5 Reduction to SAT
5.4 Completeness of the Reduction
Given a ground unifier γ of Γ w.r.t.O, we can define a valuationτ that satisfies C(Γ,O) as follows.
Let L ∈ Left and D ∈ Attr and i ∈ {0, . . . ,|T |}. We set τ([L v D]i) := 1 iff s(γ(L))←−(i)O s(γ(D)). According to Corollary 8, we thus have τ([L v D]i) = 0 for all i∈ {0, . . . ,|T |} iff γ(L)6vO γ(D). Otherwise, there is an i∈ {0, . . . ,|T |}
such that τ([LvD]j) = 1 for all j ≥i, and τ([LvD]j) = 0 for all j < i.
To define the valuation of the remaining propositional variables [X > Y] with X, Y ∈ Nv, we set τ([X > Y]) = 1 iff X >γ Y, where >γ is defined as before, i.e., X >γ Y iff γ(X)vO ∃w.γ(Y) for somew∈NR+.
Lemma 29. Let C ∈ At, D ∈ Atnv, and E be a top-level atom of γ(C) with E vsO γ(D). Then Dec(C vD) is evaluated to 1 under τ.
Proof. Consider the following cases:
• If C=D, then Dec(C vD) =>.
• If both C and D are ground, then Dec(C v D) = [C v D]|T | and γ(C) = C =E vO D =γ(D). By Corollary 8, this impliesγ(C)←−−(|T |)O γ(D), i.e., τ([C vD]|T |) = 1.
• IfC is a concept name, then we again haveγ(C)vO E vO γ(D), and thus τ([C vD]|T |) = 1.
• If C = ∃r.C0, then γ(C) is an atom, and thus E = γ(C), D = ∃s.D0, rEO s, and either γ(C0) vO γ(D0) or γ(C0) vO γ(∃t.D0) for a transitive role name t with r EO t EO s. We obtain either τ([C0 v D0]|T |) = 1 or τ([C0 v ∃t.D0]|T |) = 1, i.e., Trans(C vD) is evaluated to 1 under τ.
Lemma 30. The valuation τ satisfies C(Γ,O).
Proof. We consider the formulae introduced in Definition 24.
(I) Since γ is a unifier of Γ w.r.t. O, for every Lv? D in Γ we have γ(L)vO
γ(D). By Corollary 8, this implies γ(L) ←−−(|T |)O γ(D), and thus τ([L v D]|T |) = 1.
(II) 1) If C vO D does not hold for two ground atoms C ∈ At, D ∈ Attr, then γ(C) vO γ(D) does not hold, and thus τ([C v D]i) = 0 for all i∈ {0, . . . ,|T |}.
2) If Y ∈ Nv, B ∈ Atnv, L ∈ Left, and i, j ∈ {0, . . . ,|T |} are such that τ([L v Y]i) = τ([Y vB]j) = 1, then γ(L) ←−(i)O γ(Y) ←−(j)O γ(B), and thus γ(L) ←−−−(i+j)O γ(B). By Lemma 7, there must be a derivation of γ(L) from γ(B) that uses at most min{|T |, i+j} root steps, and thus τ([LvB]min{|T |,i+j}) = 1.
3) LetL∈Left\Nv,D∈Attr, andi∈ {0, . . . ,|T |}satisfyτ([LvD]i) = 1, i.e., γ(L)←−(i)O γ(D).
a) IfDis a ground atom andLis not a ground atom, then by Lemma 11 we have one of the following cases:
• If there is C ∈ L such that E vsO D for some top-level atom E of γ(C), then by Lemma 29 Dec(C vD) is evaluated to 1.
• Otherwise, there must be a GCI A1 u · · · uAk v B in T with γ(L) ←−−−(i−1)O A1, . . . , γ(L)←−−−(i−1)O Ak, and B vsO D. Note that this is only possible withi >0. We then have τ([L vA1]i−1) =
· · ·=τ([LvAk]i−1) = 1 and B vO D.
b) IfDis a non-variable, non-ground atom, then by Lemma 11 we have one of the following cases:
• If there isC ∈ L such that E vsO γ(D) for some top-level atom E of γ(C), then by Lemma 29 Dec(C vD) is evaluated to 1.
• If there is a GCI A1u · · · uAk vB ∈ T with γ(L) ←−(i)O B and B vsO γ(D), then τ([LvB]i) = 1 and Dec(B vD) is evaluated to 1.
(III) 1) Assume that τ([X > X]) = 1 for some variable X ∈ Nv. By the definition of>γ, this implies that γ(X)vO ∃w.γ(X) for somew∈NR+. This contradicts the assumption that O is cycle-restricted.
Moreover, if τ([X > Y]) = τ([Y > Z]) = 1, then γ(X) vO ∃ww0.γ(Z) with w, w0 ∈NR+, and thusτ([X > Z]) = 1.
2) Ifτ([X v ∃r.Y]i) = 1, thenγ(X)←−(i)O ∃r.γ(Y), which impliesγ(X)vO
∃r.γ(Y). By the definition of >γ, we thus haveτ([X > Y]) = 1.
This completes the proof of correctness of the presented reduction.
Theorem 31. Unification w.r.t. cycle-restricted ELHR+-ontologies is an NP -complete problem.
Proof. NP-hardness follows from NP-hardness of unification in EL w.r.t. the empty ontology [5]. The other direction is provided by the presented reduction to the NP-complete SAT problem.
This also shows locality of unification w.r.t. cycle-restricted ELHR+-ontologies, i.e., in this setting a unification problem has a unifier iff it has a local unifier.
Lemma 32. If a flat unification problem Γ has a unifier w.r.t. a flat, cycle-restricted ELHR+-ontology O, then it has a local unifier w.r.t. O.
Proof. If Γ has a unifier w.r.t.O, thenC(Γ,O) is satisfiable by Lemma 30. Thus, Section 5.3 shows that there is a local unifier of Γ w.r.t. O.
6 Conclusions
We have shown that unification w.r.t. cycle-restricted ELHR+-ontologies can be reduced to propositional satisfiability. This improves on the results in [2] in two respects. First, it allows us to deal also with ontologies that contain transitivity and role hierarchy axioms, which are important for medical ontologies. Second, the SAT reduction can easily be implemented and enables us to make use of highly optimized SAT solvers, whereas the goal-oriented algorithm in [1], while having the potential of becoming quite efficient, requires a high amount of additional optimization work. The main topic for future research is to investigate whether we can get rid of cycle-restrictedness.
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