Proofs of results from Section 4

Now we use the technique developed in the previous section in order to prove the claims made in Section 4.

Throughout this section we use the fact that if a type is realised then it is realised in an at most countable model (and similarly, if a set of types is precisely realisable then it is precisely realisable in an at most countable model).

Theorem 20. The following conditions are equivalent for TBoxes T1and T2inDL-Lite^N_booland a signatureΣ:

(ceb) T1Σ-concept entails T2inDL-Lite^N_bool;

(r) every T1-realisableΣQT₁∪T₂-type is T2-realisable.

Proof. (ce_b) ⇒ (r) Suppose that t is a T₁-realisableΣQT1∪T2-type which is not T₂-realisable. Then T₂|= d_C∈tC v ⊥ but T₁ 6|= d_C∈tC v ⊥, contrary to T₂beingΣ-concept entailed by T1.

(r) ⇒ (ceb) Suppose otherwise. Then there is C1 v C2 with sig(C1 v C2) ⊆ Σ such that T2 |= C1 v C2 and T₁ 6|= C1 v C2. Take the uniform interpolant T2Σof T2 with respect toΣ provided by Theorem 73. As T2 and T2Σ

areΣ-concept inseparable, we can find C⁰₁ v C⁰₂ ∈ T_2Σsuch that T1 6|= C⁰₁ v C⁰₂. And as C⁰₁ v C₂⁰ is aΣQT₂-concept inclusion in DL-Lite^N_bool, we can find a T₁-realisableΣQT1∪T2-type that is not T₂-realisable. Indeed, let I be a model of T₁with a point x such that x ∈ (C⁰₁u ¬C⁰₂)^I. Then theΣQT1∪T2-type realised at x in I is not T₂-realisable. q

Theorem 21. The following conditions are equivalent for TBoxes T1and T2inDL-Lite^N_booland a signatureΣ:

(sceb) T1stronglyΣ-concept entails T2inDL-Lite^N_bool; (qe_b) T₁Σ-query entails T2inDL-Lite^N_bool;

(sqeb) T1stronglyΣ-query entails T2inDL-Lite^N_bool;

(pr) every precisely T1-realisable set ofΣQT₁∪T₂-types is precisely T2-realisable.

40

Proof. The implications (sqe_b) ⇒ (qe_b) and (sqe_b) ⇒ (sce_b) follow immediately from definitions.

(pr) ⇒ (sqeb) Suppose there are aΣ-TBox T , Σ-ABox A and Σ-query^q(a) in DL-Lite^N_boolwith (T2∪T, A) |=^q(a) and (T1∪ T, A) 6|=^q(a). Take a model I1of (T1∪ T, A) such that I1 6|=^q(a) and letΞ be the set of ΣQT₁∪T₂-types realised in I1. By (pr),Ξ is precisely T2-realisable. Then, by Lemma 81, there exists a model I^∗ of T2 such that I^∗∼_ΣI^ω₁, and so I^∗|= (T , A) and I^∗6|=^q(a), contrary to (T2∪ T, A) |=^q(a).

(qeb) ⇒ (pr) LetΞ be the set of ΣQT₁∪T₂-types realised in a model I of T1, and let A_Ξ= {C(at) | C ∈ t, t ∈Ξ}, where at is a fresh object name for each t ∈ Ξ. It follows that I |= (T1, A_Ξ). Suppose that Ξ is not precisely T₂-realisable. Then two cases are possible:

1. If, for every model I⁰of T2, there is some t ∈Ξ that is not realised in I⁰, i.e., I⁰6|= A_Ξ, then consider the query q= ⊥: we have (T2, A_Ξ) |=^qbut (T1, A_Ξ) 6|=^q, which is a contradiction.

2. Otherwise, every model of T2must realise aΣQT₁∪T₂-type that is not inΞ. Let Θ be the set of all T2-realisable ΣQT₁∪T₂-types that are not inΞ. As Θ , ∅, we can take^q= ∃x Wt∈ΘV

C∈tC(x). Then we have (T2, A_Ξ) |=^qbut (T1, A_Ξ) 6|=^q, which is again a contradiction.

