Conjunctive Reasoning on Fuzzy Taxonomies with Order-Sorted Feature Logic

Similarity-Based Reasoning With Order-Sorted Feature Logic

“…The reader is referred to [24] for more details about OSF logic and its semantics. Fuzzy OSF logic [30] generalizes OSF logic by considering a fuzzy subsumption relation between sorts and by interpreting sorts as fuzzy sets. More precisely, the set S of sorts is ordered by a fuzzy lattice 3 • : S × S → [0, 1], and the meaning of a sort s ∈ S in an interpretation I = (∆ I , • I ) is a fuzzy set s I : ∆ I → [0, 1], where ∆ I is the domain or universe of the interpretation.…”

“…Analogously, the fuzzy sort subsumption relation can be extended to a fuzzy subsumption between OSF terms, which is a fuzzy lattice on (equivalence classes of) OSF terms [30].…”

“…With respect to computational complexity, the GLB of two OSF terms in the fuzzy subsumption lattice can be determined via their unification, through a constraint normalization procedure which is essentially the same as the one for crisp OSF logic [30]. The only difference involves the rule Sort Intersection, which in the fuzzy setting relies on the computation of the GLB of two sorts in a fuzzy lattice rather than a crisp one.…”

“…In fuzzy OSF logic, the unifier of two OSF terms is also associated with a unification degree, whose computation requires O(m(|S|+e)) time, where m = |Tags(ψ 1 ) ∪ Tags(ψ 2 )| and e is the number of edges in the graph representation of the fuzzy sort subsumption relation [30]. This computation depends on determining the subsumption degree •(s, s ) between pairs of sorts.…”

“…Intuitively, the fuzzy subsumption relation encodes the information of both the crisp subsumption and the similarity. This procedure shifts the setting to that of OSF logic with a fuzzy sort subsumption relation, or fuzzy OSF logic, whose semantics has already been developed in [30]. The advantage is that in fuzzy OSF logic it is possible to apply the same unification algorithm of (crisp) OSF logic, with essentially the same computational cost, thereby retaining the efficiency inherent to this logical framework.…”