Post on 18-Nov-2014
description
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
10:58:57 2R. Akerkar: Reasoning in DL
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)
10:58:57 3R. Akerkar: Reasoning in DL
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
10:58:57 5R. 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
10:58:57 6R. Akerkar: Reasoning in DL
(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
10:58:57 7R. Akerkar: Reasoning in DL
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
10:58:57 9R. Akerkar: Reasoning in DL
What is Description Logic (DL)p g ( ) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm
10:58:57 10R. Akerkar: Reasoning in DL
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
10:58:57 11R. Akerkar: Reasoning in DL
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.
10:58:57 12R. Akerkar: Reasoning in DL
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.
10:58:57 13R. Akerkar: Reasoning in DL
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
What is Description Logic (DL)p g ( ) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm
10:58:57 16R. 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
10:58:57 17R. Akerkar: Reasoning in DL
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
10:58:57 18R. Akerkar: Reasoning in DL
Many inference tasks can be reduced to subsumptionMany inference tasks can be reduced to subsumption reasoning
Subsumption can be reduced to satisfiability p y
10:58:57 19R. Akerkar: Reasoning in DL
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
10:58:57 20R. Akerkar: Reasoning in DL
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.
10:58:57 22R. Akerkar: Reasoning in DL
10:58:57 23R. Akerkar: Reasoning in DL
ClashClash
10:58:57 24R. Akerkar: Reasoning in DL
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.
10:58:57 25R. Akerkar: Reasoning in DL
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)
10:58:57 26R. Akerkar: Reasoning in DL
Search
10:58:57 27R. Akerkar: Reasoning in DL
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
10:58:57 28R. Akerkar: Reasoning in DL
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.
10:58:57 29R. Akerkar: Reasoning in DL
What is Description Logic (DL)p g ( ) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm
10:58:57 30R. Akerkar: Reasoning in DL
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.
10:58:57 31R. Akerkar: Reasoning in DL
SHIQ Expansion Rules
10:58:57 32R. Akerkar: Reasoning in DL
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.
10:58:57 33R. Akerkar: Reasoning in DL