Description logics can be used to specify concept definitions within a domain, a ter- minological specification, in a structured and well formed manner. Description logics are intended to be decidable subsets of first-order logic(FOL) meaning that sound and
Name Syntax Semantics top concept > ∆I bottom concept ⊥ ∅ atomic concept A AI(⊆∆I) value restriction ∀R.C {x∈∆I | ∀y (x, y)∈RI →y∈CI} existential restriction ∃R.> {x∈∆I | ∃y (x, y)∈RI}
Table 2.1: Basic (atomic) semantics
Name Syntax Semantics
atomic negation ¬A, A∈NC ∆I\AI
full negation ¬C ∆I \CI
concept intersection C1uC2 C1I∩C2I
concept union C1tC2 C1I∪C2I
full existential re-
striction ∃R.C {x∈∆ I | ∃y (x, y)∈RI∧y∈CI} at most restriction ≤nR {x∈∆I | |{y∈∆I |(x, y)∈RI}| ≤n} at least restriction ≥nR {x∈∆I | |{y∈∆I |(x, y)∈RI}| ≥n} qualified at most restriction ≤nR.C {x ∈ ∆I | |{y ∈ ∆ | (x, y) ∈ RI ∧y ∈ CI}| ≤n} qualified at least restriction ≥nR.C {x ∈ ∆I | |{y ∈ ∆I | (x, y) ∈ RI ∧y ∈ CI}| ≥n} one-of {x1, x2, . . . , xn} {x∈∆I |x=xi,1≤i≤n} has value ∃R.{x} {y∈∆I |(y, x)∈RI} inverse of R− {(x, y)∈∆I×∆I |(y, x)∈RI}
Table 2.2: Constructors semantics
complete reasoning is possible. They rose from the need to provide frame based systems with a formal semantics, aided by Hayes [69] who demonstrated that frames could be given a formal semantics by using subsets first-order logic. This allowed reasoning to be done without the need for first-order logic theorem provers [3]. The combination of formal semantics and practical reasoning has led Description Logics to become thede facto ontology language; and it is the ontology language we adopt for the rest of this thesis.
A description logic knowledge base (KB) consists of two components, the TBox(T), containing intensional knowledge, and the ABox(A), containing extensional knowl- edge [3]. The TBox defines the terminology (vocabulary) and the ABox contains as- sertions about named individuals in terms of the TBox.
The terminology comprises of concepts, denoting sets of individuals, and roles, which denote binary relations between the individuals. Complex descriptions can be assigned a name in the TBox allowing Knowledge Engineers to extend beyond atomic concepts and roles, for exampleMother≡Woman u ∃hasChild.Person.
Each DL system is distinguished by their language for building descriptions (de- scription language). Description languages are assumed to have two kinds of symbols, atomic concepts (denoted by A and B) and atomic roles (denoted by R). The semantics
of atomic concepts and roles can be seen in Table 2.1. Atomic concepts and atomic roles are the basic building blocks that are used in concept and role constructors to build complex descriptions. The letters C and D denote arbitrary more complex descriptions. The semantics of these constructors can be seen in Table 2.2.
As can be seen from Table 2.1 and 2.2 the formal semantics of a concept is considered in terms of interpretations, I, that are a non-empty set ∆I, this is the domain of interpretation, and an interpretation function. The interpretation function assigns for every atomic concept A a set AI ⊆ ∆I, this is then extended, as shown in Table 2.1 and 2.2, to cover concept description.
Let us consider an example of how interpretations work; assume we have the fol- lowing TBox that consists of two axioms from the ontology presented in Section 2.5:
AcademicvPerson
StudentvPerson
This has the following semantics in terms of interpretations:
AcademicI ⊆PersonI
StudentI ⊆PersonI
This situation can be represented using a Venn diagram, as shown in Figure 2.3. This clearly shows that the set of individuals represented by Academic and Student are subsets of Person.
Person
Academic Student
Figure 2.3: A Venn diagram representing description logic interpretations By taking different DL operators we can construct different description logics. For example, the description logic ALC (AL+C) gives us:
DL Name Constructors
ALN u,∀R.C,∃R.>,≥R,≤R
ALC u,t,¬C,∀R.C,∃R.C
SHOIN ALN ∪ ALC, R−,
role hierarchies, role transitivity, role symmetry,
(inverse) functional properties Table 2.3: Description Logics constructor subsets
• AL = Atomic negation, concept intersection, universal restrictions and limited existential quantification.
• C = Negation of arbitrary concepts.
The differences between some of the different description languages are shown in Table 2.3.
Lastly, the W3C7 have defined the Web Ontology Language (OWL) [106]. OWL comes in three flavours: OWL Lite, OWL DL and OWL Full. OWL DL is equivalent toSHOIN(D);meaning we have the following operators:
• S = ALC with transitive roles. Allowing one to create transitive properties. For example, ifJohn isWith PaulandPaul isWith RingoandisWith is transitive
thenJohn isWith Ringo.
