Rewriting Knowledge Base Satisfiability and Query Answering . 131

We next adapt the rewritings qunsat(O) and PerfectRef(ϕ,O) to incorporate the knowl-edge from the auxiliary ABoxes. Observe, however, that for constructing auxiliary ABoxes such as A_R[W,BΦ] and A_RF|o, we need to decide entailment w.r.t. different KBs already. Specifically, these tests include an index i ∈ [0, n] as parameter and regard some of the other auxiliary ABoxes. For that reason, we first propose variants of PerfectRef(ϕ,O) that similarly incorporate the given CQ ϕ and ontology but also the corresponding auxiliary ABoxes and that target our time-stamped database TDB(A).

In the following, we assume that the CQs we consider only contain individual names from N_I(Φ).

First Variants of PerfectRef(ϕ,O)

The formula PRef_(ϕ,O)(i) represents PerfectRef(ϕ,O) regarding TDB(A): Given a pa-rameteri∈[−1, n], it decides ifϕis entailed by the atemporal KBhO,A_ii. It is obtained fromPerfectRef(ϕ,O) by replacing all atomsB(t) andR(s, t) byB(t, i) andR(s, t, i), re-spectively. The correctness can be readily checked by considering the original semantics of the query (i.e., O is correctly included/rewritten w.r.t. the interpretation DB(A_i)) and the fact that TDB(A) is defined such that TDB(A)|=B(a, i) iff DB(A_i)|=B(a) for all B ∈B(O) and i ∈[0, n], and correspondingly for all relations Rin TDB(A) where R∈NR(O).

Lemma 6.26 For all i∈[−1, n], hO,A_ii |=ϕiff TDB(A)|=PRef_(ϕ,O)(i).

The formulaPRef_(ϕ,O|B^j

R|o)(i) representsPerfectRef(ϕ,O) regarding TDB(A) andB^j_R|o for all j∈[0, `] with`=|BR(O)|. We define:

• PRef_(ϕ,O|B⁰

R|o)(i) :=PRef(ϕ,O)(i).

• PRef_(ϕ,O|Bj+1

R|o )(i) for j ∈ [1, `] is obtained from PerfectRef(ϕ,O) by replacing all basic concept atomsB(t), if flexible, byB(t, i) and otherwise by

B(t, i)∨ ^{^}

a∈N_I(Φ)

(t6=a)∧ ∃p.PRef_(B(t),O|Bj R|o)(p), and all role atoms R(s, t) byR(s, t, i).

Lemma 6.27 For all j∈[0, `]and i∈[−1, n]:

hO,B_R|o^j ∪ A_ii |=ϕ iff TDB(A)|=PRef_(ϕ,O|B^j

R|o)(i).

Proof. The proof is by induction, based on the definition of the set B_R|o. For the base case, where j = 0, the definitions of B⁰_R|o and PRef_(ϕ,O|B0

R|o) yield that the equivalence hO,∅ ∪ A_ii |=ϕiff TDB(A)|=PRef(ϕ,O)(i) is to be shown, which holds by Lemma 6.26.

We regardj >0.

(⇒) We assumehO,B_R|o^j ∪ A_ii |=ϕand thus get DB(B^j_R|o∪ A_i)|=PerfectRef(ϕ,O) by Lemma 2.24. That is, there is a homomorphismπofPerfectRef(ϕ,O) into DB(B_R|o^j ∪A_i) and hence a CQ ϕ⁰ in the UCQ PerfectRef(ϕ,O) such that, for all atoms α in ϕ⁰, π(α) ∈ B_R|o^j ∪ A_i, by the query semantics and the definition of DB(B_R|o^j ∪ A_i) (see Definition 2.23). For all these atoms, we show that π is also a homomorphism of the corresponding replacement into TDB(A).¹²

• Ifαis a flexible basic concept atomB(t) or a role atomR(s, t), thenπ(α) cannot be contained in the setB^j_R|ocontaining only rigid basic concepts, by definition; hence, we get π(α) ∈ A_i. The definition of TDB(A) (see Definition 6.21) then implies thatB^DBcontains the tuple (π(t), i) or thatR^DB contains the tuple (π(s), π(t), i), respectively. Thus,π is as required.

