Description Logics as Ontology Languages for Semantic Webs

Presented by:-Somya Gupta(10305011)Akshat Malu (10305012)

Swapnil Ghuge (10305907)

Franz Baader, Ian Horrocks, and Ulrike Sattler

Presentation outline

• Introduction

• Description Logic – Concepts

• Description Logic v/s Predicate Logic

• Reasoning in Description Logic

• Summary

• What is Semantic Web?

Semantic Web is a group of methods and technologies to allow machines to understand the meaning - or "semantics" - of information on the World Wide Web.

• Use of Semantic Web

According to the original vision, the availability of machine-readable metadata would enable automated agents and other software to access the Web more intelligently. The agents would be able to perform tasks automatically and locate related information on behalf of the user.

But how do we make the same thing readable to both man and machine?

• This can be done using Ontology Languages.

• An ontology is a collection of definitions of terminologies and concepts. The shared understanding comes from the fact that all the agents interpret the concepts with respect to the same ontology.

Pre-requisites of an Ontology Language (1)

• The syntax should be both intuitive to human users and compatible with existing Web standards.

• The semantics should be formally specified to provide a shared understanding.

• Expressive power adequate enough for defining the relevant concepts in enough detail, but not too expressive to make reasoning infeasible.

Pre-requisites of an Ontology Language (2)

• Sound reasoning required for:

– Ensuring quality of Ontology

– Deriving implied relations

– Testing non-contradictory concepts

– Computing concept hierarchy

– Inter-operability and integration of various ontologies

• In nutshell, we need a well-defined semantics and a powerful reasoning tool.

Presentation outline

• Introduction

• Description Logic – Concepts

• Description Logic v/s Predicate Logic

• Reasoning in Description Logic

• Summary

What Are Description Logics?

Description logics (DL) are a family of formal knowledge representation languages. They are more expressive than propositional logic but have more efficient decision problems than first-order predicate logic.

DL Terminologies

• Concepts (formulae)

– E.g., Human, HappyParent, (Male t Female)– Concepts constructed using booleans

u, t, :,– plus restricted quantifiers

9, 8 (9hasChild.Doctor)

• Roles (modalities)

– E.g., hasChild, hasParent, hasAncestor, loves

• Individuals (Ground Terms)

– E.g., John, Mary, Italy

An Example (1)

E.g., Person all of whose children are either Doctors or have a child who is a Doctor:

Person u 8hasChild.(Doctor t 9hasChild.Doctor)

8x (Person (x) ˄ 8y(Child (y,x) ˄(Doctor(y) ˄9z (Child (z,y) D˄octor(z)))))

An Example (2)

E.g., A man that is married to a Doctor, and has at least 5 children, all of whom are Professors.

Human u ￢Female u ∃married.Doctor u(≥5 hasChild) u ∀hasChild.Professor

A man

That is married to a doctor, and

Has at least 5 children,

All of whom are professors.

Human u ￢Female u∃married.Doctor u(≥5 hasChild) u∀hasChild.Professor

Interpretation

• An Interpretation I associates

– Concepts C with sets CI and

– Roles r with binary relations rI

• The semantics of the constructors is defined through identities:– (C u D)I = CI ∩ DI

– (≥ n r)I = {d | #{e | (d,e) ∈ rI} ≥ n} (≥5 hasChild)

– (∀ r.C)I = {d | ∀e: (d,e) ∈ rI ⇒ e ∈ CI} (∀hasChild.Professor)

– …

DL Knowledge Base• A TBox is a set of “schema” axioms (sentences),

e.g.:{Doctor v Person,HappyParent ´ Person u 8hasChild.(Doctor t

9hasChild.Doctor)}

– i.e., a background theory (a set of non-logical axioms)

• An ABox is a set of “data” axioms (ground facts), e.g.:

{John:HappyParent,

John hasChild Mary}

– i.e., non-logical axioms including (restricted) use of nominals

Presentation outline

• Introduction

• Description Logic - Concepts

• Description Logic v/s Predicate Logic

• Reasoning in Description Logic

• Summary

Description Logic v/s Predicate Logic

• Anything that can be expressed in DLs can be equivalently expressed in PL with at most 3 variables. In other words, anything that can be represented with at most 3 variables in PL is expressible in DLs.

