A Tuple for Testing r-Satisfiability

The first goal is to specify a fixed tuple (A_R[W,BΦ], Q_R[W], Q^¬_R[W], R_F[W,BΦ]) as described in Section 6.1.2: it is based on two given setsW ⊆2^{p1,...,pm}such thatW ={W₁, . . . , Wk} and B_Φ ⊆ {B(a) |B ∈ B(O), a∈N_I(Φ)}, both of constant size, and r-complete iff W is r-satisfiable w.r.t. ι and K. For testing r-completeness, we subsequently also specify the KBs regarded in these tests.

10Note that we follow the approach of [Cal+05] and introduce a relation for every basic concept, instead of only for the concept names.

6.3 First-Order Rewritings of r-Satisfiability

We define the tuple such that the four sets are minimal and satisfy the r-completeness conditions. More precisely, we focus on those parts of the conditions which we can ad-dress without considering information that is not given (e.g., the mapping ι, which depends on the data). While the definition of components such as Q_R[W] and Q^¬_R[W] is straightforward, the construction of the set R_F represents a challenge. This is because the focus on data complexity imposes special limits and Condition (C6) is rather intri-cate. Regarding data complexity, it would be critical to consider the entire set R_F and ABox A_RF within our rewriting. Recall that A_RF may contain auxiliary elements tai-lored to individual elements inNI(K), which then would need to be considered explicitly within the rewriting—this is obviously impossible. To circumvent that, we first of all discern the assertions in R_F more fine-granularly, according to the kind of individual they address. In particular, the setRF is considered to be the disjoint union of three sets R_F|aux,R_F|Φ, andR_F|o(ofor “other”), each containing only assertions on the individuals fromN^aux_I ,N_I(Φ),andN_I(K)\N_I(Φ),respectively (i.e., next to the ones fromN^tree_I ). For each of these sets, we can restrict our tests to parts of Condition (C6):

• R_F|aux is constant and hence its size is not relevant. In particular, the elements of N^aux_I neither occur in the ABox type, nor in the input ABoxes, and can be uniquely associated to one of the CQs in Φ. For constructing R_F|aux in line with Condition (C6), it is thus enough to focus on instantiations of the latter CQs, individually.

• The construction of R_F|o is more involved. The issue with the size of the corre-sponding ABox is covered later in this section. To ensure that Condition (C6) is satisfied, we first model the derivation of all relevant (rigid) basic concept asser-tions, the consequences of the TKB, in our rewriting and then use them to derive those inR_F|o. Observe that the condition requires several ABoxes to be considered.

Since the elements under consideration do not occur in Φ, we can however focus on the ABox type and the input ABoxes. That is, the rigid assertions represent (the relevant part of) the ABox typeA_R[W,BΦ]; the derivation resolves transitivity and thus ensures that A_R[W,BΦ] is in line with Condition (C1). We define the set of these rigid assertions based on the below derivation as B_R|o :=B_R|o^|BR(O)|.

B_R|o⁰ :=∅,

B^j+1_R|o :={B(a)|B ∈BR(O), a∈N_I(K)\N_I(φ),∃i.0≤i≤n, hO,B_R|o^j ∪ A_ii |=B(a)}.

• The setR_F|Φis of constant size. For constructing it, we have to apply a derivation as above since the elements in N_I(Φ) might occur in the input ABoxes. Actually, these elements may occur in all the ABoxes regarded in Condition (C6), which thus all would have to be taken into account in the derivation. However, we can obviously not apply the mapping ι to define a fixed rewriting. This is why we assume a set B_Φ ⊆ {B(a) | B ∈ B(O), a ∈ N_I(Φ)} to be given instead of defining it. The test ifRF|Φsatisfies Condition (C6), in dependence of B_Φ, is thus postponed to the actual application and evaluation of the rewriting.

