1. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Introduction
Motivation for Description Logic (DL)
DL and First Order Logic (FOL)
What is Description Logic?
DL Architecture
Reasoning Structures
Applications of DL
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

2. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
Any area with information overload.
Where amount of declarative information to be processed manually exceeds that of human abilities.
Even if a large number of resources could be found, it would take an impractical number of human hours to process.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

3. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
Need for greater push for standardization of terminologies:
Semantic Web
Health Services
Bio-informatics
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

4. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
We would like to model complex concepts using modern modeling structures.
OWL, Web Ontology Language, is a computer processible syntax for expressive Description Logic.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

5. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
Propositional Logic (PL) has well defined operators and axioms creating atomic formulas.
LHS and RHS Equivalences, such as:
Commutativity
ϕ ∨ ψ Ξ
Ψ ∨ϕ
Tautology
ϕ ∨Τ Τ
Unsatisfiability
ϕ ∧  
Implication
ϕ -> ψ
 ϕ ∨ ψ
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

6. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
We can already do a lot with PL
But …
No access to the structure of atomic statements.
Consider Statements
Mary is female
John is male
Mary and John are siblings
Consider Propositions
Mary-is-female
John-is-male
Mary-and-John-are-siblings
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

7. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
First Order Logic (FOL)
The language of Predicate Logic
Atomic formulas are sets of predicates interpreted as relations between elements of a domain.
Consider Predicates
Female (Mary)
Male (John)
Siblings (Mary, John)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

8. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Motivation
Predicate Logic and FOL allows for formulas, arguments, free and bounded variables, and constants.
Allows for literals, atomic formulas (relations), and negation.
Well formed formulas, truth connectives, quantifiers over bounded variables.
Quantifiers:
Existential
x. ϕ Ξ
For some x, ϕ is true
Universal
x. ϕ
For all x, ϕ is true
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

9. Description Logic Motivation Represent relationships between objects.
More complex relationships can be defined in terms of more basic relationships or objects.
Woman Ξ
Person Π Female
Man
Person Π Woman
verses
Woman-is-a-Person-who-is-a-Female
Man-is-a-Person-who-is-not-a-Woman
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

10. Description Logic … and First Order Logic
Description Logic (DL) is derived from FOL.
DL Relationships are limited to 1-argument predicates called concepts and 2-argument predicates called roles.
FOL
Predicates
DL Relationships
Description
unary 
concepts
Woman( x )
binary
roles
Woman( x ).hasChild( y , x )
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

11. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
… and First Order Logic
Concept C and its translation π(C)(x) are equivalent in the interpretation I = (ΔI, . I ) and all a  ΔI , where Δ is the domain, and I is the interpretation.
Consider:
a  CI iff I |= π(C)(a).
a is an instance of an interpretation of WOMAN iff
a FOL translation of this concept is true for "a"
in the interpretation.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

12. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
… and First Order Logic
Translation of roles in FOL takes the form of a predicate with two arguments.
Roles relate two concepts to each other.
Roles cannot occur by themselves; only as a part of a complex concept.
This puts restrictions on syntax of FOL formulas that are allowed.
Important to guarantee decidability and good computational properties of DLs.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

13. Description Logic … and First Order Logic
Translations require mappings on two variables
: π(x) and π(y)
πx(R:C) = y:R(y; x) ∧ πy(C)
Mother(x) is true iff x truth values match those of the definition of Woman with some y a child.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

14. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
… and First Order Logic
Unary Concept translations
Binary Role translation
Description Logic
First Order Logic
πx(A) 
A(x)
πx(C Π D)
πx(C) ∧ πx(D)
πx(C  D)
πx(C) ∨ πx(D)
πx(R.C)
y.R(x; y) ∧ πy(C)
πx( R.C)
y.R(x; y)  πy(C)
Description Logic
First Order Logic
πy(A) 
A(y)
πy(C Π D)
πy(C) ∧ πy(D)
πy(C  D)
πy(C) ∨ πy(D)
πy(R.C)
 y.R(y, x) ∧ πx(C)
πy(R.C)
 y.R(y, x)  πx(C)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

15. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
… and First Order Logic
Counting Quantifier translations
Some x with at least n y’s
A simple existential quantifier is insufficient:
Description Logic
First Order Logic
πx(≥ n R) 
 ≥n y.R(x, y)
πx(≤ n R)
 ≤n y.R(x, y)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

16. Description Logic What is Description Logic?
Sub-language of First Order Logic
Model hierarchical and complex relationships.
Collection of these statements is called TBox.
Man Ξ
Person Π Woman
Father
(Person Π Woman) Π hasChild:Person
Father Ξ
(Man) Π hasChild:Person
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

17. Description Logic What is Description Logic?
The TBox is known as an ONTOLOGY (from Greek: science of beings, or science of things that exist).
Ontology used to be a sub-discipline of philosophy.
It is now at forefront of real world applications.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

18. Description Logic What is Description Logic?
Models systems which are complex through size and cyclic definitions.
“Man who has Only Sons” (Mos)
Mos Ξ
Man Π hasChild.Man
“Man who has Only Male Offspring” (Momo)
Momo Ξ
Man Π hasChild.Momo
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

19. Description Logic What is Description Logic?
DL handles cyclic definitions through fixpoints
Least fixpoint is the lowest x value where f(x) = x
Greatest fixpoint is the greatest x value where f(x) = x
Halting depends on the domain of x
Example: a binary tree’s least-point are the nodes with only terminal nodes at the branches.
BinaryTree Ξ
Tree Π (≤ 2 has-branch) Π (has-branch.BinaryTree)
BinaryEndTree
Tree Π (≤ 2 has-branch) Π (has-branch.BinaryTree)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

20. Description Logic What is Description Logic?
DL has variable-free syntax
variables are implicitly understood
There can be only two of them: x or y, but they can be reused.
DL concepts denote sets of individuals.
For concept C, and translation I.
a  CI iff I |= π(C)(a).
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

21. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
Basic description language AL
AL = attribute language
A and B are atomic concepts
C and D are concept descriptions
R represents atomic roles
Other Languages in this family are extensions of AL.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

22. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
Basic description language AL
C, D  A
atomic concept Τ
universal concept 
bottom concept A
atomic negation
C Π D
intersection
 R.
value restriction
 R.Τ
Limited existential quantification
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

23. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
Atomic Concepts and Roles
Person Π Female
Person who is not a female
Person Π hasChild.Τ
Person with at least 1 child
Person Π hasChild.
Person with no children
Person Π hasChild.Female
Person with at least one daughter
Person Π hasChild.Female
Person with only daughters
Mother  Father
Person who is a Parent
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

24. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
Limited to two-variable structures.
Handles acyclic, as well as cyclic statements.
Acyclic: LHS has no terms which appear in RHS
Mos Ξ
Man Π hasChild.Man
Cyclic: LHS has a term which appears in RHS
Momo Ξ
Man Π hasChild.Momo
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

25. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
DL defines statements by stating assertions about a subject using its vocabulary.
Type of statements which make up a DL Ontology are the ABox and the TBox formalisms.
TBox = terminology used in a domain.
ABox = assertions which make statements about particular object instances in that domain, using the terminology in TBox.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

26. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
Knowledgebase
TBox
Description Language
Reasoning
ABox
Application
Programs
Rules
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

27. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
DL Architecture
TBox
Father Ξ
Man Π hasChild.Person
Mother
Woman Π hasChild.Person
MotherWithoutDaughter
Mother Π hasChild.¬Woman
ABox
MotherWithoutDaughter(MARY)
Father(PETER)
hasChild(MARY, PETER)
hasChild(PETER, HARRY)
hasChild(MARY, PAUL)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

28. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Reasoning Structures
DL Knowledgebase stores more then definitions and assertions.
DL contains implicit knowledge which can be made explicit through inferences.
Inferences are made through a set of rules of inference.
TBox contains these statement.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

29. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Reasoning Structures
Modeling a domain requires defining a terminology, say T, which satisfies all previously defined statements about that domain.
Any new interpretation must satisfy the axioms of T, and result in a non-empty set of concepts.
Otherwise it is unsatisfiable.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

30. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Reasoning Structures
Satisfiability: A concept C is satisfiable with respect to T if there exists a model I of T such that CI is nonempty:
I |= C
Subsumption: A concept C is subsumed by a concept D with respect to T if CI  DI for every model I of T:
C T D T |= (C D) |Π |Π
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

31. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Reasoning Structures
Subsumption con’t
LHS is subsumed by the more general terms in RHS:
Mother
Woman
Father
Man
Brunch
Meal
Sport Car
Vehicle |Π |Π |Π |Π
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

32. Description Logic Reasoning Structures
Equivalence: Two concepts C and D are equivalent if C subsumes D and D subsumes C, with respect to T if CI = DI for every model I of T:
[ C ΞT D and T |= C Ξ D ] [ C D and D C ] |Π |Π
Disjointness: Two concepts C and D are disjoint with respect to T if:
CI  DI = 
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

33. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
Different extensions to AL-languages.
ALU
Union
(C  D) I
ALε
Full existential quantification
R.C
ALN
Number restrictions
≥ n R and ≤ n R
ALFL-
Non-atomic negation A
AL in combination with U, ε, N, and FL- form 8 languages which are pairwise non-equivalent.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

34. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
The special extension ALUε , represented by ALC, describe a commonly used language with axioms for union and existential quantification.
The ALC extension is the basis for SHOIN, which in turn is the basis for the Web Ontology Language OWL
OWL is formally recommended by the World Wide Web Consortium, W3C.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

35. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
OWL is a markup language given the tasks of providing syntax used to model ontologies on the web.
The Semantic Web is an attempt to organize the plethora of information available on the internet
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

36. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
On the web, instead of showing all available links and infinite levels of sub-topics, a high level “description” is given.
Description identifies the subject of the content on the web.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

37. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
Cyclic self-referenced nodes are contained within a higher level concept.
Bound by fixpoints.
This abstraction inherited from DL ensures satisfiability in the system.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

38. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
OWL Example: “Daugther” (misspelling preserved)
<owl:Class rdf:ID="Daugther">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<owl:Class rdf:ID="Child"/>
<owl:Class rdf:ID="Woman"/>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<owl:disjointWith>
<owl:Class rdf:ID="Son"/>
</owl:disjointWith> -
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

39. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
OWL Example: “Daugther” (misspelling preserved)
<owl:Class rdf:ID="Daugther">
<owl:equivalentClass>
<owl:Class>
<owl:intersectionOf rdf:parseType="Collection">
<owl:Class rdf:ID="Child"/>
<owl:Class rdf:ID="Woman"/>
</owl:intersectionOf>
</owl:Class>
</owl:equivalentClass>
<owl:disjointWith>
<owl:Class rdf:ID="Son"/>
</owl:disjointWith>
Daugther Ξ
Child Π Woman
Daugther Π Son  -
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

40. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
Description Logic can be analyzed for satisfiability with Theorem Provers (Prover9 / Vampire / Otter)
Tableau Calculus
Tableaux Calculus is a decision procedure solving the problem of satisafiability.
If a formula is satisfiable, the procedure will constructively exhibit a model of the formula.
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

41. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
Tableau Calculus con’t
The basic idea is to incrementally build the model by looking at the formula, by decomposing it in a top/down fashion.
The procedure exhaustively looks at all the possibilities, so that it can eventually prove that no model could be found for unsatisfiable formulas.
Unsatifiable formula contains a contradiction such as: p(a) Π p(a)
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

42. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
Applications of DL
Closed vs Open World Semantics
Connections to external databases require database schema mappings.
Communicating TBoxes need to be mapped in order for the data in ABoxes to be meaningful.
Closed systems represent internal databases where no mappings are required.
A complete theory in a closed system may not be complete in an open world semantic
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

43. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
References
Enrico Franconi, DESCRIPTION LOGICS - Tutorial Course Information, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy;
Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi,
Peter F. Patel-Schneider (Eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press 2003
Christine Golbreich, A SWRL/OWL Demo Ontology About Family Relationships,
OWL Ontologies, ProtegeOntologiesLibrary;
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science

44. Bart Gajderowicz, Ryerson University, Dept. of Computer Science
Description Logic
The End
Thank you
Q & A
Bart Gajderowicz
March-03-09
Bart Gajderowicz, Ryerson University, Dept. of Computer Science