Canonical Interpretations for Horn Description Logics

In this section, we focus on the Horn DLs we introduced in Section 2.1.1, EL and DL-Lite^H_horn. In a nutshell, Horn DLs do not allow to express disjunction on the right-hand side of CIs (i.e., neither directly in the syntax, nor indirectly through the seman-tics), such that the CIs can be represented as first-order Horn clauses. Reasoning in these DLs is easier than in general since it can be done using deterministic algorithms, which are often based on canonical interpretations. In what follows, we recall the well-known construction of these interpretations for knowledge bases inEL andDL-Lite^H_horn together with important properties of them and prove additional such properties.

In a nutshell, the canonical interpretation of a knowledge baseKcaptures exactly the information described by K. Since this is the knowledge to be satisfied in every model ofK, the canonical interpretation can be used for checking the consistency of K and for answering CQs w.r.t. K—by checking whether the canonical interpretation is a model of K and if it satisfies a CQ, respectively. We provide different definitions depending on the logic to facilitate later proofs. For EL, we directly consider the knowledge entailed from K, as it is done in [LTW09; KRH07], for example. Regarding DL-Lite, we explicitly construct the interpretation similar to [Cal+07b; BAC10], by applying the standard chase procedure for obtaining models of knowledge bases [DNR08]. We use this definition in order to be able to refer to the results of [BAC10], which hold for the non-standard setting with negative assertions in ABoxes, which we focus on. Note that [BAC10] extend the original definition of [Cal+07b] regarding DL-Lite^H to the logic DL-Lite^H_horn, and we further extend it. In particular, our canonical interpretation contains (unnamed) prototypical R-successors with R ∈ N⁻_R for all elements required to satisfy ∃R, by the knowledge base. In contrast, [Cal+07b; BAC10] only consider such prototypical successors if the knowledge base (i.e., the corresponding ABox) does not already identify a named individual to be such a successor. Unlike us, [Cal+07b;

BAC10] neither consider arbitrary basic concept assertions, but only concept names.

The prototypical domain elements are of the formu_% (u for “unnamed”), where% is a path %:=aR₁C₁. . . R_{`</sub>C<sub>`} over symbols occurring in the knowledge base with abeing an individual name, R₁, . . . , R_{`</sub> being roles, and C<sub>1</sub>, . . . , C<sub>`} being concepts from S(O).

We assume that the knowledge base does not already contain symbols of the form of these elements. |%|:=`denotes thelengthof a path as%. Observe that such a path also specifies the interpretation of the symbols contained in it in that it describes the

exis-tence of domain elements a, u_aR1C1, . . . , u_aR1C1...R`C`, which are related via R₁, . . . , R<sub>`</sub>, respectively; and each element u<sub>aR1C1...RmCm</sub> form≤`satisfiesC_m. Furthermore,u_%is only contained in the domain of the canonical interpretation if relations as described by

%have to be present in every model of the knowledge base. Note thatDL-Liteontologies can only enforce unnamed elements of the form u_aR1>...R

`> to exist, because they do not allow for qualified existential restriction on the right-hand side of CIs. We therefore usually write u<sub>aR1...R`</sub> in that context, to simplify notation.

Definition 2.10 (Canonical Interpretation in EL) Let K = hO,Ai be an EL knowledge base. We first define the set

∆^I_u^K :=

∞

[

j=0

∆^j_u where

∆⁰_u:={u_aRC |a∈NI(K), C ∈S(O), K |=∃R.C(a)},

∆^j+1_u :={u_%R

1C1R2C2 | ∃u_%R

1C1 ∈∆^j_u,O |=C₁ v ∃R₂.C₂}.

The canonical interpretation I_K forK is defined as follows, for alla∈N_I(A),A∈N_C, and R∈NR:

∆^IK :=N_I(A)∪∆^I_u^K, a^IK :=a,

A^IK :={a∈N_I(A)| K |=A(a)} ∪ {u_%RC ∈∆^I_u^K | O |=CvA}, R^IK :={(a, b)|R(a, b)∈ A} ∪ {(a, u_aRC)∈N_I(A)×∆^I_u^K} ∪

{(u_%, u_%RC)∈∆^I_u^K ×∆^I_u^K}.

Thefinite canonical interpretationI_K^f is defined correspondingly, but based on elements of the form u_C for all C ∈ S(O) instead of on different elements of the form u_%C. Analogously, these elements are collected in the set ∆^I_u^Kf. ♦ For simplicity, we assume in the definition regarding DL-Lite that, if the RI S vR is contained in an ontology O, then we also have ∃S v ∃R ∈ O and ∃S⁻ v ∃R⁻ ∈ O;

and that Ocontains all trivial axioms of the form BvB forB ∈B(O).

Definition 2.11 (Canonical Interpretation in DL-Lite^H_horn)Let K=hO,Ai be a DL-Lite^H_horn knowledge base. First, for all A∈N_C and P ∈N_R, we define:

A⁰:={a|A(a)∈ A},

P⁰:={(a, b)|P(a, b)∈ A} ∪

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

Then, iterate over all i ≥ 0: for all X ∈ NC∪NR define Xⁱ⁺¹ := Xⁱ; apply one of the following rules for all A ∈N_C, R, S ∈N⁻_R, andB, B₁, B₂ ∈B(O); and increment i;

(d, e) ∈(P⁻)ⁱ forP ∈N_R denotes the fact that (e, d)∈Pⁱ, and d∈(∃R)ⁱ denotes the existence of an element esuch that (d, e)∈Rⁱ:

2.1 Description Logics

• IfB₁uB₂vA∈ O,A∈N_C,and e∈B₁ⁱ ∩B₂ⁱ, then addetoAⁱ⁺¹.