These observations lead to the following definitions:

Q_R[W]:={α_j ∈ Q_Φ|W ∈ W, p_j ∈W}, Q^¬_R[W]:={α_j ∈ Q_Φ|W ∈ W, p_j 6∈W},

R_F|aux[W]:={∃S(a_y)|S∈N⁻_R(O)\N⁻_RR, a_y ∈N^aux_I ,∃W_j ∈ W, hO,A_Qji |=∃S(a_y)}, RF|Φ[B_Φ]:={∃S(a)∈ B_Φ|S ∈N⁻_R(O)\N⁻_RR},

R_F|o:={∃S(a)|S∈N⁻_R(O)\N⁻_RR, a∈N_I(K)\N_I(Φ),

∃i.0≤i≤n, hO,B_R|o∪ A_ii |=∃S(a)}.

Based on these sets, we define the corresponding auxiliary ABoxes, as in Section 6.1.2:

• A_QR[W] :=^S_ϕ∈Q

R[W]C_O(ϕ).

• A_Qj for j∈[1, k] is defined in Section 6.1.2; for convenience, we sometimes write A_QWj instead ofA_Qj.

• A_RF[W,B

Φ] :=A_RF|aux[W] ∪ A_RF|Φ[B

Φ]∪ A_RF|o; A_RF|aux,A_RF|Φ, andA_RF|o represent the ABoxes corresponding to the sets RF|aux, RF|Φ, and RF|o, respectively, and are constructed in correspondence to the definition of A_RF in Section 6.1.2. Note that, while the former sets contain only flexible assertions, the ABoxes contain only rigid ones.

• A_R[W,BΦ] is then defined as the union of the setA⁺_R[W,B

Φ], specified below, and all negative rigid role and basic concept assertions ¬α overN_I(K) for which we have α6∈ A⁺_R[W,B

Φ]. B_Φ|R denotes the set of rigid assertions inB_Φ. A⁺_R[W,B

Φ]:=B_Φ|R∪ B_R|o∪ {∃R(a)|R(a, e)∈ A_RF|Φ[B

Φ], a∈N_I(Φ)} ∪ {R(a, b)|R∈N_RR(O), a, b∈N_I(K), R(a, b)∈ A_QR[W] or

∃i.0≤i≤n,hO,A_ii |=R(a, b)}

As mentioned above, we cannot assume the arbitrary setB_Φ to be complete and to provide the part on the elements ofN_I(Φ) which follows from the TKB, especially, regarding all the assertions to be contained in A_R. Since we test the sufficiency of B_Φ when the rewriting is evaluated by using entailment tests, we then can however not detect rigid information missing in B_Φ if it is implied by the flexible assertions. For that reason, we consider A_RF|Φ[B

Φ] in the above construction and thus explicitly regard the consequences from the flexible assertions in B_Φ.

Observe that, apart fromA_RF|o andA_R[W,BΦ], which depend on the input ABoxesA, all of the ABoxes we defined are constant.

The below auxiliary lemma shows thatAR[W,B_Φ]is, in a certain sense, closed regarding the data in the given ABox sequence.

Lemma 6.22 For all B ∈BR(O), R∈N_RR(O) and a, b∈N_I(K)\N_I(Φ), we have:

• B(a)∈ AR[W,B_Φ] iff there is ani∈[0, n]such that hO,AR[W,B_Φ]∪ A_ii |=B(a),

• R(a, b)∈ A_R[W,BΦ] iff there is an i∈[0, n]such that hO,A_R[W,BΦ]∪ A_ii |=R(a, b).

6.3 First-Order Rewritings of r-Satisfiability

Proof. (⇒) This direction is trivial. (⇐) We assume that hO,A_R[W,BΦ]∪ A_ii |=B(a) holds for some i∈[0, n]. Observe that, if we have B_R|o^j =B_R|o^j+1 at some point, then we have B_R|o^j =B_R|o^j+l for all l ≥ 0. Hence, there must be some j⁰ ∈ [0,|BR(O)|] such that B^j_R|o⁰ =B_R|o^j0+l, for alll≥0, because every setB_R|o^j+1such thatB^j+1_R|o 6=B^j_R|o must contain at least one new assertion, there are only|BR(O)|relevant assertions per individual, and an assertion on a specific individual does not depend on assertions on other individuals by Lemma 2.14. GivenhO,A_R[W,BΦ]∪ A_ii |=B(a) and the fact that only the assertions on the individual a(possibly contained in a tuple of individuals) in A_R[W,BΦ] are relevant for the entailment by Lemmas 2.14 and 2.13, we can neglect all but the second and last conjuncts in the definition of A⁺_R[W,B

