1. Basics of Reasoning in Description Logics
Jie Bao
Iowa State University
Feb 7, 2006

2. An ontology of this talk

3. Roadmap What is Description Logics (DL) Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm

4. Description Logics
A formal logic-based knowledge representation language
“Description" about the world in terms of concepts (classes), roles (properties, relationships) and individuals (instances)
Decidable fragments of FOL
Widely used in database (e.g., DL CLASSIC) and semantic web (e.g., OWL language)

5. A “Family” Knowledge Base
Person include Man(Male) and Woman(Female),
A Man is not a Woman
A Father is a Man who has Child
A Mother is a Woman who has Child
Both Father and Mother are Parent
Grandmother is a Mother of a Parent
A Wife is a Woman and has a Husband( which as Man)
A Mother Without Daughter is a Mother whose all Child(ren) are not Women

6. DL for Family KB

7. DL Basics Concepts (unary predicates/formulae with one free variable)
E.g., Person, Father, Mother
Roles (binary predicates/formulae with two free variables)
E.g., hasChild, hasHudband
Individual names (constants)
E.g., Alice, Bob, Cindy
Subsumption (relations between concepts)
E.g. Female  Person
Operators (for forming concepts and roles)
And(Π) , Or(U), Not (¬)
Universal qualifier (), Existent qualifier()
Number restiction : , , =
Inverse role (-), transitive role (+), Role hierarchy

8. More for “Family” Ontology
(Inverse Role) hasParent = hasChild-
hasParent(Bob,Alice) -> hasChild(Alice, Bob)
(Transitive Role)hasBrother
hasBrother(Bob,David), hasBrother(David, Mack) -> hasBrother(Bob,Mack)
(Role Hierarchy) hasMother  hasParent
hasMother(Bob,Alice) -> hasParent(Bob, Alice)
HappyFather  Father Π 1 hasChild.Woman Π 1 hasChild.Man

9. HappyFather  Person Π 1 hasChild.Woman Π 1 hasChild.Man
DL Architecture
Knowledge Base
Tbox (schema)
HappyFather  Person Π 1 hasChild.Woman Π 1 hasChild.Man
Interface
Inference System
Abox (data)
Happy-Father(Bob)
(Example from Ian Horrocks, U Manchester, UK)

10. DL Representives ALC: the smallest DL that is propositionally closed
Constructors include booleans (and, or, not),
Restrictions on role successors
SHOIQ = OWL DL
S=ALCR+: ALC with transitive role
H = role hierarchy
O = nomial .e.g WeekEnd = {Saturday, Sunday}
I = Inverse role
Q = qulified number restriction e.g. >=1 hasChild.Man
N = number restriction e.g. >=1 hasChild

11. Roadmap What is Description Logic (DL) Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm

12. Interpretations DL Ontology: is a set of terms and their relations
Interpretation of a DL Ontology: A possible world ("model") that materalizes the ontology
Ontology:
Student  People
Student  Present.Topic
KR  Topic
DL  KR
Interpretation

13. DL Semantics
DL semantics defined by interpretations: I = (DI, .I), where
DI is the domain (a non-empty set)
.I is an interpretation function that maps:
Concept (class) name A -> subset AI of DI
Role (property) name R -> binary relation RI over DI
Individual name i -> iI element of DI
Interpretation function .I tells us how to interpret atomic concepts, properties and individuals.
The semantics of concept forming operators is given by extending the interpretation function in an obvious way.

14. DL Semantics: example I = (DI, .I) DI = {Jie_Bao, DL_Reasoning}
PeopleI=StudentI={Jie_Bao}
TopicI=KRI=DLI={DL_Reasoning}
PresentI={(Jie_Bao, DL_Reasoning)}
An interpretation that satisifies all axioms in an DL
ontology is also called a model of the ontology.

15. Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

16. Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

17. Roadmap What is Description Logic (DL) Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm

18. What is Reasoning? "Machine Understanding"
Find facts that are implicit in the ontology given explicitly stated facts
Find what you know, but you don't know you know it - yet.
Example
A is father of B, B is father of C, then A is ancestor of C.
D is mother of B, then D is female

19. Reasoning Tasks Knowledge is correct (captures intuitions)
C subsumes D w.r.t. K iff for every model I of K, CI µ DI
Knowledge is minimally redundant (no unintended synonyms)
C is equivallent to D w.r.t. K iff for every model I of K, CI = DI
Knowledge is meaningful (classes can have instances)
C is satisfiable w.r.t. K iff there exists some model I of K s.t. CI  ;
Querying knowledge
x is an instance of C w.r.t. K iff for every model I of K, xI  CI
hx,yi is an instance of R w.r.t. K iff for, every model I of K, (xI,yI)  RI
Knowledge base consistency
A KB K is consistent iff there exists some model I of K

20. Reasoning Tasks(2)
Many inference tasks can be reduced to subsumption reasoning
Subsumption can be reduced to satisfiability

21. Tableau Algorithm
Tableau Algorithm is the de facto standard reasoning algorithm used in DL
Basic intuitions
Reduces a reasoning problem to concept satisfiability problem
Finds an interpretation that satisfies concepts in question.
The interpretation is incrementally constructed as a "Tableau"

22. Short Example
given: Wife Woman, Woman Person question: if Wife Person
Reasoning process
Test if there is a individual that is a Woman but not a Person, i.e. test the satisfiability of concept C0=(WifeЬPerson)
C0(x) -> Wife(x), (¬Person)(x)
Wife(x)->Woman(x)
Woman(x) ->Person(x)
Conflict!
C0 is unsatisfiable, therefore Wife Person is true with the given ontology.

23. General Process
Transform C into negation normal form(NNF), i.e. negation occurs only in front of concept names.
Denote the transformed expression as C0, the algorithm starts with an ABox A0 = {C0(x0)}, and apply consistency-preserving transformation rules (tableaux expansion) to the ABox as far as possible.
If one possible ABox is found, C0 is satisfiable.
If not ABox is found under all search pathes, C0 is unsatisfiable.

24. NNF

25. Tableaux Expansion(Selected)
Clash

26. Termination Rules
An ABox is called complete if none of the expansion rules applies to it.
An ABox is called consistent if no logic clash is found.
If any complete and consistent ABox is found, the initial ABox A0 is satisfiable
The expansion terminates, either when finds a complete and consistent ABox, or try all search pathes ending with complete but inconsistent ABoxes.

27. Internalisation Embed the TBox in the initial ABox concept
CD is equivalent T ¬C U D (T is the "top" concept. It imeans ¬C U D is the super concept for ANY concepts)
E.g.
Given ontology: Mother  Woman Π Parent, Woman  Person
Query: Mother  Person
The intitial ABox is : ¬Mother U(Woman Π Parent) Π (¬Woman U Person) Π (Mother Π ¬Person)

28. A Expansion Example
Search

29. Tree Model
Another explanation of tableaux algorithm is that it works on a finite completion tree whose
individuals in the tableau correspond to nodes
and whose interpretation of roles is taken from the edge labels.

30. Requirments for Tab. Alg.
Similar tableaux expansions can be designed for more expressive DL languages.
A tableau algorithm has to meet three requirements
Soundness: if a complete and clash-free ABox is found by the algorithm, the ABox must satisfies the initial concept C0.
Completeness: if the initial concept C0 is satisfiable, the algorithm can always find an complete and clash-free ABox
Termination: the algorithm can terminate in finite steps with specific result.

31. Roadmap What is Description Logic (DL) Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm

32. Advanced Tableau Alg. Rich literatures in the past decade.
Advanced techniques
Blocking (Subset Blocking,Pair Locking, Dynamic Blocking)
For more expressive languages: number restriction, transitive role, inverse role, nomial, data type
Detailed analysis of complexities.
Refer to references at the end of this presentation for details

33. SHIQ Expansion Rules

34. References
F. Baader, W. Nutt. Basic Description Logics. In the Description Logic Handbook, edited by F. Baader, D. Calvanese, D.L. McGuinness, D. Nardi, P.F. Patel- Schneider, Cambridge University Press, 2002, pages
Ian Horrocks and Ulrike Sattler. Description Logics Tutorial, ECAI-2002, Lyon, France, July 23rd, 2002.
Ian Horrocks and Ulrike Sattler. A tableaux decision procedure for SHOIQ. In Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), 2005.
I. Horrocks and U. Sattler. A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation, 9(3): , 1999.