(sceb) ⇒ (pr) LetΞ be a set of precisely T1-realisableΣQT₁∪T₂-types, and let T_Ξ= > v Ft∈Ξd

C∈tC . Then, for every t ∈Ξ, we have T1∪ T_Ξ6|= dC∈tC v ⊥. Therefore, by (sceb), T2∪ T_Ξ6|= dC∈tC v ⊥, and thus there is a model I_tof T2∪ T_Ξrealising t. Clearly,L

t∈ΞI_tis a model of T2precisely realising the types inΞ. q Theorem 26. For any TBoxes T₁and T2inDL-Lite^N_hornand any signatureΣ, the following conditions are equivalent:

(qeh) T1Σ-query entails T2inDL-Lite^N_horn;

(spr) every precisely T1-realisable set ofΣQT₁∪T₂-types is sub-precisely T2-realisable.

Proof. (spr) ⇒ (qeh) Suppose there are aΣ-ABox A and a Σ-query^q(a) in DL-Lite^N_hornsuch that (T2, A) |=^q(a) and (T1, A) 6|=^q(a). Let I1 be a model of (T1, A) with I1 6|=^q(a), and letΞ be the set of ΣQT₁∪T₂-types realised in I1. By (spr),Ξ is sub-precisely T2-realisable. By Lemma 82, there is a model I^∗of T2and aΣ-homomorphism from I^∗ onto I1. But then I^∗|= A and I^∗6|=^q(a), from which (T2, A) 6|=^q(a), contrary to our assumptions.

(qeh) ⇒ (spr) LetΞ be the set of ΣQT₁∪T₂-types realised in a model I of T1, and let A_Ξ= {B(at) | B ∈ t⁺, t ∈ Ξ}, where at is a fresh object name for each t ∈ Ξ. It follows that I |= (T1, A_Ξ). Suppose thatΞ is not sub-precisely T₂-realisable. Then two cases are possible:

1. If, for every model I⁰of T2, there is t ∈Ξ with (dB∈t⁺B)^I0 = ∅, i.e., I⁰ 6|= A_Ξ, then consider the queryq= ⊥:

we have (T1, A_Ξ) 6|=^qbut (T2, A_Ξ) |=^q, which is a contradiction.

2. Otherwise, every model I⁰of T2 satisfying A_Ξmust realise aΣQT₁∪T₂-type that is not positively contained in any type fromΞ. Let Θ be the set of all such ΣQT₁∪T₂-types. AsΘ , ∅, we can take^q = ∃x Wt∈ΘV

B∈t⁺B(x).

Then (T2, A_Ξ) |=^qbut (T1, A_Ξ) 6|=^q, which is again a contradiction.

This completes the proof of the theorem. q

The following model-theoretic property of TBoxes in DL-Lite^N_hornis standard in Horn logic; see, e.g., [9].

Lemma 83. Let T be a TBox in DL-Lite^N_horn and t a T -realisable ΣQ-type. Then there exists a countable model J_T(t) of T such that J_T(t) realises t and, for every model I of T realising t, there exists aΣ-homomorphism from J_T(t) to I.

In what follows we fix some model JT(t) mentioned in the formulation of the lemma and call it the minimal model of T realising t.

Given a realisable setΞ of ΣQ-types, let the TBox TΞcontain all theΣQ-concept inclusions B1u · · · u B_kv B

in DL-Lite^N_hornsuch that B ∈ t⁺whenever B₁, . . . , B_k∈t⁺, for all t ∈Ξ. We will call T_Ξthe TBox induced byΞ. Note that (i) if, for distinctΣQ-concepts B1, . . . , B_k, there is no t ∈Ξ with B1, . . . , B_k∈t⁺then B1u · · · u B_kv ⊥ is in T_Ξ, and (ii) if B ∈ t⁺, for all t ∈ Ξ, then > v B ∈ T_Ξ. The following lemma establishes a useful criterion for deciding meet-precise T -realisability of a setΞ in terms of T ∪ TΞ-realisability:

41

Lemma 84. LetΞ be a set of ΣQ-types and t a ΣQ-type. Let Ξt = {ti∈Ξ | t⁺⊆ t⁺_i}. Then t is T_Ξ-realisable if, and only if,Ξt , ∅ and t⁺= Tt_i∈Ξtt⁺_i.

In particular,Ξ is meet-precisely T -realisable, for a TBox T , if, and only if, every type in Ξ is T ∪ T_Ξ-realisable.

Proof. (⇒) IfΞt = ∅ then d_B∈t+B v ⊥ is in T_Ξ, and so t cannot be T_Ξ-realisable. IfΞt , ∅ then, for every B⁰∈T

t_i∈Ξtt⁺_i, we haved

B∈t⁺B v B⁰in T_Ξ. So t⁺⊇T

t_i∈Ξt t⁺_i.