Φ]; a cannot occur in these sets. In addition to the assertions in B_R|o, we thus consider those entailed by some hO,A_ii,i∈[0, n]. But the basic concept assertions corresponding to those role assertions have to be contained in B¹_R|o and hence in also in B_R|o by definition. This leads to B(a) ∈ A_R[W,BΦ] by the definition of A_R[W,BΦ].

We second assumehO,A_R[W,BΦ]∪ A_ii |=R(a, b). By Definition 2.11 and Lemma 2.13, there is some S ∈ N⁻_R such that S(a, b) ∈ A_R[W,BΦ] ∪ A_i and O |= S v R. If S(a, b) ∈ A_i, then we have that hO,A_ii |= R(a, b), and hence R(a, b) ∈ AR[W,B_Φ] by the definition of A_R[W,BΦ]. Otherwise, that definition implies that there is a j ∈[0, n]

such that hO,A_ji |= S(a, b), and hence also hO,A_ji |=R(a, b), which also shows that R(a, b)∈ AR[W,B_Φ].

Given all the relevant ABoxes and additionally a mapping ι: [0, n] → [1, k], we can specify the KBs in the focus of the r-completeness tests in Definition 6.8 for all i∈[0, n+k], consideringA_i :=∅ fori > n:

K_R[W,Bⁱ

Φ]:=hO,Aⁱ_R[W,B

Φ]i, where

Aⁱ_R[W,B

Φ]:=A_R[W,BΦ]∪ A_QR[W] ∪ A_Qι(i)∪ A_RF[W,B

Φ]∪ A_i.

Before we continue regarding the rewriting, we show that we can useB_Φ as intended.

Lemma 6.23 For allW ⊆2^{p1,...,pm}such thatW ={W₁, . . . , Wk}andι: [0, n]→[1, k], there is an r-complete tuple w.r.t. W andι iff there is a set B_Φ such that the tuple

(AR[W,B_Φ], QR[W], Q^¬_R[W], RF[W,B_Φ]) is r-complete w.r.t. W and ι.

Proof. (⇐) This direction is trivial. (⇒) We assume (A_R, Q_R, Q^¬_R, R_F) to be an r-complete tuple, define B_Φ as follows, and show that (AR[W,B_Φ], Q_R[W], Q^¬_R[W], R_F[W,BΦ]) is r-complete as well:

B_Φ:={B(a)∈ A_R∪RF|B ∈B(O), a∈NI(φ)}.

We focus on the conditions in Definition 6.8. Our tuple obviously satisfies Condi-tions (C3) and (C4) by construction.

Regarding Conditions (C1), (C2), and (C5), we describe a model ofKⁱ_R[W,B

Φ]that can be homomorphically embedded into the canonical interpretation of the consistent KB that exists for the given tuple, since it satisfies Condition (C1); we denote the latter KB by K_Rⁱ. Observe that all positive assertions contained in one of the ABoxes of Kⁱ_R[W,B

Φ]

must also be contained in Kⁱ_R:

• A_QR[W] ⊆ A_QR follows from the facts that both tuples satisfy Condition (C3) and QR[W]is the minimal set satisfying that condition.

• A_RF|aux[W]∪ A_RF|Φ[B

Φ]∪ A_RF|o ⊆ A_RF is a consequence of the following observations.

By definition, every∃S(b)∈R_F|aux[W]is a consequence of a KBhO,A_Qji,W_j ∈ W.

Since the given tuple satisfies Condition (C6) and we have ι(n+j) = j in that definition, the assertion is also contained in R_F. A_RF|Φ[B

Φ] ⊆ A_RF follows from the construction. Each ∃S(b) ∈ R_F|o follows, by definition, from B_R|o together with someA_i(andO). Since the given tuple satisfies Condition (C1), A_R must contain all assertions in B_R|o; since Condition (C6) is satisfied, the assertion is thus also contained inR_F.

