Concept Definitions & Inheritance in Knowledge Base: Description Logics & Data Management , Study notes of Computer Science

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B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Description Logic(s)

• Formerly known as “terminological logic(s)”
• Idea: logic language for
  • defining concepts in terms of other concepts
  • interrelating concepts Î constraining the meaning of concepts
• DL definition of “Happy Father” (Example from Ian Horrocks, Ulrike Sattler)

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management formalized as an “ontology graph”

Purkinje cells and Pyramidal cells have dendrites that have higher-order branches that contain spines. Dendritic spines are ion (calcium) regulating components. Spines have ion binding proteins. Neurotransmission involves ionic activity ( release ). Ion-binding proteins control ion activity (propagation) in a cell. Ion-regulating components of cells affect ionic activity ( release ).

domain expert knowledge

Made usable for the system using Description Logic

Example: Domain Knowledge

to “glue” SYNAPSE & NCMIR

Data

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Source Contextualization

through Ontology Refinement

In addition to registering (“hanging off”) data relative to existing concepts, a source may also refine the mediator’s domain map...

⇒ sources can register new concepts at the mediator ... ⇒ increase your data usability

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Roots

• “Structured Inheritance Networks” [Brachman 1977]
• KL-ONE [Brachman, Schmolze 1985]
• Core ideas:
  • Building blocks: atomic concepts (unary predicates), atomic roles (binary predicates), individuals (constants)
  • Constructors for building complex concepts and roles from simpler ones
  • Automated inference for concept subsumption and instance classification (is-a/is-instance-of are not explicitly given by the user, but inferred from concept definitions/instance properties)

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002, Lyon, France, July 23rd, 2002

(TBox)

(ABox)

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Knowledge Base (DL-Style)

• Terminological Knowledge (TBox)
  • Concept Definition (naming of concepts):
  • Axiom (constraining of concepts):

=> a mediators “glue knowledge source”

• Assertional Knowledge (ABox) about Individuals
  • n27_img118 : Neuron => the concrete instances/individuals of the concepts/classes that your sources export

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Example TBox

Atomic concepts Atomic concepts = {P,F,W, M1,= {P,F,W, M1,……}}

Base concepts Base concepts = {P,F}= {P,F}

Defined concepts Defined concepts = {W, M1, M2,= {W, M1, M2, ……}}

Roles Roles = {= { h1h1 ,, h2h2 }}

Concept Definition Concept Definition

AxiomAxiom

where A atomic concept, where A atomic concept,

C, D complex concept expressionsC, D complex concept expressions

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Example TBox

• • Base conceptsBase concepts = {Person, Female}= {Person, Female} …… occur on the RHS onlyoccur on the RHS only Defined concepts Defined concepts (^) = {P, F, W,= {P, F, W, ……}}

…… occur on the LHS (& maybe RHS)occur on the LHS (& maybe RHS)

• • Base interpretationBase interpretation JJ : interpret base: interpret base concepts onlyconcepts only
• • ExtensionExtension II ofof JJ :: on same domain ason same domain as JJ and agrees (on base) withand agrees (on base) with JJ
• • TBoxTBox TT isis definitorialdefinitorial if every baseif every base interpretation has exactly one extensioninterpretation has exactly one extension that is athat is a modelmodel ofof TT

Problem / Exercise

• Let the interpretation I (Person(x)) be “x is a person”.
• Similarly, I (Female(x)) = “x is female”.
• Question: What do W, M1, etc. mean?

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Back to Reasoning with the Family ...

• concept definition: MyConcept ≡ DL-formula
• concept inclusion: MyConcept ⊆ DL-formula
• finite set of definitions is a terminology or TBox if

for every atomic concept A there is at most one

axiom whose lhs is A

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Definitorial Terminologies

• In a Tbox T we distinguish: primitive concepts (occurring only on rhs) and defined concepts (occurring on lhs)
• T is definitorial if every interpretation of primitive concepts yields exactly one model of T (and thus for the defined concepts) Î meaning of defined concepts is fixed once the primitive concepts are interpreted!
• A directly uses B in T if B appears in the rhs of the definition of A
• A uses B is the transitive closure of ‘ directly uses’
• T is cyclic if A uses A for some A; else acyclic

One can show: If T is acyclic then T is definitorial

What about this one?

B. Ludaescher, ECS289F-W05, Topics in Scientific Data Management

Expansion of Terminologies

• For acyclic T we can “unfold” concept

definitions until every defined concepts is

specified in terms of primitive concepts only

Î the expansion of a TBox T

• Example: