1. ece 627 intelligent web: ontology and beyond
lecture 16: description logic
- basic formalisms, TBox, ABox

2. description logic basic formalism
DL provides facilities to set up knowledge bases, to reason about their content, and to manipulate them
ece 627, winter ‘13

3. description logic basic formalism
TBox
Description Language
Reasoning
ABox
application
program
rules
ece 627, winter ‘13

4. description logic basic formalism
vocabulary consists of concepts, that denote sets of individuals, and roles, that denote binary relationships between individuals in addition, DL systems allow their users to build complex descriptions of concepts and roles (Tbox can be used to assign names to them)
ece 627, winter ‘13

5. description logic basic formalism
statements in the TBox and ABox can be identified with formulae in first-order logic or, in some cases, a slight extension of it
ece 627, winter ‘13

6. description logic basic formalism
typical reasoning tasks (the previous lecture): determining if a description is satisfiable or whether one description is more general than the other – subsumption
ece 627, winter ‘13

7. description logic basic formalism
important problems for ABox (the previous lecture): to find out if its set of assertions is consistent (has a model, and if individuals are instances of concept descriptions)
ece 627, winter ‘13

8. description logic basic formalism
satisfiability checks descriptions and consistency checks assertions – all to determine whether a knowledge base is meaningful at all
ece 627, winter ‘13

9. description logic basic formalism
rules – an extension of the logical core formalism – allow for adding assertions to a knowledge base
ece 627, winter ‘13

10. description logic description language
atomic concepts – A, B atomic roles – R concept descriptions – C, D description languages are distinguished by the constructors they provide – we look at the family of AL-languages (=attributive language)
ece 627, winter ‘13

11. description logic description language – AL
(atomic concept) (universal concept) (bottom concept) (atomic negation) (intersection) (value restriction) (limited exist. quant.)
ece 627, winter ‘13

12. description logic description language – AL
examples: atomic concepts – Person and Female Person Female Person Female Person hasChild. Person hasChild.Female
ece 627, winter ‘13

13. description logic semantics of AL–language
to define semantics of AL-concepts – we consider interpretations I that consist of a non-empty set and an interpretation function for every atomic concept A: and a binary relation:
ece 627, winter ‘13

14. description logic semantics of AL–language
ece 627, winter ‘13

15. description logic semantics of AL–language
examples:
ece 627, winter ‘13

16. description logic family of AL–languages
to obtain more expressive languages – we add further constructor to AL: union (U): full existential quantifier (E):
ece 627, winter ‘13

17. description logic family of AL–languages
number restrictions (N) negation (C) AL{U}{E}{N}{C} and others …
ece 627, winter ‘13

18. description logic AL–language and predicate logic
the semantics of concepts identifiers are fragments of first-order predicate logic – atomic concepts and roles are unary and binary predicates any concept C and be translated into a predicate logic formula FC(x), with one free variable x such that elements of satisfying FC(x) is exactly CI
ece 627, winter ‘13

19. description logic AL–language and predicate logic
examples: variable free syntax of DLs is much more concise
ece 627, winter ‘13

20. description logic terminology
forming complex descriptions of concepts: terminological axioms – make statements about how concepts or roles are related to each other definitions – specific axioms
ece 627, winter ‘13

21. description logic terminology - axioms
in the most general case inclusion: C D (R S) equality: C D (R S) C, D are concepts, R, S are roles
ece 627, winter ‘13

22. description logic terminology - axioms
an interpretation I satisfies an inclusion C D if , and it satisfies an equality C D if
ece 627, winter ‘13

23. description logic terminology - axioms
if T is a set of axioms then I satisfies T iff I satisfies each element of T, if I satisfies a set of axiom then we say that it is a model of this set of axioms
ece 627, winter ‘13

24. description logic definitions
an equality whose left-hand side is an atomic concept is a definition definitions are use to introduce symbolic names for complex descriptions Mother Woman hasChild.Person
ece 627, winter ‘13

25. description logic TBox …
a finite set of definitions T is called a terminology or TBox
ece 627, winter ‘13

26. description logic TBox …
ece 627, winter ‘13

27. description logic TBox …
name symbols – symbols that appear on the left-hand side of axioms (also called defined concepts) base symbols – symbols that appear only on the right-hand side of axioms (also called primitive concepts)
ece 627, winter ‘13

28. description logic TBox …
if we know what the base symbols stand for – then the meaning of of the name symbols is completely determined*
*additionally terminology has to be definitorial
ece 627, winter ‘13

29. description logic TBox …
an important (and related to the issue of being definitorial) question – are definitions of a terminology cyclic or not? Human’ Animal hasParent.Human’ (… is cyclic iff there exists an atomic concept that uses itself)
ece 627, winter ‘13

30. description logic TBox …
ece 627, winter ‘13

31. description logic TBox …
there are terminologies with cycles that are definitorial …
ece 627, winter ‘13

32. description logic TBox …with inclusion axioms
for some concepts we will not be able to define them completely – we will state necessary conditions for concept membership using an inclusion Woman Person
ece 627, winter ‘13

33. description logic TBox …with inclusion axioms
if we do this – the terminology loses its definitorial impact, even if it is acyclic, then …
ece 627, winter ‘13

34. description logic ABox …
the second component of a knowledge base a world description or ABox
ece 627, winter ‘13

35. description logic ABox …
in ABox we describe a specific state of affairs of a given application domain we introduce individuals, by giving them names, and we asserts properties of these individuals
ece 627, winter ‘13

36. description logic ABox …
we denote names as a, b, c using concepts C and roles R, we can make assertions of the following two kinds in the ABox C(a) R(b,c)
ece 627, winter ‘13

37. description logic ABox …
C(a) – this concept assertion used to stat that a belongs to (the interpretation of C) R(b,c) – this role assertion states that c is a filler of the role R for b
ece 627, winter ‘13

38. description logic ABox …
Father(PETER) – means that PETER is a father hasChild(MARY,PAUL) – means that Paul is a child of Mary
ece 627, winter ‘13

39. description logic ABox …
examples: MotherWithoutDaughter(MARY) hasChild(MARY, PETER) hasChild(MARY, PAUL) Father(PETER) hasChild(PETER, HARRY)
ece 627, winter ‘13

40. description logic ABox …
the semantic of ABox is “open-world semantics” – we cannot assume that the knowledge in the knowledge base is complete (on the contrary to “closed-world” semantics of classical databases)
ece 627, winter ‘13

41. description logic ABox …
the semantic is given to ABoxs by extending interpretations to individual names
ece 627, winter ‘13

42. description logic ABox …
now, an interpretation I not only maps atomic concepts and roles to sets and relations, but in addition it maps each individual name a to and element we assume that distinct individual names denote distinct objects
ece 627, winter ‘13

43. description logic ABox …
the interpretation I satisfies the concept assertion C(a) if , and it satisfies the role assertion R(a, b) if
ece 627, winter ‘13

44. description logic ABox …
an interpretation satisfies the ABox A if it satisfies each assertion in A in this case we say that I is a model of the assertion or the ABox the concept assertion
ece 627, winter ‘13

45. description logic ABox …
I satisfies an ABox A with respect to a TBox T if in addition to being a model of A, it is a model of T
ece 627, winter ‘13

46. description logic inference …
and more …. … to be continued (sometime)
ece 627, winter ‘13