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Basics of Reasoning in Description Logics

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Jie Bao

Iowa State University

Feb 7, 2006

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

- 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)

- 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

- 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

- (Inverse Role) hasParent = hasChild-
- hasParent(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

Knowledge Base

Tbox (schema)

HappyFather Person Π 1 hasChild.Woman Π 1 hasChild.Man

Interface

Inference System

Abox (data)

Happy-Father(Bob)

(Example from Ian Horrocks, U Manchester, UK)

- 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

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

- 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

- 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.

- 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,

Source: Description Logics Tutorial, Ian Horrocks and Ulrike Sattler, ECAI-2002,

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

- "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

- Knowledge is correct (captures intuitions)
- C subsumes D w.r.t. K iff for every modelI of K, CI µ DI

- Knowledge is minimally redundant (no unintended synonyms)
- C is equivallent to D w.r.t. K iff for every modelI of K, CI = DI

- Knowledge is meaningful (classes can have instances)
- C is satisfiable w.r.t. K iff there exists some modelI of K s.t. CI;

- Querying knowledge
- x is an instance of C w.r.t. K iff for every modelI of K, xICI
- hx,yi is an instance of R w.r.t. K iff for, every modelI of K, (xI,yI) RI

- Knowledge base consistency
- A KB K is consistent iff there exists some modelI of K

- Many inference tasks can be reduced to subsumption reasoning
- Subsumption can be reduced to satisfiability

- 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"

- given: Wife Woman, Woman Personquestion: 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=(WifeΠ¬Person)
- 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.

- 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.

Clash

- 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.

- 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 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)

Search

- 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.

- 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.

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

- 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

- 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.