• Let α be a rigid basic concept atom B(t). If π(α) ∈ B_R|o^j , then the defini-tion of B_R|o^j , especially the fact that B^j_R|o contains only individual names from N_I(K) \ N_I(Φ), yields that π(t) 6∈ N_I(Φ) and that there is a k ∈ [0, n] such that hO,B^j−1_R|o ∪ A_ki |= π(α). By the induction hypothesis, the latter implies TDB(A) |= PRef_(π(α),O|B^j−1

R|o)(k), which together with the former yields that π is also a homomorphism of the replacement into TDB(A) and thus as required.

If π(α)∈ A_i, thenB^DB contains the tuple (π(t), i) by the definition of TDB(A).

(⇐) We argue correspondingly and consider a given homomorphism π and a satisfied disjunct ofPRef_(ϕ,O|Bj

R|o)(i), which itself is a conjunction and, before our replacements, originally had been a CQ obtained from PerfectRef(ϕ,O). We consider an arbitrary conjunct α in that conjunction and show that π is also a homomorphism of the atom that has been replaced by the conjunct into DB(B_R|o^j ∪ A_i).

• Ifαhas replaced a role atomR(s, t) or flexible basic concept atomB(t), then it is of the formR(s, t, i) or, respectively,B(t, i). By the assumption that TDB(A)|=π(α) and the definition of TDB(A), we then directly get that R(π(s), π(t)) ∈ A_i or, respectively, B(π(t))∈ A_i. Hence, π is as required w.r.t. the original atom and the database.

• Let α be the replacement of a rigid basic concept atom B(t). If the first dis-junct B(t, i) in α is satisfied, we can argument as in the previous case. Other-wise, we have TDB(A) |= ∃p.PRef(B(π(t)),O|B^j−1

R|o )(p) and π(t) 6∈ N_I(Φ). But this yields hO,B_R|o^j−1 ∪ A_pi |= B(π(t)) by the induction hypothesis. Then, we have

12We assume the notion of homomorphism to be extended to the replacements, which may be nested disjunctions of rewritings (i.e., PRef_(ϕ,O|Bj+1

R|o )(i) is a disjunction whose disjuncts are conjunctions where each conjunct my be an inequality, a disjunction or, in turn, be recursively defined as to be of that form) and equality assertions in the obvious way.

6.3 First-Order Rewritings of r-Satisfiability

B(π(t)) ∈ B^j_R|o, by the definition of B^j_R|o, and hence again get that π is as re-quired.

The formulaPRef_(ϕ,O|AR[W,B

Φ])(i) representsPerfectRef(ϕ,O) regarding TDB(A) and A_R[W,BΦ]. It is obtained from PerfectRef(ϕ,O) by replacing

• all basic atoms B(t), if flexible, byB(t, i) and otherwise by



• all role atoms R(s, t), if flexible, byR(s, t, i) and otherwise by

∃p.PRef_(R(s,t),O)(p)∨

Given the definition of A_R[W,BΦ]and the rewritings introduced above, it can be readily checked that the adapted query decides if ϕis entailed by the KBhO,A_R[W,BΦ]∪ A_ii.

Lemma 6.28 hO,A_R[W,BΦ]∪ A_ii |=ϕ iff TDB(A)|=PRef(ϕ,O|AR[W,BΦ])(i).

Proof. Since we construct the rewriting based onPerfectRef(ϕ,O), Lemma 2.24 yields hO,A_R[W,BΦ]∪ A_ii |=ϕiff DB(A_R[W,BΦ]∪ A_i)|=PerfectRef(ϕ,O).

(⇒) We assume hO,A_R[W,BΦ]∪ A_ii |=ϕ, which hence means that there is a disjunct ϕ⁰ in the UCQPerfectRef(ϕ,O) and a homomorphism π ofϕ⁰ into DB(A_R[W,BΦ]∪ A_i).

