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Basics of Reasoning in
Description Logics
Jie Bao
Iowa State University
Feb 7, 2006
An ontology of this talk
Topic
People
Knowledge Representation
Student
Description Logic
present
Jie Bao
DL reasoning
Roadmap
What is Description Logics (DL)
Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm
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)
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
DL for Family KB
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
More for “Family” Ontology
(Inverse Role) hasParent = hasChildhasParent(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
DL Architecture
Knowledge Base
Abox (data)
Happy-Father(Bob)
Interface
HappyFather  Person Π 1
hasChild.Woman Π 1 hasChild.Man
Inference System
Tbox (schema)
(Example from Ian Horrocks, U Manchester, UK)
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
Roadmap
What is Description Logic (DL)
Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm
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:
Interpretation
Student  People
Student  Present.Topic
KR  Topic
DL  KR
People
Topic
Knowledge Representation
Student
Description Logic
present
Jie Bao
DL reasoning
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.
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.
Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,
CSE-291: Ontologies in Data & Process Integration
Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,
CSE-291: Ontologies in Data & Process Integration
Roadmap
What is Description Logic (DL)
Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm
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
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
Reasoning Tasks(2)
Many inference tasks can be reduced to
subsumption reasoning
Subsumption can be reduced to satisfiability
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"
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.
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.
NNF
Tableaux
Expansion(Selected)
Clash
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.
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)
A
Expansion
Example
Search
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.
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.
Roadmap
What is Description Logic (DL)
Semantics of DL
Basic Tableau Algorithm
Advanced Tableau Algorithm
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
SHIQ Expansion Rules
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. PatelSchneider, Cambridge University Press, 2002, pages 47100.
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.