Syntax, Semantics, and Standard Reasoning

As described in Chapter 1, description logics focus onconcepts, which are interpreted as sets; roles, which are interpreted as binary relations; and individual names, which are interpreted as constants. Accordingly, DLsignaturesΣ = (N_I,N_C,N_R) are based on three kinds of non-logical symbols representing constants (i.e., zero-ary function symbols), unary, and binary predicates, respectively: individual names N_I,concept names N_C, and role names N_R, all of which are non-empty, pairwise disjoint sets. The various DLs then differ in the allowed logical symbols and in the way the axioms of the theories are built.

In the following, we introduce the syntax of the axioms in the description logicELand several members of theDL-Litefamily and, based on the axioms, specify the notion of a DL theory, theknowledge base. In the remainder of the section, we cover the semantics and standard reasoning tasks.

Definition 2.1 (Syntax of Axioms in EL) Let Σ = (N_I,N_C,N_R) be a DL signature.

InEL, the sets of roles over Σ and concepts over Σ are defined, respectively, by the following grammars:

R::=P C ::=> |A| ∃R.D|DuE

whereA∈N_C,P ∈N_R, andD andE are concepts, in their turn; a concept of the form A,>,∃R.>, or∃R.A is abasic concept.

In what follows, letA1, A2, A3 ∈NC∪{>},R∈NR, Bbe a basic concept, andCandD be concepts. EL axioms are the following kinds of expressions: concept inclusions (CIs) of the form CvD, and assertions of the form C(a) andR(a, b), where a, b∈N_I. A CI is in normal form if it has one of the following forms:

A1uA2 vA3, A1v ∃R.A₂, B vA1. ♦ We sometimes use the abbreviation ∃R₁. . . R_{`</sub>.C for the concept ∃R<sub>1</sub>. . . .∃R<sub>`}.C. Note that concept inclusions in EL or more expressive DLs are also called general concept inclusions (GCI), which expresses the fact that the inclusion may contain arbitrary concept expressions on the left-hand side—historically, first so-called primitive concept definitions with only concept names on the left-hand side were considered [Baa+07].

The logics of theDL-Lite family all extend the base formalismDL-Lite_core, in which CIs with complex concept expressions on the left-hand side cannot be expressed. In this work, we focus on several of the logics presented in [Art+09], which differ in the kind of concept inclusions, the Boolean operators allowed in the concept expressions, and if role inclusion axioms (also role hierarchies) are allowed. Similar to concept inclusion axioms, the latter are of the form S v R and express that the role S is more specific than the role R. DL-Lite fragments that allow for such inclusions are labeled with the superscript H.

Definition 2.2 (Syntax of Axioms in DL-Lite) Let Σ = (N_I,N_C,N_R) be a DL signature. In DL-Lite, the sets of roles over Σ and basic concepts (also concepts) over Σ are defined, respectively, by the following grammars:¹

R::=P |P⁻ B ::=A| ∃R.>

whereA∈NC,P ∈NR, and·⁻ denotes the inverse role operator. N⁻_R denotes the set of roles.

DL-Lite axiomsare the following kinds of expressions: concept inclusions (CIs)of the form

B1u · · · uBmvBm+1t · · · tBm+n (∗) where B₁, . . . , Bm+n are concepts;² role inclusions (RIs) of the form S v R, where R, S ∈ N⁻_R; and assertions of the form B(a) andP(a, b), where B is a basic concept, P ∈NR, anda, b∈NI.

Forc∈ {core,horn,krom,bool}, we denote by DL-Lite_c the logic that does not allow for role inclusions and restricts concept inclusions of the form (∗) as follows:

• m, n are arbitrary if c=bool,

• m+n≤2 if c=krom,

• n≤1 ifc=horn,

1In the literature, aroleis sometimes an expression that may be prefixed by negation [Cal+07b].

2Both sides of CIs may be empty.