That is, for all atoms α inϕ⁰, we have π(α)∈ A_R[W,BΦ]∪ A_i by Definition 2.23.

• If α is flexible, then we have π(α) ∈ A_i since A_R[W,BΦ] contains only rigid as-sertions. But then, π also fits our rewriting for that case, which addresses the corresponding relation in TDB(A).

• If α is rigid and π(α) ∈ A_i, then the definition of B_R|o yields π(α) ∈ B_R|o if we consider a basic concept atom. Thus, TDB(A) satisfies either PRef_(π(α),O|B

R|o)(i) or PRef_(π(α),O)(i), depending on the kind of atom, by Lemmas 6.26 and 6.27.

Hence, our rewriting is as required for this case.

For the remaining case where π(α) ∈ A_R[W,BΦ], it can readily be checked that the rewriting covers all parts referenced in the definition of A_R[W,BΦ], again using Lemmas 6.26 and 6.27.

(⇐) We argument correspondingly and consider a given homomorphismπ and a sat-isfied disjunct in the UCQ PRef_(ϕ,O|AR[W,B

Φ])(i), which itself is a conjunction. That is, it is a CQ obtained from PerfectRef(ϕ,O) where the atoms are replaced according to our above construction. We focus on an arbitrary such replacement for an atom α and show that we have π(α) ∈ A_R[W,BΦ]∪ A_i. By the definition of the replacements, the replacement must be a disjunction, and at least one of its disjuncts must be satisfied

under π. If this disjunct refers to an assertion contained in B_Φ|R,A_RF|Φ[B

The Final Versions of qunsat(O) and PerfectRef(ϕ,O)

We finally come to the rewritings we target. Recall that, for checking the conditions for r-completeness (see Definition 6.8), we need to decide KB consistency and CQ entailment by including, next to the input ABoxes A= (A_i)_0≤i≤n, several auxiliary ABoxes based on the tuple defined at the beginning of this section. Specifically, we focus on the KBs hO,Aⁱ_R[W,B whereasA_Qι(i) andA_idepend on the considered time pointi. Since the auxiliary ABoxes are not part of the input (of the r-satisfiability problem), we in the following adapt the FO formulas q_unsat(O) and PerfectRef(ϕ,O) such that the two problems can be solved by evaluating the respective formula over the input ABoxes alone (i.e., also without the ontology), which are captured by TDB(A).

For all setsW ⊆2^{p1,...,pm},W ∈ W, andB_Φ ⊆ {B(a)|B ∈B(O), a∈NI(Φ)}, which are constant, we create the rewritingsqunsat(O|W,W,B_Φ)(i) andPRef_{(ϕ,O|W,W,BΦ)}(i), based on q_unsat(O)^℘ and PerfectRef(ϕ,O)^℘, respectively;¹³·^℘ represents a function that applies the rewriting·^℘to every CQ in the given UCQ. That is, we adopt the approach proposed by Lemma 6.25; in line with this, we also consider A^℘_R

F|o instead of A_RF|o. As in the previous section, we provide replacements for the atoms in these two queries.

Since the database addressed by these queries contains only the individuals occur-ring in the input ABoxes, but we want to include auxiliary elements from N^aux_I , N^tree_I , and N^pro_I , we especially have to consider the quantifiers, which quantify only over the individuals in the original database. Note that this issue is not relevant regarding the rewritings we proposed in the previous section because we assume all individual names occurring in Φ to also occur in the input ABoxes and, in that section, focus on auxiliary ABoxes that do not contain names from N^aux_I , N^tree_I , and N^pro_I . To this end, we now consider each disjunct

q=∃x₁, . . . , x`.ϕ(x1, . . . , x`)∧ϕfilter(x1, . . . , x`)

contained in the original queries—because of the filter condition included by ·^℘, we do not have UCQs anymore—, where ϕ(x1, . . . , x`) is a conjunction of atoms. The idea

13Note that we here use the subscript partW, W,BΦ to describe the dependence of the rewriting on the given parameters, whereas we have used the additionally included ABoxes above. For brevity, we here do not useAR[W,B_Φ]∪ AQ_R[W]∪ AQ_ι(i)∪ AR_F|aux[W]∪ AR_F|Φ[B

Φ ]∪ A^℘_R

F|o.

