The document discusses description logics (DL), which are formal logic-based knowledge representation languages used to represent knowledge in terms of concepts, roles, and individuals. It covers the semantics of DL, basic tableau algorithms for reasoning, and more advanced tableau algorithms for more expressive DL languages. The key points are: - DL allows knowledge to be represented through concepts, roles, and individuals. Tableau algorithms are commonly used for reasoning. - The semantics of DL are defined using interpretations over a domain. Tableau algorithms work by trying to construct an interpretation that satisfies a concept. - Basic tableau algorithms expand concept descriptions into a tableau using rules until a clash is found, proving unsatisfiability, or no

1. Rajendra Akerkar
Vestlandsforsking, Norway
R. Akerkar: Reasoning in DL 10:58:57 1

2.  What is Description Logics ( )
p g (DL)
 Semantics of DL
 Basic Tableau Algorithm
 Advanced Tableau Algorithm
R. Akerkar: Reasoning in DL 10:58:57 2

3.  g g
A formal logic-based knowledge
representation language
◦ “Description" about the world in terms of concepts
(classes), roles (
( l ) l (properties, relationships) and
ti l ti hi ) d
individuals (instances)
 Decidable fragments of FOL
g
 Widely used in database (e.g., DL CLASSIC)
and semantic web (e.g., OWL language)
R. Akerkar: Reasoning in DL 10:58:57 3

4. Person include Man(Male) and
Woman(Female),
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
g
whose all Child(ren) are not Women
R. Akerkar: Reasoning in DL 10:58:57 4

5. 10:58:57 R. Akerkar: Reasoning in DL 5

6.  Concepts (unary predicates/formulae with one free variable)
◦ E.g., Person, Father, Mother
E P F th M th
 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
R. Akerkar: Reasoning in DL 10:58:57 6

7.  (Inverse Role) hasParent = hasChild-
◦ hasParent(Bob,Alice) -> hasChild(Alice, Bob)
 (Transitive Role)hasBrother
◦ h B h (B b D id) h B h (D id M k)
hasBrother(Bob,David), hasBrother(David, Mack)
-> hasBrother(Bob,Mack)
 (Role Hierarchy) hasMother  hasParent
◦ hasMother(Bob,Alice) -> hasParent(Bob, Alice)
 HappyFather  Father Π hasChild.Woman
ppy
Π hasChild.Man
R. Akerkar: Reasoning in DL 10:58:57 7

8. Knowledge Base
Tbox (schema)
HappyFather  Person Π 
ystem
hasChild.Woman Π hasChild.Man
face
Inference Sy
Interf
Abox (data)
Happy-Father(Bob)
(Example taken from Ian Horrocks, U Manchester, UK)
R. Akerkar: Reasoning in DL 10:58:57 8

9.  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
hasChild Man
 N = number restriction e.g. >=1 hasChild
R. Akerkar: Reasoning in DL 10:58:57 9

10.  What is Description Logic ( )
p g (DL)
 Semantics of DL
 Basic Tableau Algorithm
 Advanced Tableau Algorithm
R. Akerkar: Reasoning in DL 10:58:57 10

11. DL Ontology: is a set of terms and their
gy
relations
Interpretation of a DL Ontology: A possible
world ("model") that materializes the
ontology
Ontology:
Student  People
Student  Present Topic
Present.Topic
KR  Topic
DL  KR
10:58:57 R. Akerkar: Reasoning in DL 11

12.  DL semantics defined by interpretations: I = (I, .I),
where
◦ I is the domain (a non-empty set)
◦ .I is an interpretation function that maps:
 Concept (class) name A -> subset AI of I
 Role (property) name R -> binary relation RI over I
 I di id l name i -> iI element of I
Individual l t f
 Interpretation function .I tells us how to interpret
atomic concepts, properties and individuals.
p ,p p
◦ The semantics of concept forming operators is given by
extending the interpretation function in an obvious way.
R. Akerkar: Reasoning in DL 10:58:57 12

13.  I = (I, .I)
 I = {Raj, DL_Reasoning}
 PeopleI=StudentI={Raj}
 TopicI=KRI=DLI={DL_Reasoning}
 PresentI={(Raj, DL_Reasoning)}
An interpretation that satisifies all axioms in an DL
ontology is also called a model of the ontology.
R. Akerkar: Reasoning in DL 10:58:57 13

14. Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002
R. Akerkar: Reasoning in DL 10:58:57 14

15. Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002
R. Akerkar: Reasoning in DL 10:58:57 15