2.1 Description Logics

• m+n≤2 and n≤1 if c=core.

If role inclusions are allowed in addition, this is indicated by the superscriptH, and we

obtain the four DLs denoted by DL-Lite^H_c . ♦

InDL-Lite, the abbreviation∃Ris usually used to abbreviate concepts of the form∃R.>, whereRis a role. As usual, we generally denote the empty conjunction (u) by>and the empty disjunction (t) by ⊥. We may further use the abbreviationsB1u · · · uBmv ¬B forB₁u · · · uBmuB v ⊥,d

Bfor the conjunctionB₁u · · · uBm ifB={B₁, . . . , Bm}, P⁻(a, b) :=P(b, a), and (P⁻)⁻:=P forP ∈N_R and a, b∈N_I.

In some constructions (in Chapters 4 and 6), we also consider negated assertions of the form ¬α, where α is an assertion; if this is the case, it is mentioned explicitly.

Definition 2.3 (Syntax of Knowledge Bases) Let DL be a description logic. An ontologywritten inDLis a finite set of concept and (if allowed inDL) role inclusions, and anABox is a finite set of assertions of concept and role names. Together, an ontologyO and an ABoxA form aknowledge base (KB)K:=O ∪ A, writtenK=hO,Ai. ♦ We sometimes also refer to the ABox as fact base or simply as the data.³ In this work, we assume every knowledge base to be such that all concept and role names occurring in the ABox also occur in the ontology. Given the (standard) semantics introduced below, it can readily be checked that this assumption is without loss of generality

For a given KBK:=O ∪A, we denote byN_I(K) andN_I(A) the set of individual names that occur in K and A, respectively (i.e., in EL and DL-Lite, we have NI(K) =NI(A));

by N_C(O) andN_R(O) the sets of, respectively, concept names and role names occurring in K; by N⁻_R(O) the set of roles occurring inK if it is inDL-Lite; and byS(O) the set of all concepts that occur in O. Note that the latter set includes all sub-concepts of complex concept expressions. A concept over O is a concept constructed (only) from the concept and role names occurring in O; observe that it does not necessarily have to be contained in S(O). Moreover, B(O) denotes the set of all basic concepts that can be built fromN_C(O),>, and the roles occurring inO, andB^¬(O) denotes the set B(O) extended by negation, meaning B^¬(O) := {B,¬B |B ∈ B(O)}. Note that, regarding DL-Lite,S(O) and B(O) nearly coincide; yet, B(O) always contains>and the concept

∃P⁻ for all P ∈N_R(O).

The semantics of DLs is commonly specified in a model-theoretic way, based on inter-pretations. General logical notions like consistency and entailment can hence be defined as usual.

Definition 2.4 (Semantics) An interpretation I = (∆^I,·^I) for a description logic signature Σ = (N_I,N_C,N_R) consists of a non-empty set ∆^I, the domain of I, and an interpretation function ·^I, which assigns to every A ∈ NC a set A^I ⊆ ∆^I, to every P ∈ N_R a binary relation P^I ⊆ ∆^I ×∆^I, and to every a ∈ N_I an element a^I ∈ ∆^I such that, for all a, b ∈ N_I with a 6= b, we have a^I 6= b^I (unique name assumption (UNA)). This function is extended to all roles and concepts as described in the first part of Figure 2.1.

3In correspondence with the notion of ABox, ontologies are often separated intoTBoxesandRBoxes, containing the concept and role inclusions, respectively.

Name Syntax Semantics

inverse role R⁻ {(y, x)∈∆^I×∆^I |(x, y)∈R^I}

top concept > ∆^I

bottom concept ⊥ ∅

negation ¬C ∆^I\C^I

conjunction CuD C^I∩D^I disjunction CtD C^I∪D^I

exist. restriction ∃R.C {x∈∆^I | ∃y∈C^I,(x, y)∈R^I} concept inclusion C vD C^I ⊆D^I

role inclusion RvS R^I ⊆S^I concept assertion B(a) a^I ∈B^I role assertion R(a, b) (a^I, b^I)∈R^I

Figure 2.1: Semantics of role expressions, concept expressions, and axioms for an inter-pretation I= (∆^I,·^I).

