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Title: Basics of Reasoning in Description Logics

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

2
An ontology of this talk
3
• What is Description Logics (DL)
• Semantics of DL
• Basic 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(?) , 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) -gt hasChild(Alice, Bob)
• (Transitive Role)hasBrother
• hasBrother(Bob,David), hasBrother(David, Mack)
-gt hasBrother(Bob,Mack)
• (Role Hierarchy) hasMother ? hasParent
• hasMother(Bob,Alice) -gt hasParent(Bob, Alice)
• HappyFather ? Father ? ?1 hasChild.Woman ? ?1
hasChild.Man

9
DL Architecture
Knowledge Base
Tbox (schema)
HappyFather ? Person ? ?1 hasChild.Woman ? ?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
• SALCR ALC with transitive role
• H role hierarchy
• O nomial .e.g WeekEnd Saturday, Sunday
• I Inverse role
• Q qulified number restriction e.g. gt1
hasChild.Man
• N number restriction e.g. gt1 hasChild

11
• What is Description Logic (DL)
• Semantics of DL
• Basic 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 -gt subset AI of DI
• Role (property) name R -gt binary relation RI over
DI
• Individual name i -gt 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
• PeopleIStudentIJie_Bao
• TopicIKRIDLIDL_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
• What is Description Logic (DL)
• Semantics of DL
• Basic 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
• 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
• 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(Wife?Person)
• C0(x) -gt Wife(x), (Person)(x)
• Wife(x)-gtWoman(x)
• Woman(x) -gtPerson(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 ? Parent, Woman ?
Person
• Query Mother ? Person
• The intitial ABox is Mother U(Woman ? Parent)
? (Woman U Person) ? (Mother ? 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
• What is Description Logic (DL)
• Semantics of DL
• Basic Tableau Algorithm

32
• Rich literatures in the past decade.
• 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 47-100.
• 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)385-410, 1999.