6.3 First-Order Rewritings of r-Satisfiability

is to duplicate q several times such that we have 2<sup>`</sup> versions q<sub>0</sub>, . . . , q<sub>2</sub>`−1 of it, and to augment the quantification of variables to consider also the elements inN^aux_I ∪N^tree_I ∪N^pro_I , a constant number. We then replace q by the disjunction q0∨ · · · ∨q₂^`₋₁.

Formally, for allj ∈[0, `−1] andk∈[0,2<sup>`</sup>−1], we consider the quantified variable x_j inq_k with

k=b0∗2⁰+. . .+bj ∗2^j+. . .+b`−1∗2<sup>`−1</sup> and drop the quantification if bj = 0. We then define

qk:=∃⁰x1. . . .∃⁰x`.rep_{(ϕ∧ϕfilter|W,W,BΦ)}(i)

• The remaining kinds of atoms are covered in Figures 6.1 and 6.2.

Observe that neither the two above case distinctions nor the tables conditions ref-erencing A_QR[W],A_QW,A_RF|aux[W], and A_RF|Φ[B

Φ] depend on the input ABoxes. Fur-thermore, the definitions basically consider two different cases, depending on whether the atom contains a variable which gets unquantified by our adaptation of the query.

If this is the case, then it has to be mapped to an auxiliary element, and hence corresponding assertions (i.e., for the atom under consideration) can only occur in A_QR[W] ∪ A_QW ∪ A_RF|aux[W] ∪ A_RF|Φ[B

Φ]; otherwise, the input ABoxes and A_R[W,BΦ] are also taken into account—by using the query PRef(ϕ,O|AR[W,BΦ]). The variables that get unquantified by our adaptation and are associated to an element from N^pro_I represent however an exception. For them, the input ABoxes have to be considered to ensure that corresponding prototypical elements actually occur in A^℘_R

F|o. Also recall that the same definitions apply fori=−1, where we assume the considered ABox A_i to be empty.

The next lemma establishes the correctness of our translation of the original UCQs overKⁱ_R[W,B

Conditions rep(B(t)|W,W,B_Φ)(i) aj ∈N^aux_I ,B(aj)∈ A_QR[W]∪ A_QW true

aj ∈N^tree_I ,B(aj)∈ A_RF|aux[W]∪ A_RF|Φ[B

Φ] true

a_j ∈N^pro_I , h∅,A_∃Si |=B(aj), and aj =a_[S]S% ∃x. ^{^}

a∈NI(Φ)

(x6=a)

!

∧

∃p.PRef(∃y.S(x,y),O|AR[W,BΦ])(p)

Otherwise false

(a)t∈Q0(k) (t=xj)

Conditions rep(B(t)|W,W,B_Φ)(i)

B 6∈BR(O)

_

B(a)∈A_QW, a∈N_I(Φ)

(t=a)

!

∨B(t, i)

Otherwise

_

B(a)∈A_Q

R[W], a∈N_I(Φ)

(t=a)

!

∨PRef(B(t),O|AR[W,BΦ])(i)

(b)t∈Q1(k)∪NI(q)

Figure 6.1: The replacement definition for all B ∈B(O) and t∈NI(q)∪NV(q).

6.3 First-Order Rewritings of r-Satisfiability

• DB(Aⁱ_R[W,B that TDB(A) satisfies one of the queries in the disjunction representing the correspond-ing rewritcorrespond-ing. By the definition of ·^℘, the semantics of disjunction, and Lemma 6.25, we can equivalently assume DB(Aⁱ_R[W,B definition, our translation of (q⁰)^℘ further contains a query q_k that is an adaptation of q^℘ and such that

k=b₀∗2⁰+. . .+bj ∗2^j+. . .+b`−1∗2<sup>`−1</sup> where b_j = 0 iffπ(x_j)∈N^aux_I ∪N^tree_I ∪N^pro_I for allj ∈[0, `−1].

