Unification in the Description Logic EL

Abstract

Abstract. The Description Logic E L has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, E L is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in E L is decidable. More precisely, E L-unification is NP-complete, and thus has the same complexity as E L-matching. We also show that, w.r.t. the unification type, E L is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.

Unification in modal and description logics

“…The description logic FL 0 has the constructors top concept, conjunction, and value restriction, whereas in EL value restriction is replaced by existential restriction. Equivalence in these two logics can be axiomatized by the following equational theories [13,11,72]. For FL 0 one takes the equational axioms stating associativity, commutativity, and idempotency of , the fact that is a unit for , and the additional axioms ∀r.…”

“…The exact complexity of deciding the solvability of a general Boolean unification problem was determined in [5]: it is PSpace-complete. 11 This complexity result is based on results from [15] obtained in the context of combining decision procedures for unification in the union of equational theories over disjoint signatures.…”

“…In this setting, it frequently happens that the same (intuitive) concept is introduced several times, often with slightly differing descriptions. Testing for equivalence of concepts is not always sufficient to find out 11 In contrast, this problem is NP-complete for elementary Boolean unification problems and Π p 2 -complete for Boolean unification problems with constants.…”

“…the inverse subsumption order : given ELconcept descriptions C, D, we define C > is D iff C < D. In [11] it is shown that the strict order > is defined this way is well-founded. Consequently, its multiset extension > m is also well-founded [14].…”

“…In addition, it obviously always terminates. In [11], completeness is shown by proving that, for every solvable flat EL unification problem Γ, there is a way of guessing in Step 1 subsets S X of the non-variable atoms of Γ such that the depends on relation determined in Step 2 is acyclic and the substitution σ computed in…”