1. A Tutorial Summary of Description Logic and Hybrid Rules
Jing Mei, Harold Boley
October, 2005

2. Description Logics d
R(a,d) b a
R(a,b)
DL: Family of (decidable) sublanguages of First-Order Logic
Concept
Atomic concepts: C, C1, C2, D
Boolean combinations: negation, intersection, union
Restrictions: existential, universal, number qualifications
Role
Atomic roles: R, S
Property: transitive, symmetric, functional, inverse-functional
Axiom
TBox: subsumption
ABox: assertion e
R(a,e) a
R(a,b) b

3. Rules Language Head  Body term: constant, variable, function symbol
atom: predicate symbol applied to terms
literal: atom + negative atom
extended literal: literal + negation as failure literal
Head  Body
Without function symbols in atoms H and Bi
Datalog: H  B1  …  Bm
Datalog: H1  …  Hn  B1  …  Bm
Datalog : H1  …  Hn  [naf] B1  …  [naf] Bm
With function symbols in atoms H and Bi
Horn programs: H  B1  …  Bm
Normal programs : H  [naf] B1  …  [naf] Bm
Extended programs : H1  …  Hn  [naf][neg] B1  …  [naf][neg] Bm

4. Hybrid Knowledge Base Hybrid Rules: H  B1  …  Bm & Q1  …  Qn
Hybrid KB: K = (S, R)
S: a finite set of DL axioms in the ontology language
R: a finite set of hybrid rules
Hybrid Rules: H  B1  …  Bm & Q1  …  Qn
H, Bi are ordinary atoms for rules
Qj are queries to the DL component S