16.  What is Description Logic ( )
p g (DL)
 Semantics of DL
 Basic Tableau Algorithm
 Advanced Tableau Algorithm
R. Akerkar: Reasoning in DL 10:58:57 16

17.  "Machine Understanding" g
 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
R. Akerkar: Reasoning in DL 10:58:57 17

18.  Knowledge is correct (captures intuitions)
◦ C subsumes D w.r.t. K iff for every model I of K, CI µ DI
wrt K
 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 i meaningful ( l
K l d is i f l (classes can h have instances)
i t )
◦ 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
R. Akerkar: Reasoning in DL 10:58:57 18

19. Many inference tasks can be reduced to subsumption
reasoning
Subsumption can be reduced to satisfiability
p y
10:58:57 R. Akerkar: Reasoning in DL 19

20.  g
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
p p
question.
◦ The interpretation is incrementally constructed as a
"Tableau"
Tableau
R. Akerkar: Reasoning in DL 10:58:57 20

21.  given: Wife Woman, Woman Person
question: if Wife Person
 Reasoning process
◦ T t if th
Test there is a individual th t i a W
i i di id l that is Woman b t not
but t
a Person, i.e. test the satisfiability of concept
C0=(WifeЬPerson)
◦ C0(x) -> Wife(x), (¬Person)(x)
◦ Wife(x)->Woman(x)
◦W ( ) >P
Woman(x) ->Person(x) ( )
◦ Conflict!
◦ C0 is unsatisfiable, therefore Wife Person is true
with the given ontology.
R. Akerkar: Reasoning in DL 10:58:57 21

22.  Transform C into negation normal form(NNF),
i.e. negation occurs only in front of concept
i ti l i f t f t
names.
 Denote the transformed expression as C0, the
p
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 f
f d d ll h h
found under all search pathes,
C0 is unsatisfiable.
R. Akerkar: Reasoning in DL 10:58:57 22

23. R. Akerkar: Reasoning in DL 10:58:57 23

24. Clash
R. Akerkar: Reasoning in DL 10:58:57 24

25.  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.
l hi f d
 If any complete and consistent ABox is
found,
found the initial ABox A0 is satisfiable
 The expansion terminates, either when
finds a complete and consistent ABox or
ABox,
try all search pathes ending with complete
but inconsistent ABoxes.
R. Akerkar: Reasoning in DL 10:58:57 25

26.  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 f ANY concepts)
t for t )
 E.g.
◦ Given ontology: Mother  Woman Π Parent
Parent,
Woman  Person
◦ Query: Mother  Person
y
◦ The intitial ABox is : ¬Mother U(Woman Π Parent)
Π (¬Woman U Person) Π (Mother Π ¬Person)
R. Akerkar: Reasoning in DL 10:58:57 26

27. Search
R. Akerkar: Reasoning in DL 10:58:57 27

28.  Another explanation of tableaux algorithm
is that it works on a finite completion tree
whose
◦ i di id l i th t bl
individuals in the tableau correspond t nodes
d to d
◦ and whose interpretation of roles is taken from
the edge labels.
g
R. Akerkar: Reasoning in DL 10:58:57 28

29.  Similar tableaux expansions can be
designed for more expressive DL
d i df i
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
algorithm
satisfies the initial concept C0.
◦ Completeness: if the initial concept C0 is
satisfiable, the algorithm can always fi d an
i fi bl h l ih l find
complete and clash-free ABox
◦ Termination: the algorithm can terminate in
finite steps with specific result.
R. Akerkar: Reasoning in DL 10:58:57 29

30.  What is Description Logic ( )
p g (DL)
 Semantics of DL
 Basic Tableau Algorithm
 Advanced Tableau Algorithm
R. Akerkar: Reasoning in DL 10:58:57 30

31.  Rich literatures in the past decade.
 Advanced techniques
◦ Blocking (Subset Blocking, Pair Locking, Dynamic
Blocking)
◦ For more expressive languages: number
restriction, inverse role, transitive role, nomial,
data type
◦ Detailed analysis of complexities.
R. Akerkar: Reasoning in DL 10:58:57 31

32. SHIQ Expansion Rules
R. Akerkar: Reasoning in DL 10:58:57 32

33.  F. Baader, W. Nutt. Basic Description Logics. In the Description
Logic Handbook, edited by F Baader, D. Calvanese, D.L.
Handbook 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.
R. Akerkar: Reasoning in DL 10:58:57 33