Reasoning in Description Logics

Rajendra AkerkarVestlandsforsking, Norway

10:58:57 1R. Akerkar: Reasoning in DL

What is Description Logics (DL)p g ( ) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm


A formal logic-based knowledge g grepresentation language ◦ “Description" about the world in terms of concepts

( l ) l ( ti l ti hi ) d(classes), roles (properties, relationships) and individuals (instances)

Decidable fragments of FOLg Widely used in database (e.g., DL CLASSIC)

and semantic web (e.g., OWL language)


Person include Man(Male) and Woman(Female)Woman(Female), A Man is not a WomanA Father is a Man who has ChildA Father is a Man who has ChildA Mother is a Woman who has ChildBoth Father and Mother are ParentBoth Father and Mother are ParentGrandmother is a Mother of a ParentA Wife is a Woman and has a Husband(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 Women10:58:57 4R. Akerkar: Reasoning in DL

Concepts (unary predicates/formulae with one free variable)E P F th M th◦ E.g., Person, Father, Mother

Roles (binary predicates/formulae with two free variables)◦ E.g., hasChild, hasHudband

Individual names (constants) 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 : Number restiction : ◦ Inverse role (-), transitive role (+), Role hierarchy


(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 (Role Hierarchy) hasMother hasParent◦ hasMother(Bob,Alice) -> hasParent(Bob, Alice)

HappyFather Father Π hasChild.Woman ppy Π hasChild.Man


Knowledge Base

Tbox (schema)HappyFather Person Π

hasChild.Woman Π hasChild.Man

yste

m

face

Abox (data) ren

ce S

y

Inte

rf

Happy-Father(Bob)

Infe

(Example taken from Ian Horrocks, U Manchester, UK)10:58:57 8R. Akerkar: Reasoning in DL

ALC: the smallest DL that is propositionally closedclosed◦ Constructors include booleans (and, or, not),

Restrictions on role successors SHOIQ = OWL DL

S=ALCR+: ALC with transitive roleH = role hierarchyO = nomial .e.g WeekEnd = {Saturday, Sunday}I = Inverse roleQ = qulified number restriction e.g. >=1 hasChild ManhasChild.Man N = number restriction e.g. >=1 hasChild


What is Description Logic (DL)p g ( ) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm


DL Ontology: is a set of terms and their gyrelationsInterpretation of a DL Ontology: A possible world ("model") that materializes the ontology

Ontology:

Student PeopleStudent Present TopicStudent Present.TopicKR TopicDL KR


DL semantics defined by interpretations: I = (I, .I),

where◦ I is the domain (a non-empty set) ◦ .I is an interpretation function that maps: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 i iI l t f I Individual name i -> iI element of I

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.


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 DLontology is also called a model of the ontology.


Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-200210:58:57 14R. Akerkar: Reasoning in DL

Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-200210:58:57 15R. Akerkar: Reasoning in DL

"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 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


Knowledge is correct (captures intuitions)C subsumes D w r t K iff for every model I of K CI µ DI◦ 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

K l d i i f l ( l h i t ) 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 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) RIR

Knowledge base consistency◦ A KB K is consistent iff there exists some model I of K


Many inference tasks can be reduced to subsumptionMany inference tasks can be reduced to subsumption reasoning

Subsumption can be reduced to satisfiability p y


Tableau Algorithm is the de facto standard greasoning 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


given: Wife Woman, Woman Personquestion: if Wife Person

Reasoning processT t if th i i di id l th t i W b t t◦ Test if there is a individual that is a Woman but not a Person, i.e. test the satisfiability of concept C0=(WifeΠ¬Person)0◦ C0(x) -> Wife(x), (¬Person)(x)◦ Wife(x)->Woman(x)

W ( ) >P ( )◦ Woman(x) ->Person(x)◦ Conflict!◦ C0 is unsatisfiable, therefore Wife Person is trueC0 is unsatisfiable, therefore Wife Person is true

with the given ontology.10:58:57 21R. Akerkar: Reasoning in DL

Transform C into negation normal form(NNF), i ti l i f t f ti.e. negation occurs only in front of concept names.

Denote the transformed expression as C0, the p 0,algorithm starts with an ABox A0 = {C0(x0)}, and apply consistency-preserving transformation rules (tableaux expansion) to the ABox as farrules (tableaux expansion) to the ABox as far as possible.

If one possible ABox is found, C0 is satisfiable.f f d d ll h h If not ABox is found under all search pathes, C0 is unsatisfiable.


ClashClash


An ABox is called complete if none of the expansion rules applies to it.

An ABox is called consistent if no logic l h i f dclash is found.

If any complete and consistent ABox is found the initial ABox A is satisfiablefound, the initial ABox A0 is satisfiable

The expansion terminates, either when finds a complete and consistent ABox orfinds a complete and consistent ABox, or try all search pathes ending with complete but inconsistent ABoxes.


Embed the TBox in the initial ABox concept CD is equivalent T ¬C U D (T is the

"top" concept. It imeans ¬C U D is the super t f ANY t )concept for ANY concepts)

E.g. Given ontology: Mother Woman Π Parent◦ Given ontology: Mother Woman Π Parent, Woman Person◦ Query: Mother Persony◦ The intitial ABox is : ¬Mother U(Woman Π Parent) Π (¬Woman U Person) Π (Mother Π ¬Person)


Search


Another explanation of tableaux algorithm is that it works on a finite completion tree whose

i di id l i th t bl d t d◦ individuals in the tableau correspond to nodes ◦ and whose interpretation of roles is taken from

the edge labels.g


Similar tableaux expansions can be d i d f i DLdesigned for more expressive DL languages.

A tableau algorithm has to meet three A tableau algorithm has to meet three requirements◦ Soundness: if a complete and clash-free ABox

is found by the algorithm the ABox mustis found by the algorithm, the ABox must satisfies the initial concept C0.◦ Completeness: if the initial concept C0 is

i fi bl h l i h l fi dsatisfiable, the algorithm can always find an complete and clash-free ABox◦ Termination: the algorithm can terminate in

finite steps with specific result.


Rich literatures in the past decade. Advanced techniques◦ Blocking (Subset Blocking, Pair Locking, Dynamic

Blocking)Blocking)◦ For more expressive languages: number

restriction, inverse role, transitive role, nomial, data type◦ Detailed analysis of complexities.


SHIQ Expansion Rules


F. Baader, W. Nutt. Basic Description Logics. In the Description Logic Handbook edited by F Baader D Calvanese D LLogic 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, 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.


Reasoning in Description Logics

Education

Transcript of Reasoning in Description Logics