We show that, if it is restricted to N_V(q_k)∪N_I(q_k), π is a homomorphism of q_k into TDB(A).¹⁴ For that, we consider all conjunctsαofq^℘and replacementsrep_(α|W,W,BΦ)(i) introduced by our adaptation and prove that π(α) is satisfied by the database iff π(rep_(α|W,W,BΦ)(i)) is satisfied by TDB(A). Observe that this approach focusing on those atoms in q^℘ that contain quantified variables or individual names, and the cor-responding replacements, is correct since the queries are Boolean. Furthermore, since the case that π(α) is not satisfied by the database can only occur w.r.t. the filtering conjunct, it actually suffices to show implication regarding all atoms with predicates dif-ferent frompro. By the assumption, the database satisfies the assertion π(α), obtained fromα by replacing the variable(s)xinαbyπ(x). By Definition 2.23, we thus get that π(α)∈ Aⁱ_R[W,B

Φ]

℘.

Regarding the case that α = pro(xj), the correspondence of the definitions of the predicateproandrep_(α|W,W,BΦ)(i) directly yields the equivalence. That is,π(α) evaluates to true iff rep_(α|W,W,B

Φ)(i) = true, which means that the restriction of π (trivially) fits in this regard. Note that the shape of these replacements yields that the adaptation of ϕ_filter is basically a conjunction of equality statements (see Definition 6.24).

Regarding the other kinds of atoms, we focus on the types of the mapped ele-ments. As mentioned above, the definitions of the replacement formulas overall de-pend on whether the regarded atom contains a variable x_j, j ∈ [0, `−1], such that π(x_j) ∈ N^aux_I ∪N^tree_I ∪N^pro_I . For that reason, we assume α to be an arbitrary atom containing such a variable xj, in all but the last of the below cases, where we show that the above mentioned implication holds. The scheme of these proofs is the same: the kinds of the considered elements (i.e., regarding all elements π maps a term of α to) constrain the ABoxes to be considered, and the definition of rep_(α|W,W,BΦ)(i) is based on exactly those ABoxes.

• Letπ(xj)∈N^aux_I ∪N_I(A_RF|aux[W]). That is, we also regard elements fromN^tree_I of the forma_ax%, wherea_x ∈N^aux_I . We hence must haveπ(α)∈ A_QR[W]∪A_QW∪A_RF|aux[W]

14Note that we have not defined the notion of homomorphism w.r.t. disjunction (∨) and (in)equality predicates in Definition 3.5, but the corresponding extension should be obvious.

6.3 First-Order Rewritings of r-Satisfiability

because the input ABoxes andA_R[W,BΦ]only regard elements ofN_I(K) and neither A_RF|Φ[B

Φ] norA^℘_R

F|o contain elements of this kind. For the case that π(x_j)∈N^tree_I , we obtain π(α) ∈ A_RF|aux[W] by analogous arguments and the fact that neither A_QR[W] norA_QW contains elements from N^tree_I .

If α does not contain a variable that is quantified within q_k, then we directly get rep_(α|W,W,BΦ)(i) = true, which is trivially satisfied by π. For basic concept atoms, this holds because we only have to consider π(x_j) and A_RF|aux[W] does not contain basic concept assertions on elements of N^aux_I . Regarding role atoms, observe that π cannot map to elements of N^pro_I , since they do not occur within A_QR[W]∪ A_QW∪ A_RF|aux[W], and thatA_RF|aux[W] does not contain role assertions with individuals fromN_I(K). The latter is relevant regarding the case where t∈N_I(q) (in the definition of the replacement), since we have NI(qk) ⊆ NI(Φ) and assume N_I(Φ)⊆N_I(K).

If α next to x_j contains a variable y that is quantified within q_k—which means we regard the case s ∈ Q₀(k), t ∈ Q₁(k)—, then we have π(y) ∈ N_I(K), by the semantics, and that α is a role atom. Let α = R(xj, y), R ∈ N⁻_R. The fact that A_RF|aux[W] does not contain role assertions with individuals from N_I(K) yields R(π(x_j), π(y)) ∈ A_QR[W] ∪ A_QW. Since rep_(α|W,W,BΦ)(i), for all assertions R(π(xj), b) ∈ A_QR[W] ∪ A_QW with b ∈ NI(Φ), contains the disjunct (y = b) and elements fromN_I(K)\N_I(Φ) do not occur in this ABox, it also contains the disjunct (y =π(y)). Henceπ is obviously as required.