(⇐) If there is no model of T_Ξrealising t, then T_Ξ |= dB∈t⁺B v F

¬B∈t\t⁺B, whence, by Theorem 24, there is

¬B⁰∈ t \ t⁺such that T_Ξ|= dB∈t⁺v B⁰, and sod

B∈t⁺B v B⁰is in T_Ξ. Therefore, B⁰∈ t⁺, which is impossible. q

Lemma 85. LetΞ be a T -realisable set of ΣQ-types. If a ΣQ-type t is TΞ-realisable then t is T -realisable.

Proof. By Lemma 84, there are t1, . . . , tk ∈Ξ such that t⁺= Tit⁺_i. As the tiare all realised in some model I of T and the intersection of models is a model for a Horn KB, t is T -realisable. q

We are now in a position to prove the following criterion:

Theorem 27. For any TBoxes T₁and T2inDL-Lite^N_hornand any signatureΣ, the following conditions are equivalent:

(sceh) T1stronglyΣ-concept entails T2inDL-Lite^N_horn; (sqeh) T1stronglyΣ-query entails T2inDL-Lite^N_horn;

(mpr) every precisely T1-realisable set ofΣQT1∪T2-types is meet-precisely T2-realisable.

Proof. The implication (sqeh) ⇒ (sceh) is trivial.

(mpr) ⇒ (sqe_h) Suppose there are a Σ-TBox T , a Σ-ABox A and a Σ-query^q(a) in DL-Lite^N_horn such that (T₂∪ T, A) |=^q(a) but (T₁∪ T, A) 6|=^q(a). Let I be a model of (T₁∪ T, A) with I 6|= ^q(a), and letΞ be the set ofΣQT₁∪T₂-types realised in I. Since I |= T_Ξ, every type inΞ is T_Ξ-realisable. LetΞ^∗be the set of all T_Ξ-realisable ΣQT₁∪T₂-types. Consider

J = I ⊕ M

t∈Ξ^∗

J_TΞ(t).

AsΞ is T1∪ T -realisable, by Lemma 85, every T_Ξ-realisable type is T1∪ T -realisable, and so J |= T1∪ T . Clearly, we have J |= A and, as there is a Σ-homomorphism from J onto I, J 6|= ^q(a). Also, observe that J precisely realisesΞ^∗: indeed, J realises every T_Ξ-realisableΣQT₁∪T₂-type and, conversely, everyΣQT₁∪T₂-type realised in J is T_Ξ-realisable. By (mpr) and Lemma 84, there exists a model I⁰of T2realising all the types inΞ^∗and such that eachΣQT₁∪T₂-type realised in I⁰is T_Ξ^∗-realisable. In fact, I⁰realises precisely the setΞ^∗: sinceΞ^∗is T_Ξ-realisable, by Lemma 85, every T_Ξ^∗-realisable type t is T_Ξ-realisable, and so t ∈Ξ^∗. We then apply Lemma 81 to J andΞ^∗and find a model I^∗of T2such that I^∗∼_ΣJ^ω. It follows that I^∗is a model of (T2∪ T, A) such that I^∗6|=^q(a), which is a contradiction.

(sceh) ⇒ (mpr) LetΞ be a set of precisely T1-realisableΣQT₁∪T₂-types. Then T1∪ T_Ξ6|= dB∈t⁺B vF

¬B∈t\t⁺B, for each t ∈ Ξ. Therefore, for each t ∈ Ξ and each ¬B⁰ ∈ t \ t⁺, we have T1∪ T_Ξ 6|= dB∈t⁺B v B⁰, whence, by (sceh), T2∪ T_Ξ6|= dB∈t⁺B v B⁰. As an intersection of models of a Horn KB is also a model of this KB, we obtain T₂∪ T_Ξ 6|= dB∈t⁺B vF

¬B∈t\t⁺B, and thus t is T2∪ T_Ξ-realisable, for each t ∈Ξ. Take the disjoint union J of all models Itof T2∪ T_Ξrealising t, for t ∈Ξ. Clearly, J realises all the types in Ξ and each ΣQT₁∪T₂-type realised in J is T2∪ T_Ξ-realisable. Therefore, by Lemma 84,Ξ is meet-precisely T2-realisable. q

42