On Complexity

In this report, we study the complexity of TCQ entailment via the satisfiability problem, which has the same complexity as the complement of the entailment problem [BBL15c]. We consider two kinds of complexity measures: combined complexity and data complexity. For the combined complexity, all parts of the input, meaning the TCQ φ and the entire temporal knowledge base K, are taken into account. In contrast, for the data complexity, the TCQ φ and the ontology O are assumed to be constant, and thus the complexity is measured only w.r.t.

the data, the sequence of ABoxes.

Satisfiability UCQ Answering

Logic Combined Complexity Combined Complexity Data Complexity [ACKZ09] [BAC10, BMP13, BMP14] [ACKZ09]

DL-Lite^[_core^|H] NLogSpace NP inAC⁰

DL-Lite^[_horn^|H] P NP inAC⁰

DL-Litekrom NLogSpace inExpTime co-NP

DL-Lite^H_krom NLogSpace ? co-NP

DL-Lite_bool NP ExpTime-hard co-NP

DL-Lite^H_bool NP 2-ExpTime co-NP

Table 2.9: Known results for the atemporal setting

Table 2.9 summarizes known complexity results for atemporal problems in the DL-Lite family, which are important for our work. We consider some complexity classes from the world of circuits:

AC⁰ ⊆NC¹ ⊆AC¹, which relate to the machine classes

LogTime⊆ALogTime⊆NLogTime⊆LogSpace⊆NLogSpace⊆P such that AC⁰ ⊆ LogTime, ALogTime = DLogTime-uniform NC¹, and NLogSpace ⊆ AC¹ ⊆ P. Note that the class AC⁰ is of special interest for query answering in DL-Lite. This is because problems whose data complexity is in AC⁰ can be solved by encoding them as first order (FO) queries over finite structures. Such problems are therefore also called first order rewritable.

Recall that we assumed all concept and role names in the ABox to also occur in the ontology. If this was not the case, we could simply add trivial axioms like A v A or ∃R v ∃R to O in order to satisfy this requirement. Although this formally increases the size of O, these axioms do not affect the semantics of O, and can thus be ignored in all reasoning problems involvingO. Hence, complexity results without this assumption remain valid in our setting.

3 Atemporal Canonical Models and Conjunc-tive Queries Revisited

In this section, we recall and extend known definitions and results, which we use in our proofs later in the report.

3.1 Canonical Models for Horn CIs

We considerDL-Lite^H_hornand subsets of this logic and specify the notion of canoni-cal interpretationfor a knowledge base. This interpretation can be used for decid-ing consistency of the knowledge base and for answerdecid-ing CQs, because it contains those (prototypical) elements whose presence is enforced by the knowledge base.

Then, it suffices to check whether the canonical interpretation is a model of a given knowledge base and if it satisfies a CQ, respectively. We use a construc-tion based on the so-calledchase [AHV95], similar to that proposed in [CDL⁺07]

and [BAC10]; the latter extend the original definition of [CDL⁺07] to the logic DL-Lite^H_horn, and we further extend it. In particular, our canonical interpreta-tion contains (unnamed) prototypical R-successors, R ∈N⁻_R, for all elements the knowledge base requires to satisfy ∃R; in contrast, [CDL⁺07, BAC10] only con-sider such prototypical successors if the knowledge base (i.e., the corresponding ABox) does not already identify a named individual to be such a successor. Un-like us, [CDL⁺07, BAC10] do also not consider arbitrary basic concept assertions, but only concept names.

We use the notation caR1...R<sub>`</sub>, fora∈NI and R1, . . . , R` ∈N⁻_R, which is a domain element that acts as a prototypical R<sub>`</sub>-successor of a, if ` = 1, and of c_aR1...R`−1, otherwise. For simplicity, we below assume that if R₁ v R₂ is contained in an ontology O, then we also have ∃R1 v ∃R2 ∈ O and ∃R⁻₁ v ∃R⁻₂ ∈ O; and that O contains all trivial axioms of the form B vB for B ∈BC(O).

Definition 3.1 (Canonical interpretation). Let K = hO,Ai be a DL-Lite^H_horn -knowledge base. We start defining the following sets, for all A∈NC, P ∈NR,

A⁰ :={a|A(a)∈ A} and P⁰ :={(a, b)|P(a, b)∈ A} ∪

{(a, c_aP)| ∃P(a)∈ A} ∪ {(c_aP⁻, a)| ∃P⁻(a)∈ A}.

Further, we define corresponding sets, for all i > 0, by applying the below rules.

We denote with (e, e⁰) ∈ (P⁻)ⁱ the fact that (e⁰, e) ∈ Pⁱ. Similarly, e ∈ (∃R)ⁱ denotes that there is an e⁰ such that (e, e⁰)∈Rⁱ.

• If R₁ vR₂ ∈ O and (e, e⁰)∈Rⁱ₁, then we add (e, e⁰) to Rⁱ⁺¹₂ .

