [14,15] etc.…”