• All positive assertions inA_R[W,BΦ]have to be positive inA_R, too, by the definition of AR[W,B_Φ] and the observations in the previous items. More precisely, we have B_Φ|R⊆ A_R,B_R|o ⊆ A_R, and that all positive assertions inA_R[W,BΦ]\ B_Φ|R\ B_R|o are implied by a KB hO,A_RF|Φ[B

Φ] ∪ A_QR[W] ∪ A_ii, i ∈ [0, n]. Since the latter assertions occur inKⁱ_R, which is consistent by assumption, the ABox typeA_Ralso contains the latter rigid consequences.

Hence, any difference between Kⁱ_R and Kⁱ_R[W,B

Φ] (i.e., focusing on the assertions in Kⁱ_R[W,B

Φ] and disregarding additional assertions in K_Rⁱ) must be due to negative rigid assertions in AR[W,B_Φ] that occur positively in A_R (because A_R is an ABox type) and may cause the inconsistency of Kⁱ_R[W,B

Φ]. By providing a model forKⁱ_R[W,B

Φ], we show that such assertions cannot exist. Since the given tuple satisfies Condition (C1) and Kⁱ_R contains all positive assertions occurring inKⁱ_R[W,B

Φ], the KB [Kⁱ_R[W,B

Φ]]⁺, obtained from Kⁱ_R[W,B

Φ] by dropping the negative assertions, is also consistent. We focus on the canonical interpretation I of that KB and show that it also satisfies Kⁱ_R[W,B

Φ]. We consider negative role and basic concept assertions in Kⁱ_R[W,B

Φ].

Φ], and A_i and argue with the definitions of these ABoxes.

Let S be rigid, first. We can disregard A_Qι(i) and A_RF[W,B

Φ], since all rigid asser-tions in the former ABox are also contained in A_QR[W] and because the ABoxes A_RF[W,B

Φ] consists of do not contain assertions on two elements ofNI(K). Further, (rigid) role assertions that contain only elements of NI(K) are positively contained inA_R[W,BΦ]if they occur inA_QR[W] or are a consequence of a KBhO,A_ji,j∈[0, n].

Together with the fact that S(a, b)∈ A_QR[W] impliesR(a, b)∈ A_QR[W], this yields that the occurrence ofS(a, b) in one of the latter ABoxes causesR(a, b)∈ A_R[W,BΦ].

6.3 First-Order Rewritings of r-Satisfiability

Since A_R[W,BΦ] is an ABox type (i.e., only oneR(a, b) or¬R(a, b) is contained in it), that contradicts the assumption.

If S is flexible, S(a, b) occurs in A_i or A_Qι(i), which implies that W_ι(i) ∈ W. By the definition ofA_R[W,BΦ]andA_QR[W], based onQ_R[W]and Definition 6.2, we then get R(a, b) ∈ A_R[W,BΦ] orR(a, b)∈ A_QR[W], which also implies R(a, b)∈ A_R[W,BΦ] and thus a contradiction to the assumption.

• Let¬B(a)∈ A_R[W,BΦ]. Ifa∈N_I(Φ), this implies¬B(a)∈ A_Rby the definitions of A_R[W,BΦ] based onB_Φ|R andB_Φ based on A_R. SinceA_R is an ABox type and the given tuple satisfies Condition (C1), Lemma 2.12 yieldsI⁰6|=B(a), assumingI⁰ to be the canonical interpretation of Kⁱ_R. By our above observation on the positive assertions in Kⁱ_R[W,B

Φ], this interpretation must also satisfy [Kⁱ_R[W,B

Φ]]⁺. Hence, B(a) cannot be a consequence of that KB, and Lemma 2.13 yields I 6|=B(a).

For the case a6∈N_I(Φ), we again proceed by contradiction and assume I |=B(a).