• Let π(x_j) ∈ N^tree_I ∩N_I(A_RF|Φ[B

Φ]). That is, we regard elements from N^tree_I of the forma_b%, whereb∈N_I(Φ)—the elements from N^tree_I that remain to be considered.

We then must haveπ(α)∈ A_RF|Φ[B

Φ] since that kind of elements only occurs in this ABox. The arguments for showing that π satisfiesrep_(α|W,W,BΦ)(i) are analogous to the ones applied for the case where π(x_j) ∈ N^tree_I in the previous item. The only difference now is that A_RF|Φ[B

Φ] can obviously contain role assertions with individuals from N_I(K).

• For π(x_j) ∈ N^pro_I , we get π(α) ∈ A^℘_R

F|o, since the other ABoxes do not contain elements from N^pro_I , and can assume aj to be of the form a_[S]%. Observe that, then, no term is mapped to an element of N_I(Φ) sinceA^℘_R

F|o does not contain such elements (1). By the definition ofA^℘_R

F|o,a_[S]%is (unambiguously) associated to the ABox A_∃S, one of the components of A^℘_R

F|o, which do not share elements of N^pro_I . This yields π(α)∈ A∃S (2). The existence of the component A∃S further implies that there is an individual a∈NI(K)\NI(Φ) such that∃S(a)∈RF|o. By the defi-nition of the latter set, there then is an index psuch thathO,B_R|o∪ A_pi |=∃S(a).

Lemma 6.27 thus yields that TDB(A) satisfies PRef(∃y.S(x,y),O|B_R|o)(p), which is a disjunct of PRef(∃y.S(x,y),O|AR[W,BΦ])(p). Note that, regarding role atoms, (1) implies that the case t∈N_I(q) never applies.

If α does not contain a variable that is quantified within q_k, then (2),a6∈N_I(Φ), and TDB(A) satisfyingPRef(∃y.S(x,y),O|AR[W,BΦ])(p) together yield that the replace-mentrep_(α|W,W,BΦ)(i) is (trivially) satisfied under π.

Ifαcontains a variabley that is quantified withinq_k, we need to showπ(y) = [S], in addition. In this case, we haveπ(y)∈N_I(K) and that αis of the formR(x_j, y), R ∈N⁻_R. This means that the assertionR(π(xj), π(y)) is contained inA^℘_R

F|o and, especially, in A_∃S(a), by the definition of A^℘_R

F|o and the above observation that

∃S(a)∈RF|o. The (tree) shape ofA_∃S(a) and the definition ofA_∃S, based on that ABox, then imply R(π(xj),[S])∈ A_∃S (and%=S),S ∈N⁻_R.

• We consider the case where the variable(s) and term(s) in α are not mapped to auxiliary elements of N^aux_I ∪ N^tree_I ∪N^pro_I , which means they are mapped to elements of N_I(K). If α is a flexible atom, then we have π(α) ∈ A_QW ∪ A_i since all other ABoxes contain only rigid assertions. In caseπ(α)∈ A_QW, we can apply arguments corresponding to (a subset of) those used for π(xj) ∈N^aux_I in the first item. The other case,π(α)∈ A_i, is covered by Definitions 2.23 and 6.21. That is, the definition of TDB(A) based onA_i yields that π is as required.

If α is rigid, then we have π(α) ∈ A_R[W,BΦ]∪ A_QR[W] ∪ A_QW ∪ A_i because A_RF contains no assertion without auxiliary elements from N^tree_I . We can also dis-regard A_QW since its rigid assertions are part of A_QR[W], by definition, given that we assume W ∈ W. Moreover, A_QR[W] is addressed directly in the replace-ment; that part is especially complete since A_QR[W] does not contain elements of N_I(K)\N_I(Φ). The other two ABoxes are covered byPRef_(α,O|AR[W,B

Φ])(i), accord-ing to Lemma 6.28.