• IfB v ∃R∈ O and e∈Bⁱ:

– ife∈NI(A), then add (e, u_eR) to Rⁱ⁺¹; – ife=u_%, then add (e, u_%R) to Rⁱ⁺¹.

• If∃RvA∈ O, (d, e)∈Rⁱ, then add dtoAⁱ⁺¹.

• IfS vR∈ O and (d, e)∈Sⁱ, then add (d, e) toRⁱ⁺¹. The set ∆^I_u^K collects the above introduced unnamed individuals.

The canonical interpretation I_K for K is then defined as follows, for all a ∈ N_I(A), A∈N_C, and P ∈N_R:

∆^IK :=N_I(A)∪∆^I_u^K, a^IK :=a, A^IK :=

∞

[

i=0

Aⁱ, P^IK :=

∞

[

i=0

Pⁱ. ♦

Note that the above assumptions about additional axioms in the ontology ensure that, whenever there is a named elementa∈(∃R)ⁱ for somei≥0, thenahas anR-successor of the form u_aR in the canonical interpretation, and similar for the unnamed elements.

In the remainder of this section, we recall and prove results about how the canonical interpretation may simplify reasoning. To this end, we assume K = hO,Ai to be a consistent knowledge base in EL or DL-Lite, depending on the context. In the latter case,A may particularly include negated assertions.

RegardingEL, we refer to the results from [LTW09], where the canonical interpreta-tion is however defined slightly differently, based on the finite one. More precisely, the possible paths over the domain elements of the latter that start at the named individ-uals represent the domain elements of the former interpretation. It is easy to see that this approach yields an interpretation corresponding to the one from Definition 2.10.

Regarding DL-Lite, note that the two above mentioned differences, w.r.t. basic con-cept assertions and the additional successor individuals we consider, do not have special effects on reasoning. This is why we below refer to the results of [BAC10] without providing detailed proofs.

An important property of the canonical interpretation I_K, which can be checked easily, is that it is a model ofK. IfK was inconsistent, then this would obviously not be the case. The converse of this statement is however harder to show—for DL-Lite; EL KBs cannot be inconsistent. The proof proposed by [Cal+07b; BAC10] is three-fold:

• First, it is shown thatI_K is a model of allpositive inclusions inO, which are CIs whose right-hand side is not⊥; all other CIs are called negative inclusions.

• For checking satisfiability ofDL-Lite^H_hornKBs, negative inclusions are critical: if a negative inclusion in the ontology is violated by assertions of the ABox, then the knowledge base is inconsistent and hence unsatisfiable. In particular, an interac-tion of positive and negative inclusions may lead to such an inconsistency. This is why a special closureof the negative inclusions contained in Ois regarded, which represents all negative inclusions implied by the ontology.

The second step then consists of showing that K is consistent iff the assertions of the ABox do not contradict this closure.

• Third and last, it is shown that the latter is the case iff I_K is a model of K.

These observations show that neither basic concept assertions nor “unnecessary” pro-totypical successors have special effects on the outcomes of the proof in [BAC10]. We hence can similarly state the result.

Lemma 2.12 ([BAC10, Lem. 3, Thm. 4],[LTW09, Prop. 1])I_K|=KandI_K^f |=K.

A result equally important for our work is about queries.

Lemma 2.13 ([BAC10, Thm. 9],[LTW09, Prop. 4]) For every Boolean UCQ ϕ, we have K |=ϕiff I_K|=ϕ.

In the following Lemmas 2.14 and 2.15, we describe the concepts satisfied by the domain elements of the canonical model and the subset of prototypical elements, re-spectively. Note that the following Lemma 2.14 is restricted to DL-Lite.

Lemma 2.14 Regarding Definition 2.11, all e ∈ ∆^IK, and all B ∈ B(O), we have e∈B^IK iff O |=dB vB, where B is defined as follows, based on the minimal number i for which there is a symbol X such that e∈Xⁱ:

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

Proof. (⇐) For all C ∈ B, we know that e ∈ C^IK, by the definitions of B and I_K. Hence, Lemma 2.12 yields the claim.

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

Regardingj > i, we assume that the claim holds for allC ∈B(O) for whiche∈C^j−1. We consider the rule in Definition 2.11 that causes eto be contained inB^j.

• If it is because of a CId

B⁰ vB ∈ O, then we havee∈C^j−1for allC ∈ B⁰. Hence, the induction hypothesis together with the semantics yields O |= dB v dB⁰. Because of the considered CI, this leads to O |=d

B vB.

For the other kinds of CIs, the proof works correspondingly.

• If it is because of the last rule, some SvR∈ O, and (e, d)∈S^j−1 ((d, e)∈S^j−1), thenB must be of the form B =∃R⁽⁻⁾. Further, (e, d)∈S^j−1 ((d, e)∈S^j−1) im-pliese∈(∃S⁽⁻⁾)^j−1, for which the induction hypotheses yieldsO |=d

B v ∃S⁽⁻⁾. By our assumption (see the part above Definition 2.11), we have ∃S v ∃R ∈ O (∃S⁻v ∃R⁻∈ O) and thus get O |=d

B v ∃R⁽⁻⁾.

The next lemma describes the concepts the new domain elements in ∆^I_u^K satisfy in a straightforward way and hence shows that an element of the form u_%RC ∈ ∆^I_u^K can indeed serve as a prototypical R-successor. Regarding DL-Lite, the lemma directly follows from Definition 2.11 and Lemma 2.14. For EL, it is easy to prove by induction on the structure of concepts.

Im Dokument Using Ontology-Based Data Access to Enable Context Recognition in the Presence of Incomplete Information (Seite 27-31)