Lemma 2.14 then yields that there are positive assertions about ain the ABoxes AR[W,B_Φ]∪ A_RF|o ∪ A_j, j ∈ [0, n], that together imply B; the other ABoxes do not contain assertions on such individuals. By that lemma, we can also disregard A_RF|osinceA_R[W,BΦ]contains the relevant assertions. Note that assertions inA_RF|o only on elements ofNI(K) are always basic concept assertions. Then, Lemma 6.22 implies that we can actually focus onA_R[W,BΦ]alone and obtainB(a)∈ A_R[W,BΦ]. This contradicts the assumption since A_R[W,BΦ] is an ABox type.

SinceI is a model of [K_R[W,Bⁱ

Φ]]⁺by Lemma 2.12 and we have shown that it satisfies all negative assertions inAR[W,B_Φ], it is also a model of Kⁱ_R[W,B

Φ]. Hence, our tuple satisfies Condition (C1).

If one of Conditions (C2) and (C5) is contradicted, then Lemma 2.13 yields that there is a homomorphism of the CQ that causes the contradiction into I. Again, the above observation that the positive assertions contained in Kⁱ_R[W,B

Φ]must be contained inKⁱ_R is important. By Definition 2.11 and the semantics, every such homomorphism into I is also a homomorphism into the canonical interpretation of the positive part of Kⁱ_R. This contradicts the assumption that Kⁱ_R satisfies Conditions (C2) and (C5), again by Lemma 2.13.

It remains to consider Condition (C6), and we discern regarding the elements in focus. Observe that, w.r.t. the ABoxes considered in that condition, the individual names occurring inR_F|aux[W] can only occur withinA_QR[W]∪^S_W∈WA_QW, and those in R_F|o only in A_R[W,BΦ]∪^S_0≤i≤nA_i.

• We regard the assertions in R_F|aux[W]. (⇐) For every ∃S(a_y) ∈ R_F|aux[W], and thusa_y ∈N^aux_I , the definition of R_F|aux[W] directly yields that there is a W_j ∈ W such that hO,A_Qji |= ∃S(a_y). This solves the claim given that Condition (C6) considersι to be such thatj=ι(n+j).

(⇒) If there is a W ∈ W such thathO,A_QR[W]∪ A_QWi |=∃S(a_y),a_y ∈N^aux_I , then Lemma 2.14 implies that∃S(a_y) can only follow from assertions involvingay. But a_y can be associated to a unique query ϕ_j ∈ Q_Φ that contains the variabley and corresponding ABox A_ϕj; no other such ABoxes contains assertions on a_y. This implies ϕj ∈QR[W]. By the definition of QR[W], there is a W⁰ ∈ W withpj ∈W⁰

and, in particular, A_Q

W0 implies all assertions on a_y inA_QR[W]. This shows that

∃S(a_y) already follows from A_Q

W0 (andO), which yields∃S(a_y)∈R_F|aux[W].

• We regard the assertions in R_F|o. (⇐) If ∃S(b) ∈ R_F|o, then there is an in-dex i∈[0, n] such that hO,B_R|o∪ A_ii |=∃S(b). By the definition of AR[W,B_Φ], we have B_R|o ⊆ A_R[W,BΦ], and hence obtain that hO,A_R[W,BΦ]∪ A_ii |=∃S(b).

(⇒) IfhO,A_R[W,BΦ]∪A_ii |=∃S(b), then the definition ofA_R[W,BΦ]and Lemma 2.14 yield that the assertion follows from rigid assertions in B_R|o and several rigid role assertions of the form R(b, a), R ∈ N⁻_R(O), each of which follows from a KB hO,A_ji, j ∈ [0, n]. But, for those rigid role assertions, we have that the corresponding rigid basic concept assertions ∃R(b) are contained in B_R|o by con-struction. Hence ∃S(b) follows from B_R|o (andO), and is thus contained in R_F|o, by the definition of that set.

• We regard the assertions inR_F|Φ[BΦ]. Since the given tuple satisfies Condition (C6) and, by the definition of RF|Φ[B_Φ], RF and RF|Φ[B_Φ] coincide regarding NI(Φ), we have that there is an i∈[0, n] such that hO,A_R∪ A_QR∪ A_Qι(i)∪ A_ii |=∃S(a) iff

∃S(a)∈R_F|Φ[BΦ] for all a∈N_I(Φ).

