If S is r-satisfiable w.r.t. ι and O, then there is an r-

We first show that the satisfaction of a witness query for a CQ α ∈ Q_φ implies the satisfaction of α.

Lemma 4.10. Let α∈ Q_φ, ψ be a witness query for α, and I be a model of O.

Then, I |=ψ implies I |=α.

Proof. Let π be the homomorphism of ψ into I, and consider first the case that ψ is a tree witness query w.r.t. some tree witness f and B ∈Con(α,f). We define a homomorphism π⁰ of α into I using an auxiliary mapping π⁰: range(f) → ∆^I, for which we then set π⁰(y) := π⁰(f(y)) for all y ∈ N_V(ψ). We start by defining π⁰(%) := π(y_%) for all % ∈ range(f) that are not rigidly witnessed in ψ. Observe that if % is rigidly witnessed in ψ, then the variable y_% does not occur in ψ;

furthermore, % = is never rigidly witnessed and hence at least y must occur in ψ.

We defineπ⁰ on the remaining elements%∈range(f) by induction on the structure of f. We first consider the case that % is directly rigidly witnessed, i.e., no prefix of % is rigidly witnessed.

• If %= (R,C), then there must be a B∃R⊆ BC_R(O) with O |=d

B∃R v ∃R and B∃R(y)⊆ ψ. Since π is a homomorphism of ψ into I and I |=O, we get π⁰() =π(y)∈(d

B∃R)^I ⊆(∃R)^I. Hence, there must exist an element e∈∆^I with (π⁰(), e)∈R^I. We set π⁰(R,C) := e.

• If % = %₁·(R₁,C₁)·(R₂,C₂), then there is a set B∃R2 ⊆ BC_R(O) such that O |=dB_∃R2 v ∃R₂ andB_∃R2(y_%1·(R1,C1))⊆ψ. As above, we can defineπ⁰(%) to be theR-successor ofπ⁰(%₁·(R₁,C₁)) that must exist inI because of the above conditions.

For the induction step, let %=%₁ ·(R₁,C₁)·(R₂,C₂) be such that %₁·(R₁,C₁) is rigidly witnessed andπ⁰(%·(R₁,C₁)) has already been defined. By the construction above, we know thatπ⁰(%·(R₁,C₁))∈(∃R⁻₁)^I. By the last condition ofCon(α,f), there must be an R₂-successor of π⁰(%·(R₁,C₁)) in I, which we can choose as π⁰(%).

Consider now any concept atom A(y)∈α.

• If f(y) = , then we know that either (i) A ∈ N_RC and A(y) ∈ ψ, or (ii) O |=d

B|_R |=A. In both cases, it follows that π⁰(y) =π(y)∈A^I.

• If f(y) = % ·(R,C), then either (i) f(y) is rigidly witnessed in ψ⁰, or (ii) there is a set B_A ⊆ BC_R(O) with O |= d

B_A v A and B_A(y_f(y)) ⊆ ψ. In the second case, we get π⁰(y) = π(y_f(y)) ∈ A^I as above. In the first case, the above construction of π⁰ implies that π⁰(y) ∈ (∃R⁻)^I. By the third condition of Con(α,f), we conclude thatπ⁰(y)∈A^I.

For a role atom S(y, y⁰)∈α, by Definition 3.7 we have to distinguish two cases.

• If f(y) = %, f(y⁰) = % · (R,C), and O |= S⁰ v S for some S⁰ ∈ C, then by Definition 4.7 either (i) f(y⁰) is rigidly witnessed in ψ, or (ii) S⁰(y_f(y), y_f(y⁰₎) ∈ ψ. In the first case, we know from the definition of π⁰ that (π⁰(y), π⁰(y⁰)) ∈ R^I, and hence also (π⁰(y), π⁰(y⁰)) ∈ (S⁰)^I ⊆ S^I by the second condition of Definition 3.7. In case (ii), we directly obtain (π⁰(y), π⁰(y⁰)) = (π(y_f(y)), π(y_f(y⁰₎))∈(S⁰)^I ⊆S^I.

• The other case follows dual arguments, exchanging f(y) and f(y⁰), and re-placing S by S⁻.

This shows that π⁰ is a homomorphism of α intoI.