• If d

B v B ∈ O and e∈(B⁰)ⁱ, for all B⁰ ∈ B, then – we add e to Bⁱ⁺¹ if B ∈N_C;

otherwise, we have B =∃R, R ∈N⁻_R, and, – if e∈N_I(K), we add (e, c_eR) to Rⁱ⁺¹; – else if e=c_%, we add (e, c_%R) to Rⁱ⁺¹.

We collect the newly introduced individuals of the form c_% in the set ∆^I_u^O, and define the canonical interpretationI_KforKas follows, for alla∈N_I(A),A∈N_C, and P ∈N_R:

∆^IO :=N_I(A)∪∆^I_u^O, a^IO :=a,

A^IO :=

∞

[

i=0

Aⁱ, and P^IO :=

∞

[

i=0

Pⁱ.

Note that the above assumptions about additional axioms in O ensure that, whenever a∈(∃R)ⁱ, thena has an R-successor of the form c_aR.

The rules given in the above definition correspond to the three rules proposed in [BAC10]. Further, the two above mentioned differences, regarding basic concept assertions and the additional successor individuals we consider, do not have spe-cial effects on reasoning. This is why we below sometimes refer to the results of [BAC10] without providing detailed proofs.

If K is inconsistent, then it is obvious that IK cannot be a model of K. The converse of this statement is a little harder to show.

The proof proposed by [CDL⁺07, BAC10] is three-fold. First, it is shown thatIK

is a model of all positive inclusions (PIs) in O, which are CIs whose right-hand side is not ⊥. All other CIs are called negative inclusions. In order to check satisfiability of DL-Lite^H_horn-KBs, negative inclusions must be considered. That is, if a negative inclusion in the ontology is violated by assertions of the ABox, then the knowledge base is inconsistent and hence unsatisfiable. Furthermore, an interaction of positive and negative inclusions may cause inconsistency. For these reasons, all negative inclusions implied by the ontology have to be considered and the so-called closure of the negative inclusions contained in O is regarded. The second step then consists of showing that Kis consistent iff the assertions of the ABox do not contradict this closure. Third and last, it is shown that the latter is the case iffIK is a model ofK. The following proposition is a direct consequence of the above observations.

Proposition 3.2 ([BAC10, Lemma 3, Thm. 4]). Let K=hO,Ai be a consistent DL-Lite^H_horn-knowledge base, possibly including negated assertions. Then I_K |=K.

The next proposition describes which basic concepts the elements of ∆^IK satisfy, in dependence of the ABox.

Proposition 3.3. Let K = hO,Ai be a consistent DL-Lite^H_horn-knowledge base, e∈∆^IK,i be the minimal number for which there is a symbolS such thateoccurs

in Sⁱ,

B :={A∈N_C(O)|e∈Aⁱ} ∪ {∃R|R ∈N⁻_R(O), (e, e⁰)∈Rⁱ}

be the set of corresponding basic concepts, and B ∈ BC(O). Then, we have e∈B^IK iff O |=d

B v B.

Proof. For (⇐), we know that e ∈ (B⁰)^IK for all B⁰ ∈ B due to the definition of I_K. Hence, Proposition 3.2 yields the claim.

For (⇒), letjbe the minimal index for whiche ∈B^j, which means thatj ≥i. We show the claim by induction onj. If j =i, then B ∈ B, and hence O |=dB vB trivially holds.

Ifj > i, assume that the claim holds for allB⁰ withe ∈(B⁰)^j−1. We consider the rule application which caused e to be contained in B^j.

• If it was caused by R₁ v R₂ ∈ O, then B =∃R⁽⁻⁾₂ and e∈(∃R⁽⁻⁾₁ )^j−1. By the induction hypothesis, O |=d

B v ∃R⁽⁻⁾₁ v ∃R⁽⁻⁾₂ .

• If it was caused by a CI d

B⁰ vB ∈ O, then we know that e∈(B⁰)^j−1 for allB⁰ ∈ B⁰. By the induction hypothesis, O |=dB vdB⁰ vB.

The next proposition describes the basic concepts the new domain elements in

∆^I_u^K satisfy in a straightforward way and hence shows that an element of the form c_%R ∈∆^I_u^O can indeed serve as a prototypical R-successor. The proposition directly follows from Definition 3.1 and Proposition 3.3.

Proposition 3.4. Let K = hO,Ai be a consistent DL-Lite^H_horn-knowledge base.

Then, for all elements c%R ∈ ∆^I_u^K and all B ∈ BC(O), we have c%R ∈ B^IK iff O |=∃R⁻ vB.

We conclude the section referring to a result which is rather important for us since we focus on query answering.

Proposition 3.5 ([BAC10, Thm. 9]). For every UCQ ψ and every consistent DL-Lite^H_horn-knowledge base K = hO,Ai, possibly including negated assertions, we have K |=ψ iff IK|=ψ.