Description Logics as Ontology Languages for Semantic Webs

1
Description Logics as Ontology Languages for
Semantic Webs
Franz Baader, Ian Horrocks, and Ulrike Sattler

• Presented by-
• Somya Gupta(10305011)
• Akshat Malu (10305012)
• Swapnil Ghuge (10305907)

2
Presentation outline

• Introduction
• Description Logic Concepts
• Description Logic v/s Predicate Logic
• Reasoning in Description Logic
• Summary

3

• 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.

4

• 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?

5

• 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.

6

• 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.

7

• 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.

8
Presentation outline

• Introduction
• Description Logic Concepts
• Description Logic v/s Predicate Logic
• Reasoning in Description Logic
• Summary

9

• 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.

10
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

11
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)Doctor(z)))))

12
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
13
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 n DI
• ( n r)I d e (d,e) ? rI n (5
  hasChild)
• (? r.C)I d ?e (d,e) ? rI ? e ? CI
  (?hasChild.Professor)

14
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.
• JohnHappyParent,
• John hasChild Mary
• i.e., non-logical axioms including (restricted)
  use of nominals

15
Presentation outline

• Introduction
• Description Logic - Concepts
• Description Logic v/s Predicate Logic
• Reasoning in Description Logic
• Summary

16
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) ? Black (x))
• Cat u Black

17
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.

18
Presentation outline

• Introduction
• Description Logic Concepts
• Description Logic v/s Predicate Logic
• Reasoning in Description Logic
• Summary

19
Reasoning
Subsumption 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.
20
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/consi
  stency of KB
• Tableaux algorithms work by trying to construct a
  concrete example (model) consistent with KB
  axioms
• Start from ground facts (ABox axioms)
• Explicate structure implied by complex concepts
  and TBox axioms
• Syntactic decomposition using tableaux expansion
  rules
• Infer constraints on (elements of) model

21
Tableaux Reasoning (1)

• E.g., TBoxHappyParent Person u
  8hasChild.(Doctor t 9hasChild.Doctor)

• AboxJohnHappyParent,
• John hasChild Mary,
• Mary Doctor,
• Wendy hasChild Mary,
• Wendy marriedTo John

Person 8hasChild.(Doctor t 9hasChild.Doctor)
22
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
• Person v 9hasParent.Person,
• JohnPerson

23
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

• Efficient algorithms in practice for very
  expressive DLs?

Expressivity sufficient?
Reasoning feasible
versus
24
Presentation outline

• Introduction
• Description Logic - Concepts
• Description Logic v/s Predicate Logic
• Reasoning in Description Logic
• Summary

25
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)

26
References

• 1F. Baader, I. Horrocks, U. Sattler.
  Description Logics as Ontology Languages for the
  Semantic Web. In Mechanizing Mathematical
  Reasoning, LNAI 2605, pages 228-248, 2005.
• 2I. 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.
• 3E. Graedel. Guarded fragments of first-order
  logic A perspective for new description logics?
  In Proc. of the 1998 Description Logic Workshop
  (DL98). CEUR Electronic Workshop Proceedings,
  http//ceur-ws.org/Vol-11/, 1998.
• 4F. Baader and U. Sattler. An overview of
  tableau algorithms for description logics. In
  Tableaux 2000, LNAI 1847, pages 413-439, 2000.
• 5U. 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.
• 6A. Borgida. On the Relationship between
  Description Logic and Predicate Logic Queries.
  CIKM 94- 11/94 Gaitherburg, MD, USA.ACM 1994

27
Thank you for listening
Any questions?