Consider now the second case of Definition 4.7, i.e., there are R ∈ N⁻_R(O), B ⊆ BC_R(O), and a tree witness f for α such that B is a witness of ∃R w.r.t. O, {∃R} ∈ Con(α,f), and ψ = ∃x.B(x). From the first and the last property and the fact thatI |=ψ, we know that I contains an elemente that satisfies∃R. We define π⁰ in analogy to the first case above, and begin by setting π⁰() :=e. We can define the remainder of the homomorphism by induction on range(f) as in the case for the indirectly rigidly witnessed elements above, by treating as if it was rigidly witnessed. Only in the first step, instead ofπ⁰() being an instance of some ∃R⁻₁ in I, we know that it satisfies ∃R, and hence we obtain the required role successors by the second condition of Con(α,f).

Since all elements can be treated as rigidly witnessed, the remaining proof is a special case of the arguments above, except in the case of a concept atomA(y)∈α with f(y) = . But then we know that π⁰(y) ∈ (∃R)^I, and hence π⁰(y) ∈ A^I by the first condition of Con(α,f).

We can now use this lemma to prove the first direction of Lemma 4.9, namely that, ifS is r-satisfiable w.r.t. ιand O, then there is an r-complete tuple w.r.t. S and ι.

For this purpose, let J₁, . . . ,J_k,I₀, . . . ,I_nbe the interpretations over the domain

∆ that exist according to the r-satisfiability of S (cf. Definition 4.2). We as-sume w.l.o.g. that ∆ containsN_Iand that all individual names are interpreted as themselves in all of these interpretations. We first define the tuple (A_R, Q_R, Q^¬_R) as follows:

A_R :={B(a)|a∈N_I(K), B ∈BC_R(O), a^J1 ∈B^J1} ∪ {¬B(a)|a∈N_I(K), B ∈BC_R(O), a^J1 ∈/ B^J1} ∪ {R(a, b)|a, b∈N_I(K), R∈N_RR(O),(a, b)∈R^J1} ∪ {¬R(a, b)|a, b∈N_I(K), R∈N_RR(O),(a, b)6∈R^J1};

Q_R :={α_j ∈ Q_φ|X ∈ S, p_j ∈X}; and Q^¬_R :={α_j ∈ Q_φ|X ∈ S, p_j 6∈X}.

Obviously,A_Ris an ABox type forO,Q_Rsatisfies Condition (R3), andQ^¬_Rsatisfies Condition (R4).

Our goal is to modify the given interpretations into models of the knowledge bases K_Rⁱ, but need to ensure that the individuals of the form ax are always interpreted by the same elements of the common domain. For this purpose, we consider the canonical modelsIKα,α∈Q_R, for the KBsK_α :=hO,A_αi, whereA_α is obtained by instantiating all variables x in α by the corresponding individual names a_x ∈ N^aux_I . The goal is to add, to each I_i (J_i) with p_j ∈ X_ι(i) (p_j ∈ X_i), the whole interpretation I_αj, which includes a homomorphic image of α_j that involves the same domain elements (from NI(K)∪N^aux_I ) in each interpretation.

Note that K_α is consistent since α ∈ Q_R, and hence there must be one I_i (J_i) that satisfies α and thus is a model of K_α.⁶

For those interpretationsI_i (J_i) that do not satisfy α, we nevertheless know that they satisfy its rigid consequences CO(α) since all these interpretations respect the rigid names andαis satisfied by at least one of them. Hence, we can also add the rigid consequences ofI_Kα to every interpretation without losing the property that they satisfy O (and potentially A_i). For this purpose, we consider similar

6If two variables are mapped by the homomorphism to the same domain element, we obtain a model respecting the UNA by creating a copy of this element that satisfies exactly the same concept names and participates in the same role connections as the original element.

interpretationsI_K^R_α, which are inductively defined as in Definition 3.1, but starting instead with the following interpretation for all symbols X ∈N_C∪N_R:

X⁰ :=







X^IKα if X ∈N_RC∪N_RR,

∅ otherwise.

This means that they must behave exactly as IKα on the rigid names, but the interpretation of the flexible names contains only those tuples that are implied by the rigid information. Each I_K^R_α is a model ofO andCO(α) (see Proposition 3.2).

Note that the domains of I_K^R_α and IKα coincide since all elements c_% that would be created by the iteration in Definition 3.1 for I_K^R_α have already been created by the one for IK_α, and hence are already contained in the initial interpretation above.

The crucial properties of the above introduced interpretations are the following:

• IK_αj can be homomorphically embedded into eachI_i (J_i) for which we have p_j ∈X_ι(i) (p_j ∈X_i) since there must be a homomorphism ofα_j into I_i (J_i) and this entails the existence of domain elements that must satisfy at least the symbols satisfied by the elements of IK_αj.

• I_K^R

αj can be homomorphically embedded into each I_i (J_i), because there must be some J_x, 1 ≤ x ≤ k, that satisfies α_j and hence its rigid conse-quences (i.e., those of IK_αj) are satisfied in all of these interpretations.

