Joining Gödel and Zadeh Fuzzy Logics in Fuzzy Description Logics

Abstract

Ontologies have succeeded as a knowledge representation formalism in many domains of application. Nevertheless, they are not suitable to represent vague or imprecise information. To overcome this limitation, several extensions to classical ontologies based on fuzzy logic have been proposed. Even though different fuzzy logics lead to fuzzy ontologies with very different logical properties, the combined use of different fuzzy logics has received little attention to date. This paper proposes a fuzzy extension of the Description Logic [Formula: see text] — the logic behind the ontology language OWL 2 — that joins Gödel and Zadeh fuzzy logics. We analyze the properties of the new fuzzy Description Logic in order to provide guidelines to ontology developers to exploit the best features of each fuzzy logic. The proposal also considers degrees of truth belonging to a finite set of linguistic terms rather than numerical values, thus being closer to real experts' reasonings. We prove the decidability of the combined logic by presenting a reasoning preserving procedure to obtain a crisp representation for it. This result is generalized to offer a similar reduction that can be applied when any other finite t -norms, t -conorms, negations or implications are considered in the logic.

Reasoning in Expressive Description Logics under Infinitely Valued Gödel Semantics

“…The details of these concepts are explained in Section 4. In contrast to the reductions for finitely valued Gödel FDLs [6,7], our reduction does not produce an exponential blowup in the nesting depth of concepts in the input ontology.…”

“…This differs from previous reductions for finitely valued FDLs [7,9,31] in that we not only consider cut-concepts of the form p C with p ∈ V O , but also relationships between different concepts. 8 We use the abbreviations α β := β α , α < β := ¬ α β , and similarly for = and >.…”

“…Note that we use the involutive negation to define at-most restrictions. In [7], they are defined using the residual negation: n s.C := ( (n + 1) s.C) → ⊥. This has the effect that the value of n r.C is always either 0 or 1 (see the semantics below).…”

“…We can simulate other common role axioms in G-SROIQ [7,22] Other common reasoning problems for FDLs, such as concept satisfiability and subsumption can be reduced to consistency [12]: the subsumption between C and D to degree q w.r.t. a TBox T and an RBox R is equivalent to the inconsistency of ({ a:C → D < q }, T , R), and the satisfiability of C to degree q w.r.t.…”

A Fuzzy Extension to the OWL 2 RL Ontology Language

“…Important examples are, extensions to tableaux algo-G. Stoilos et al rithms [11,17], implemented in the system FiRE [18], tableaux coupled with mixed integer linear programming algorithms [19,20], implemented in the system fuzzyDL [8], algorithms that reduce fuzzy DLs to crisp DLs [21,22,23], implemented in the DeLorean system [24], SMT-based methods [25], and the direct crispification method [26]. In addition, several optimisations have also been proposed [18,27].…”