• H = Role hierarchy. Allowing one to create subproperties of properties. For example,hasAuthor has the subproperty hasCoAuthor.
• O = Nominals. Allowing one to state that a class is restricted to a given set of individuals. For example,BeatlesMemberisone of the set{John, Paul, Ringo
and George}, where this set is a set of instance names.
• I = Inverse properties. Allowing one to state an inverse property of a property. For example,isAuthoris the inverse ofhasAuthor
• N = Cardinality restrictions. Allowing one to restrict the number of times an individual can have a certain property. For example, aPersoncan have at most
one dateOfBirth.
• D = Datatypes. Allowing one to assign a datatype to a property. For example, the range ofname must be a string.
There are two other variants of OWL, these are:
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• OWL Lite. Is a syntactic subset of OWL DL and has expressivity SHIF(D) [75]. This results in the following: prohibits unions and complements, does not allow individuals to occur in the description of a class, and limits cardinalities to 0 or 1. This results in more efficient reasoning than OWL DL, but less expressive power.
• OWL Full. Contains OWL DL, but has more expressive power than Description Logics. OWL Full was designed to be fully upwardly compatible with RDF and RDFS [3] (see Section 2.4). For example, it is possible to represent a resource simultaneously as a set of individuals and an individual. This kind of construct means reasoning in OWL Full is undecidable [3].
OWL provides a standardised syntax, based on XML, for representing description logic ontologies. This facilitates the sharing of ontology specifications as it allows humans and computers to successfully give the syntax of the ontology specification the appropriate semantics.
There is a proposed W3C recommendation for OWL 28. OWL 2 is a compatible extension to OWL providing some new features, such as providing extra restrictions for datatypes and qualified cardinality restrictions.
2.3.1 Open World Assumption
DLs adopt the open world assumption which informally means that no single knowledge base has complete knowledge and therefore cannot assume a closed world. A closed world is one where any statement that is not known to be true is assumed to be false, the notion being that everything is known. The adoption of the open world assumption affects what kind of inferences can be made. Furthermore, it is a fair assumption to make when considering open and distributed environments (see Section 6.2.3), such as the Semantic Web, where it would be unwise to assume an agent knew everything.
Let us consider the following example to highlight this important distinction. As- sume we have the following statement.
Statement: ‘Turing’ is a ‘Machine’
Now if we ask the following question.
Question: ‘John’ is a ‘Machine’?
The answer from a closed and open world would be as follow.
Closed World: No
Open World: Unknown
So, the closed world system is able to state that‘John’is not a‘Machine’because nowhere does it state that he is. However, this absence of information in the open world means that we could equally derive that both ‘John’is a ‘Machine’ and that
‘John’is not a‘Machine’, and so we cannot definitively answer the question.
This raises an interesting problem for Knowledge Engineers who choose to use DLs because it means that one cannot fully specify what can be said about a concept; it is only possible to constrain the individuals that will be entailed to be an instance of the concept.
2.3.2 Reasoning with DLs
The following reasoning services9 are supported by DLs [3]:
TBox
• Concept Satisfiability. Can the concept definition admit instances? Cis satisfiable iff there is some modelI ofT such that
CI 6=∅
• Concept Subsumption. Is one concept subsumed by another concept? T |=CvD
ifCI ⊆DI for every modelIofT. Concept equivalence and concept disjointness can be reduced to concept subsumption as follows:
– T |=C ≡D↔ T |=C vD, DvC – CI∩DI =∅ ↔ T |= (CuD)v ⊥
ABox
• Consistency. Is the ABox satisfiable with respect to some TBox? A is consistent iff there exists some modelI of T andA
• Instance Checking. Is an individual an instance of a concept? ais an instance of
C iff for every modelI of T and A
aI ∈CI
• Realization. For all individual in the ABox compute their most specific concept names with respect to some TBox such that A |= C(a) and C is minimal with respect to the subsumption ordering.
There are now several software implementations for carrying out these reasoning tasks, such as Pellet [120], KAON [143] and FACT++ [135].
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2.3.3 When is a DL TBox an Ontology?
Having a collection of DL axioms does not mean that you have an ontology; an ontology is more than a collection of axioms; it is the result of some form of conceptualization (see Section 2.1.1). Thus, a TBox is an ontology when it actually represents a con- ceptualization specified by a Knowledge Engineer. For example, consider the following axioms: AvBtC and P oliticianvLef ttRight. Under the semantics of DL these two axioms are equivalent; the semantics of DL place no semantics on the label of a concept, it is entirely possible that two concepts with different labels can be inferred to be the same concept. However, the intended meaning of the second axiom is the result of some process of conceptualization and the resulting ontological commitment. That is the second axiom contains labels that are intended to convey meaning beyond the semantics of DL and relate to the domain being modelled by the axiom.