These facts imply that it is safe to add these new interpretations to I_i (J_i), as they already contain elements that behave in exactly the same way.

We now modify the interpretationsIi(Ji) into models I_i⁰ (J_i⁰) ofKⁱ_R(Kⁿ⁺ⁱ_R ), with the exception of A_RF, as follows:

• The common domain ∆ is extended by the union of the domains of IKα, α∈Q_R (which are also the domains ofI_K^R_α). Note that these domains may overlap in N_I.

• The individual names from N^aux_I are interpreted as themselves.

• For each j, 1 ≤j ≤m, and I_i (J_i) with p_j ∈X_ι(i) (p_j ∈X_i), we interpret all symbols on the domain ofIK_αj exactly as inIK_αj. Note that there are no role connections between the old and the new domains except betweenNI(K) and N^aux_I .

• If p_j ∈X_ι(i) (p_j ∈X_i), we interpret all symbols as inI_K^R

αj. We then have the following:

• All interpretationsI_i⁰ (J_i⁰) satisfyOand A_R since the new domain elements do not exhibit new behavior that was not already present inI_i (J_i) and the interpretation of basic concepts on the elements of N_I does not change.

• Each I_i⁰ still satisfies A_i because of the same reason.

• Each interpretation is a model of A_QR since that ABox consists exactly of the ABoxesC_O(α),α∈Q_R, which are satisfied by the new domain elements because of I_K^R_α, which is contained in IK_α.

• Each I_i⁰ (J_i⁰) satisfies AQ_ι(i) (AQi) since they consist exactly of the ABoxes A_αj with p_j ∈X_ι(i) (p_j ∈X_i), which are satisfied by IK_αj.

• For each I_i (J_i) and p_j ∈X_ι(i) (p_j ∈X_i), we know thatI_i⁰ 6|=α_j (J_i⁰ 6|=α_j) since any homomorphism of α_j intoI_i⁰ (J_i⁰) would allow us to find one into I_i (J_i) as well, which contradicts the assumption thatI_i |=χ_ι(i) (J_i |=χ_i).

Hence, we know that I_i⁰ |=χ_ι(i) (J_i⁰ |=χ_i).

• All interpretationsI₀⁰, . . . ,I_n⁰ and J₁⁰, . . . ,J_k⁰ respect the rigid names.

If φ is not r-simple, then we also have to define RF and extend the above inter-pretations to models of A_RF:

RF :={∃S(b)|S ∈NR(O)\NRR, b∈NI(K)∪N^aux_I ,

i∈ {0, . . . , n+k}, hO,A_R∪ A_QR∪ A_Qι(i) ∪ A_ii |=∃S(b)}.

Since the interpretations I_i⁰ (J_i⁰) are models of these knowledge bases, for each

∃S(b)∈RF we know that one of these models satisfies the assertion. Hence, the rigid consequences described in A_RF must already be satisfied by some domain elements in the common domain (in all of these interpretations). We can define the interpretation of the elements a_b% ∈N^tree_I accordingly, but again may have to copy elements if the UNA would be violated otherwise. Note that Condition (R6) is obviously satisfied by our definition of R_F. These modified interpretations I_i⁰ (J_i⁰) prove now that Condition (R1) is also satisfied.

Regarding Condition (R2), assume that there are an index i, 0 ≤ i ≤ n (or n + 1 ≤ i ≤ n +k), and p_j ∈ X_ι(i) such that Kⁱ_R |= α_j, and thus I_i⁰ |= α_j (J_i−n⁰ |=α_j). But this contradicts the fact that I_i⁰ |=χ_ι(i) (J_i−n⁰ |=χ_i−n).

The proof of Condition (R5) is also by contradiction. We assume that there arei, 0≤i≤n (n+ 1≤i≤n+k), a tree-shaped CQαj ∈Q^¬_R, and a witness query ψ for α_j w.r.t. K_Rⁱ such that Kⁱ_R |= ψ, and thus I_i⁰ |= ψ (J_i−n⁰ |= ψ). However, α_j ∈ Q^¬_R yields that there is an X_x ∈ S, 1 ≤ x ≤ k, such that p_j 6∈ X_x, and thus J_x⁰ 6|=αj. By Lemma 4.10, we know that J_x⁰ 6|=ψ. But this contradicts the facts that ψ contains only rigid names and J_x⁰ and I_i⁰ (J_i−n⁰ ) respect the rigid names.

This concludes the proof of the first direction of Lemma 4.9.

4.2.4 If there is an r-complete tuple w.r.t. S and ι, then S is