SAT Encoding of Unification in $\mathcal{EL}$

Extending Unification in $\mathcal{EL}$ to Disunification: The Case of Dismatching and Local Disunification

“…In this way, we reduce dismatching to local disunification. We then provide two different NP-algorithms for the latter problem by extending the rule-based unification algorithm from [1] and adapting the SAT encoding of unification problems from [2]. These algorithms are more efficient than the brute-force "guess and then test" procedure on which our argument for Fact 3.3 was based.…”

“…Once no more rules can be applied, we obtain a flat disunification problem Γ of which the extended substitution γ is a (possibly non-local) solution. To obtain a local solution, we denote by At, Var, and At nv the sets as defined in Section 3 and define the assignment S induced by γ as in [2]:…”

“…We thus developed a goal-oriented unification algorithm for EL, which is more efficient since nondeterministic decisions are only made if they are triggered by "unsolved parts" of the unification problem. Another option for obtaining a more efficient unification algorithm is a translation to satisfiability in propositional logic (SAT): in [2] it is shown how a given EL-unification problem Γ can be translated in polynomial time into a propositional formula whose satisfying valuations correspond to the local unifiers of Γ. Intuitively, a unifier of two EL concepts proposes definitions for the concept names that are used as variables: in our example, we know that, if we define Head_injury as Injury ⊓ ∃finding_site.Head and Severe_finding as ∃severity.Severe, then the two concepts (1.1) and (1.2) are equivalent w.r.t.…”

Recent Advances in Unification for the EL Family

“…Previous algorithms for EL-unification w.r.t. the empty ontology were based on an even simpler characterization of subsumption that only had to take into account the structure of the compared concept descriptions [8,9,10]. Each of the following algorithms is based on one of those earlier algorithms and generalizes it using one of the characterizations from [5] and [6].…”

“…A satisfying propositional valuation of these clauses yields an acyclic assignment S, and thus a local substitution, in the following way: S X contains all non-variable atoms D for which [X D] is true. To account for ELH R + -ontologies, the original reduction from [9] was modified in [6] using the mentioned characterization of subsumption. More precisely, the clauses encoding the properties of subsumption were extended to allow to take GCIs and role inclusions into account.…”