It can readily be checked that the above cases cover all kinds of elements π can map to. These observations show that π is a homomorphism of all our replacements into TDB(A), and thus TDB(A)|=q_k. Hence, TDB(A) satisfies also our adaptation of (q⁰)^℘, which was to be shown.

(⇐) We only sketch the proof for this direction since it is very similar to the above one. Since it neither differs for the two items to be proven, we again assume q⁰ to be one of the two UCQs that are rewritten. By the semantics and the definition of the rewritings, we have a query q_kin the given rewriting that is an adaptation of a queryq^℘ in (q⁰)^℘ and satisfied in TDB(A). We thus have a homomorphismπ of q_k into TDB(A) (i.e., w.r.t. the individual names and (existentially quantified) variables inqk) and show that we can extend it adequately to cover the terms occurring in q^℘. Because, q_k is a disjunction by construction, one of its disjuncts must be satisfied, by the semantics. We regard the individual names a_j fromN^aux_I ∪N^tree_I ∪N^pro_I associated to the variablesx_j in q^℘ and to that disjunct. In particular, we extend π such that π(xj) =aj for all these variables and subsequently show that this definition satisfies our purpose.

We ignore the “filtering” conjunct ofq^℘, for now, and consider an arbitrary conjunct representing the replacement for an atomB(t) inq^℘, in the disjunct under consideration, satisfied under π by assumption.

• Ifπ(t)∈N^aux_I ∪N^tree_I , thentruemust be the replacement andB(π(t)) contained in A_QR[W]∪ A_QW∪ A_RF|aux[W]∪ A_RF|Φ[B

Φ]. By Definition 2.23, it can readily be checked that π then is as required.

• Ifπ(t)∈N^pro_I , then there are anS ∈N⁻_R(O)\N_RR(i.e., ABoxes of the formA_∃Sare only defined for such roles), an individuala∈NI(K)\N_I(Φ), and an indexi∈[0, n]

6.3 First-Order Rewritings of r-Satisfiability

such that hO,A_R[W,BΦ]∪ A_ii |=∃S(a) by Lemma 6.28. By Lemma 6.22, all rigid assertions about a that are consequences of that KB are contained in AR[W,B_Φ]. By the definition of the latter, such assertions with a then are either contained in B_R|o or role assertions and consequences of a KB hO,A_ji, j ∈ [0, n]; none of the other parts of A_R[W,BΦ] contains an element a 6∈ N_I(Φ), by their definitions.

Since B_R|o includes the basic concept assertions corresponding to the former role assertions, by definition, andA_R[W,BΦ]cannot contain flexible assertions, we hence gethO,B_R|o∪ A_ii |=∃S(a) by Lemma 2.14. This means that∃S(a) is part ofR_F|o, by definition, which yields A_∃S ⊆ A^℘_R

F. We thus can again apply Definition 2.23, to show that π is as required.

• For the case whereπ(t)∈NI(K), Definition 2.23 can be applied regarding the first disjuncts, referring toQ_R[W]. Regarding the others, the definition of TDB(A) and, respectively, Lemma 6.28 confirm the claim that the database satisfies B(π(t)).

The proof is similar for role atoms.

We lastly consider the “filtering” conjunctϕ_filter of q^℘, which is a conjunction of im-plications. By our extension ofπ in accordance with the individual names associated to the disjunct under consideration and the definitions of the predicateproand the replace-ment of the corresponding atoms (i.e., those containing pro), we have that each atom occurring in ϕ_filter is satisfied in the database under the extended π iff its replacement is true and hence evaluates to true in TDB(A) underπ. Thus, our extension of π is as required. Again, the proof for the equality atoms is correspondingly.

6.3.3 Rewriting r-Satisfiability

Based on the specific FO rewritings of KB satisfiability and UCQ entailment developed in the previous sections, we next define the FO formulas that capture r-satisfiability based on r-completeness.