(⇒) This direction then directly follows from the above observation that all posi-tive assertions in AR[W,B_Φ] occur inA_R and A_QR[W] ⊆ A_QR.

(⇐) Since A_R is an ABox type, the fact that the given tuple satisfies Condi-tion (C1) yields that all basic concept asserCondi-tions that can be derived from asser-tions in A_R or A_QR are also positively contained in A_R; note that these ABoxes both contain only rigid assertions. Moreover, B_Φ contains all these rigid basic concepts on elements of N_I(Φ) which are contained in A_R, by definition. Hence, Lemma 2.14 yields that ∃S(a) ∈R_F|Φ[BΦ] implies that there is an i∈ [0, n] such that hO,AR[W,B_Φ]∪ A_QR[W]∪ A_Qι(i)∪ A_ii |=∃S(a).

Thus, Condition (C6) is also satisfied.

In order to be able to specify rewritings, we next propose an approach to handle the critical size of the ABox A_RF|o.

Introducing Prototypes

To untangle the knowledge captured in the ABox A_RF|o and separate it from the in-put data as much as possible, we introduce prototypes¹¹. Specifically, instead of the names occurring in A_RF|o, we consider prototypical, fresh individual names of the form [S], a_[S]S, a_[S]S%, etc., meaning [S] is used instead of a concrete individual name from N_I(K). We collect all these new names in the set N^pro_I .

The ABoxA_∃S for allS∈N⁻_R(O)\N_RR represents a prototypical version of an ABox A_∃S(b) with b∈NI(K) (see Section 6.1.2) and is obtained fromA_∃S(b) as follows, where b∈N_I(K) and a_b%, a_b%S ∈N^tree_I :

• everyR(b, a_bS) is replaced byR(b, a_[S]S),

11Not to be confused with the prototypical elements in canonical interpretations.

6.3 First-Order Rewritings of r-Satisfiability

• everyR(a_b%, a_b%S) is replaced byR(a_[S]%, a_[S]%S),

• everyB(a_b%) is replaced by B(a_[S]%).

For query answering regarding the adapted ABox, we propose the rewriting ·^℘.

Definition 6.24 (·^℘)Letprobe a unary predicate that identifies exactly the elements of N^pro_I . For a CQ ϕ:=∃~x.ψ, define the query ϕ^℘:=∃~x.ψ∧ψ_filter with

F|o is defined as the union of the set^S_∃S(a)∈R

F|oA_∃S and the set that, for each a∈N^pro_I occurring in the latter set, contains the assertion pro(a). ♦ The below lemma captures the intent of the rewriting.

Lemma 6.25 Let ϕ be a CQ and A ∪ A_RF|o be one of the ABoxes Aⁱ_R[W,B prototypes, which exist in the domain of DB(A ∪ A^℘_R

F|o) by the definition of A^℘_R

F|o. Let R(s, t), S(t, u) ∈ ϕ;R, S ∈ N⁻_R(O); π⁰(s) 6∈ N^pro_I ; and π⁰(t) ∈ N^pro_I , such that the precondition of ψfilter evaluates to true under π⁰; otherwise, the implication clearly holds. Recall that, an ABox Aⁱ_R[W,B

Φ] as considered (see the definition in the previous section) only via A_RF|o refers to elements such as π(t) of N^tree_I that are associated to named individuals—here,π(s)—fromN_I(K)\N_I(Φ) (i.e., elements which are replaced by

·^℘). By Definition 2.23, the assumption thus impliesR(π(s), π(t)), S(π(t), π(u))∈ A_RF|o. Together with the assumption thatπ⁰(s)6∈N^pro_I ; the fact that·^℘replaces all ofN^tree_I , and only those, by prototypes; andπ⁰(s) =π(s), given by the definition ofπ⁰, this means that π(s)6∈N^tree_I . Sinceπ(s) occurs in the ABox A_RF|o, we hence haveπ(s)∈N_I(K)\N_I(Φ).