An interpretation I satisfies (or is a model of) an axiom α, written I |= α, if the corresponding condition given in Figure 2.1 is satisfied. I satisfies (or is a model of) a knowledge base K , writtenI |=K, if it satisfies all axioms contained in it.

A knowledge baseKisconsistent(orsatisfiable) if it has a model, and it isinconsistent (or unsatisfiable) otherwise. K entails an axiom α, written K |= α, if all models of K also satisfyα. This terminology and notation is extended to (single) axioms, ontologies, and ABoxes by regarding each as a (singleton) knowledge base. ♦ We denote the fact that an interpretation I does not satisfy a KB K by I 6|= K and, similarly, non-entailment by K 6|=α. In accordance with Figure 2.1, the negated asser-tions of the form ¬B(a) and¬R(a, b), which we sometimes consider, are satisfied in an interpretationI = (∆^I,·^I) if, respectively, a^I 6∈B^I and (a^I, b^I)6∈R^I hold.

Regarding two domain elementsdandeand an interpretationIsuch that (d, e)∈R^I, d is anR-predecessor of e, andean R-successor of d. Note that the terms “individual elements”, “domain elements”, “elements”, and “individuals” are used interchangeably for the elements of an interpretation domain. If the terms are prefixed by “named”, then we refer to those elements of the domain that are used to interpret individual names.⁴ In what follows, the signature of an interpretation is generally not mentioned explicitly if it is irrelevant or clear from the context.

In some constructions, we apply the DLELO_⊥, which extends ELby allowing⊥and so-callednominals in concept expressions. ⊥is interpreted as the empty set and can be used in an ontology for expressing disjointness of concepts. Nominals are concepts of the form {a}, based on some individual namea, and interpreted as singleton sets that contain the corresponding named individual. Further, note that DL-Lite_R, which is the DL closest to the OWL 2 QL profile, extends DL-Lite^H_core in that it allows to express disjointness of roles.

4In the literature, the term “individuals” sometimes only refers to those elements of the domain that are used to interpret individual names.

2.1 Description Logics

Given the semantics, observe that allowing conjunction on the right-hand side of CIs does not increase expressivity since any CI of the form CvDuE can be split into two CIsC vDandCvE in a KB without affecting the semantics. For similar reasons, we can assume all CIs to have maximally two conjuncts on the left-hand side and maximally two disjuncts on the right-hand side.

Ontologies and knowledge bases on the whole are usually not only used for modeling domain knowledge and querying it, but also for deriving logical consequences that are not explicitly stated in the stored knowledge. This (typically automatic) process is called reasoning and comprises several standard reasoning problems. Below, we list those relevant for our work; for a larger overview, we refer to [Baa+07].

Definition 2.5 (Standard Reasoning Problems) Let K be a DL knowledge base, and let C and D be concepts. The standard reasoning problems in DLs include the following:

• Concept Subsumption: DoesK |=C vDhold?

• Concept Satisfiability: Is there an interpretation I such that I |=K andC^I 6=∅?

• Consistency Checking: IsK consistent?

• Instance Checking: DoesK |=C(a) hold? ♦

It is well-known that these reasoning tasks are reducible to each other in linear time in any DL that allows for concept name assertions and CIs expressing disjointness, of the form CuDv ⊥ (e.g., a conceptC is not satisfiable w.r.t. a KBK ifK |=C v ⊥; that CI is equivalent to Cu > v ⊥).

In contrast,conjunctive query answering and entailment are non-standard reasoning problems. These problems are important for TCQ answering since TCQs are based on conjunctive queries. We therefore introduce them next.