For allW ⊆2^{p1,...,pm},W ∈ W,and B_Φ⊆ {B(a)|B ∈B(O), a∈NI(Φ)}:

• f_(C1)(i) :=¬qunsat(O|W,W,B_Φ)(i);

• f_(C2)(i) :=^V_p

j∈W¬PRef_{(ϕj,O|W,W,BΦ)}(i);

• f_(C5)(i) :=^V ϕ∈Q^¬_R, ψwitness query

forϕw.r.t.O

¬PRef_(ψ,O|W,W,B

Φ)(i).

We integrate these formulas into the following abbreviation and subsequently describe how r-satisfiability can be tested.

rSatW,W,B_Φ(i) :=f_(C1)(i)∧f_(C2)(i)∧f_(C5)(i).

Lemma 6.30 For allW ⊆2^{p1,...,pm} where W ={W₁, . . . , Wk}, ι: [0, n]→[1, k], and B_Φ⊆ {B(a)|B ∈B(O), a∈N_I(Φ)}, the tuple

(A_R[W,BΦ], Q_R[W], Q^¬_R[W], R_F[W,BΦ]) is r-complete w.r.t. W and ι iff the following hold:

• For all i∈[0, n], we have TDB(A)|=rSatW,W_ι(i),B_Φ(i).

• For all W ∈ W, we have TDB(A)|=rSatW,W,B_Φ(−1).

• For all S ∈ N⁻_R(O)\N⁻_RR and a ∈ N_I(Φ), we have ∃S(a) ∈ B_Φ iff there is an i∈[0, n]such that TDB(A)|=PRef(∃S(a),O|W,W_ι(i),B_Φ)(i).

Proof. We consider Definition 6.8. The tuple generally satisfies some of the conditions by construction: A_R[W,BΦ] is an ABox type; and QR[W] and Q^¬_R[W] are in accordance with Conditions (C3) and (C4). Furthermore, Lemmas 6.29 and 2.24 show that the formulas in rSatW,W,B_Φ(i) as considered in the first two items cover Conditions (C1), (C2), and (C5) adequately.

It remains to prove the equivalence between the satisfaction of Condition (C6) and the last item. For R_F|aux[W] and R_F|o, we have shown in the proof of Lemma 6.23 that they satisfy Condition (C6) by construction, independent of that item. We therefore focus on R_F|Φ[BΦ].

(⇐) If the equivalence in the last item holds, then the definition of R_F|Φ[BΦ]based on B_Φ and Lemmas 6.29 and 2.24 yield that ∃S(a)∈RF|Φ[B_Φ] iff there is ani∈[0, n] such that hO,A_R[W,BΦ]∪ A_QR[W]∪ A_Qι(i) ∪ A_i∪ A_RF|Φ[B

Φ]i |=∃S(a); note that R_F|aux[W] and R_F|o do not contain relevant assertions. However, by Lemma 2.14, all parts of A_RF|Φ[B relevant to obtain such a conclusion are contained in AR[W,B_Φ], given the definition ofΦ]

that ABox and that of A_RF|Φ[B

Φ]. Note that the latter does not contain basic concept assertions on elements of N_I(Φ). We thus get that ∃S(a) ∈ R_F|Φ[BΦ] iff there is an i∈[0, n] such thathO,A_R[W,BΦ]∪ A_QR[W]∪ A_Q

ι(i) ∪ A_ii |=∃S(a), as required.

(⇒) It is easy to see that, given the definition of RF|Φ[B_Φ]and Lemmas 6.29 and 2.24, Condition (C6) implies the condition in the last item.

6.4 Data Complexity

In this section, we show that the low data complexity of query answering in DL-Lite does not increase dramatically in our temporal setting, for which we prove ALogTime -completeness. Though, FO rewritability thus is lost. As it is shown next, this holds already for the case without rigid names and regarding DL-Lite_core. The matching containment result is presented thereafter and based on several results from the previous sections.

Im Dokument Using Ontology-Based Data Access to Enable Context Recognition in the Presence of Incomplete Information (Seite 141-152)