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Basics of Reasoning in Description Logics. Jie Bao Iowa State University Feb 7, 2006. An ontology of this talk. Roadmap. What is Description Logics (DL) Semantics of DL Basic Tableau Algorithm Advanced Tableau Algorithm. Description Logics.

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Basics of reasoning in description logics
Basics of Reasoning in Description Logics

Jie Bao

Iowa State University

Feb 7, 2006


  • What is Description Logics (DL)

  • Semantics of DL

  • Basic Tableau Algorithm

  • Advanced Tableau Algorithm

Description logics
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
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 basics
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
More for “Family” Ontology

  • (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

Dl architecture
DL Architecture

Knowledge Base

Tbox (schema)

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


Inference System

Abox (data)


(Example from Ian Horrocks, U Manchester, UK)

Dl representives
DL Representives

  • ALC: the smallest DL that is propositionally closed

    • Constructors include booleans (and, or, not),

    • Restrictions on role successors


    • 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


Dl semantics
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
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.

Roadmap Sattler, ECAI-2002,

  • What is Description Logic (DL)

  • Semantics of DL

  • Basic Tableau Algorithm

  • Advanced Tableau Algorithm

What is reasoning
What is Reasoning? Sattler, ECAI-2002,

  • "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
Reasoning Tasks Sattler, ECAI-2002,

  • 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, xICI

    • 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

Reasoning tasks 2
Reasoning Tasks(2) Sattler, ECAI-2002,

  • Many inference tasks can be reduced to subsumption reasoning

  • Subsumption can be reduced to satisfiability

Tableau algorithm
Tableau Algorithm Sattler, ECAI-2002,

  • 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
Short Example Sattler, ECAI-2002,

  • 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.

General process
General Process Sattler, ECAI-2002,

  • 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 Sattler, ECAI-2002,

Tableaux expansion selected
Tableaux Expansion(Selected) Sattler, ECAI-2002,


Termination rules
Termination Rules Sattler, ECAI-2002,

  • 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 Sattler, ECAI-2002,

  • 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
A Expansion Example Sattler, ECAI-2002,


Tree model
Tree Model Sattler, ECAI-2002,

  • 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
Requirments for Tab. Alg. Sattler, ECAI-2002,

  • 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 Sattler, ECAI-2002,

  • What is Description Logic (DL)

  • Semantics of DL

  • Basic Tableau Algorithm

  • Advanced Tableau Algorithm

Advanced tableau alg
Advanced Tableau Alg. Sattler, ECAI-2002,

  • 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 Sattler, ECAI-2002,

References Sattler, ECAI-2002,

  • 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.