By the construction ofA_RF|o and the two role atoms contained in it, we thus must have π(u) =π(s) orπ(u)∈N^tree_I . This yields that eitherπ⁰(u) =π⁰(s) orπ⁰(u)∈N^pro_I by our is because atoms containing terms mapped to elements of N^pro_I can only be satisfied by DB(A ∪ A^℘_R

F|o) through assertions inA^℘_R

F|o, by Definition 2.23 and the fact that elements ofN^pro_I do not occur inA;A^℘_R

F|oonly contains individuals ofN^pro_I andN_I(K)\N_I(Φ); and

we assume ϕto be connected. Furthermore, the elements ofN^pro_I inA^℘_R

F|o are connected in tree structures (see the definition of A_RF|o in Section 6.1.2) and, for each such struc-ture, there is at least one corresponding structure in A_RF|o by the definition of A^℘_R

F|o. We hence can define π to map to the corresponding tree elements in one such structure which correspond to the ones that π⁰ maps to.

It remains to consider the case thatπ⁰maps to elements of bothN^pro_I andN_I(K)\N_I(Φ).

First, we defineπ asπ⁰ w.r.t. all terms that are not mapped to elements ofN^pro_I . Note that A^℘_R

F|o does not contain assertions which do not contain elements of N^pro_I (see the definition of A_RF|o). By Definition 2.23, we thus haveα∈ A for all atomsα∈ϕwhere π⁰(α) does not contain elements of N^pro_I . This particularly means that our definition of π is as required w.r.t. all those atoms.

For definingπon the terms not yet considered, we regard a termsinϕthat is mapped to an element ofN_I(K)\N_I(Φ) and occurs in a role atom R(s, t) together with a term t mapped to an element ofN^pro_I ; since we assumeπ⁰ to map to elements of bothN^pro_I and NI(K)\NI(Φ) and ϕ to be connected, such a role atom must exist. By Definition 2.23 and the fact that A does not contain elements of N^pro_I , given by assumption, we have R(π⁰(s), π⁰(t))∈ A^℘_R the introduction of prototypes yielding elements of the form a_[S]S does not change the fact that all assertions in A^℘_R

F|o on both π⁰(s) and a_[S]S must stem from ∈ A_∃S(π⁰_(s)). Note that we also get π⁰(t) =a_[S]S. We define π(t) =a_π⁰_(s)S. Since the filter conjunct is satisfied, we have that all role atoms R⁰(t, u) in whicht occurs contain only termsu that are either mapped to π⁰(s) or to an element π⁰(u)∈N^pro_I . The former kind of role atoms is satisfied by this definition of πby the above observation that all role assertions inA^℘_R

F|o that contain bothπ⁰(s) anda_[S]S stem from A_∃S(π0(s)).

We regard a role atom of the other kind. Since this case is similar to the induction case, we abstract from the given information by assuming π⁰(t) = a_[S]%, π(t) = a_b%

to be defined, and ∃S(b) ∈ R_F|o. Again, by Definition 2.23 and the fact that A does not contain elements of N^pro_I , by assumption, we have R⁰(π⁰(t), π⁰(u)) ∈ A^℘_R

F|o. By the definition of A^℘_R

F|o based on ABoxes of the formA_∃S⁰,S⁰ ∈N⁻_R, which do not overlap in the elements of N^pro_I they contain, we get that R⁰(π⁰(t), π⁰(u))∈ A∃S since the shape of π⁰(t) indicates that it is contained in that ABox. By the same argument, we then get R⁰⁰(π⁰(t)), π⁰(u)) ∈ A_∃S for all R⁰⁰(π⁰(t), π⁰(u)) ∈ A^℘_R

6.3 First-Order Rewritings of r-Satisfiability

If we apply this lemma in the following, we assumeN^tree_I to only contain the auxiliary elements used in A_RF|aux[W] and A_RF|Φ[B

Φ], meaning that we disregard the auxiliary ele-ments fromA_RF|o, which it contains according to the original definition. Also note that N^tree_I does not contain elements from N^pro_I .

Im Dokument Using Ontology-Based Data Access to Enable Context Recognition in the Presence of Incomplete Information (Seite 132-141)