• There are certain relatively simple definitions of unary predicates that are expressible as conjunctive queries, but which cannot be expressed as concepts in DLs.– E.g. 9 x (Cat(x) ᴧBlack (x))

– Cat u Black

Description Logic v/s Predicate Logic (2)

• The source of expressive weakness of DLs is exactly their computational strength: the absence of variables.

• Features like at-most (≤) and at-least (≥) in DLs cannot be expressed in PL with a bounded number of variables.

Presentation outline

• Introduction

• Description Logic – Concepts

• Description Logic v/s Predicate Logic

• Reasoning in Description Logic

• Summary

ReasoningSubsumption: Is C a sub-concept of D?

C v D iff CI ⊆ DI for all interpretations I.

Satisfiability: Is the concept description C non-contradictory?

C is satisfiable iff there is an I such that CI ≠ Ø.

Consistency: Is the ABox A non-contradictory?

A is consistent iff it has a model.

Instantiation: Is ‘e’ an instance of C w.r.t. the given ABox A?

A |= C(e) iff eI ∈ CI for all models I of A.

Using Standard DL Techniques

• Key reasoning tasks reducible to KB satisfiability

– E.g., C v D w.r.t. KB K iff K [ {x:(C u :D)} is not satisfiable

C D :C t D• DL systems typically use (highly optimised) tableaux

algorithms to decide satisfiability/consistency of KB

• Tableaux algorithms work by trying to construct a concrete example (model) consistent with KB axioms:

1. Start from ground facts (ABox axioms)

2. Explicate structure implied by complex concepts and TBoxaxioms

a) Syntactic decomposition using tableaux expansion rules

b) Infer constraints on (elements of) model

Tableaux Reasoning (1)• E.g., TBox{HappyParent ´ Person u 8hasChild.(Doctor t

9hasChild.Doctor)}

Person

8hasChild.(Doctor t 9hasChild.Doctor)

Abox{John:HappyParent, John hasChild Mary,Mary:: Doctor,Wendy hasChild Mary,Wendy marriedTo John}

Tableaux Reasoning (2)

• Stop when no more rules applicable or there is an obvious contradiction

• Cycle check (blocking) often needed to ensure termination

– E.g., KB:

{Personv 9hasParent.Person,John:Person}

Focus of DL Research•Decidability/Complexity of reasoning

•Requires restricted description language

•Application relevant concepts must be definable

•Some application domains require very expressive DLs

Reasoning feasible

Expressivity sufficient?

versus

Efficient algorithms in practice for very expressive DLs?

Presentation outline

• Introduction

• Description Logic - Concepts

• Description Logic v/s Predicate Logic

• Reasoning in Description Logic

• Summary

Summary

• Allowing machines to understand the meaning or “semantics” of information on WWW, we can avoid overwhelming the user with the sheer volume of information becoming available.

• A semantic Web can be implemented using Ontology Languages, based on Description Logics.

• Description Logic is a trade-off between expressivity and the decidability of reasoning.

• While DL is less expressive than FOL, it certainly has much stronger reasoning. (provability in FOL is un-decidable)

References

[1]F. Baader, I. Horrocks, U. Sattler. Description Logics as Ontology Languages for the Semantic Web. In Mechanizing Mathematical Reasoning, LNAI 2605, pages 228-248, 2005.

[2]I. Horrocks, P.F. Patel-Schneider, D.L. McGuiness, C.A. Welty. OWL: a Descrition Logic based Ontology Language for the Semantic Web. In The Description Logic Handbook: Theory, Implementation and Applications, Cambridge University Press, 2nd edition, pages 458-486, 2007.

[3]E. Graedel. Guarded fragments of first-order logic: A perspective for new description logics? In Proc. of the 1998 Description Logic Workshop (DL’98). CEUR Electronic Workshop Proceedings, http://ceur-ws.org/Vol-11/, 1998.

[4]F. Baader and U. Sattler. An overview of tableau algorithms for description logics. In Tableaux 2000, LNAI 1847, pages 413-439, 2000.

[5]U. Sattler, D. Calvanese , R.Molitor. Relationships with Other Formalisms. . In The Description Logic Handbook: Theory, Implementation and Applications, Cambridge University Press, 1st edition, pages 137-178, 2003.

[6]A. Borgida. On the Relationship between Description Logic and Predicate Logic Queries. CIKM ‘94- 11/94 Gaitherburg, MD, USA.ACM 1994

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