Description Logics:
Logic foundation of Semantic Web
Semantic Web - Spring 2006
Computer Engineering Department
Sharif University of Technology
Outline
First order logic and Models
 Introduction to Description Logics
 Reasoning on Description Logics

2
Prepositional Logic
3
Truth Tables
4
First Order Logic (FOL)
5
Models
6
Important Equivalences
7
Example of a model
8
Knowledge Representation with FOL
9
Knowledge Representation with FOL
10
What Are Description Logics?

A family of logic based Knowledge Representation
formalisms
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Based on concepts and roles
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Descendants of semantic networks and KL-ONE
Frame like
Key features of DLs are:
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Formal semantics
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Decidable fragments of FOL
Closely related to Propositional Modal & Dynamic Logics
Provision of inference services
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Concepts are interpreted as sets of objects.
Roles are interpreted as binary relations between objects.
Sound and complete decision procedures for key problems
Implemented systems (highly optimised)
Trade-off between expressive power and computational
complexity.
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12
Description Logic Family
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Particular languages mainly characterised by:
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Set of constructors for building complex concepts and roles
from simpler ones.
Set of axioms for asserting facts about concepts, roles and
individuals.
Simplest logic in this family is named AL (attributive language)
Others are specified by adding some suffixes like U NC:

ALC

ALCU
…
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Description logic AL
14
Fundamental Equivalences
15
Interpretation (model)
16
Example of a model
17
AL Constructors at Work
18
Additional Constructors (1)
19
Additional Constructors (2)
20
Some Examples

The “Happy Father” concept:
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A person who has at most one child or has at least 3 children
from which at least one of them is female.
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Some more examples!
Π
22
Classes
23
Semantic Networks
24
Rule Constructors
25
Examples

Expressing what we stated in slides 9 and 10 with DL:
 There is a lecturer who teaches INFS4201
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Guido teaches every course

Bob teaches some courses
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DL as fragments of Predicate Logic
27
Lisp like style for DL
28
Normal Forms
29
Representing Knowledge in DL
30
DL Architecture
31
Terminologies or TBoxes
32
Terminologies or Tboxes (cont.)
33
Reasoning about TBoxes
34
Reduction to Subsumption
35
Reduction to Unsatisfiability
36
Reducing Unsatisfiability
37
Inference services
38
Inference service: concept
satisfiability
39
Inference services based on
satisfiability
40
Inference service: concept
subsumption
41
Concept examples
42
Example taxonomy
43
World description: ABox
44
ABox inference services
45
Abox inference services (cont.)
46
ABox example
47
TBox taxonomy plus individuals
48
Open world assumption
49
Reasoning Procedures
50
Structural Subsumption
51
Examples
52
Example (do it yourself !)
53
Tableaux
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The Tableaux Algorithm is a decision procedure
solving the problem of satisfiability.
If a formula is satisfiable, the procedure will
constructively exhibit a model of the formula.
The basic idea is to incrementally build the model
by looking at the formula, by decomposing it in a
top/down fashion. The procedure exhaustively
looks at all the possibilities, so that it can
eventually prove that no model could be found
for unsatisfiable formulas.
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Tableaux Algorithm
55
Negation Normal Form
56
Completion Rules: the AND rule
57
The AND rule
58
The OR rule
59
The SOME rule
60
The FORALL rule
61
Clash
62
Completion rules for the logic ALC
63
Example inference
64
Example inference
65
References

Chapters 1 and 2 of DLHB.
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