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The Semantic Web – WEEK 10: Introduction to Description Logics

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### The Semantic Web –WEEK 10: Introduction to Description Logics

We are back down to here!

The “Layer Cake” Model –

[From Rector & Horrocks Semantic Web cuurse]

Recap

- First order logic is an
- expressive
- precise
- well-researched
- representation language family, and has systematic semi-decidable proof procedures like resolution refutation for automated reasoning.
- BUT FOL has drawbacks – it is too perhaps too expressive and unstructured

The Semantic Web

Description Logic

is a FAMILY of languages which have been used to give a semantics to

- OO modelling languages such as E-R diagrams and UML class diagrams

- Diagrammatic representations such as ‘Semantic Nets’

- Ontology languages such as OWL

DL relates to

- formal methods in software engineering,
- database
- AI

The Semantic Web

Description Logic

is centred around CLASSES

F O Logic allows the user to make assertions about sets of individuals ..

Eg Ax (P(x) => …)

Ex (P(x) & …)

DL allows the user to NAME SETS OF INVIDUALS for which some property is true and COMBINE these with other.

P = { x | P(x) }

The Semantic Web

Description Logics from F.O.Logic

F.O.Logic

Universe

Constant name

One-place predicate

Two-place predicate

> 2 place predicate

Variables

Functions

Connectives

Quantifiers

Description Logic

Domain

Individual

Concept (or Class)

Role (or Property)

no equivalent!!

no equivalent!!

no equivalent!!

Restricted use

Restricted use

The Semantic Web

Description Logics from F.O.Logic: Concepts

F.O.Logic

One place predicate C

Ax P(x) => Q(x)

Ax P(x) => ¬ Q(x)

Ax P(x) Q(x)

Ax C(x) P(x) & Q(x)

Ax C(x) P(x) V Q(x)

Description Logic

A concept C {x | wff}

P Q Q subsumes P

P ¬ Q Q and P are disjoint

P Q P is equivalent to Q

C P Q

C P U Q

- Examples
- Postgrads are (defined as) students who have a first degree
- Professors are also Doctors

The Semantic Web

Description Logics from F.O.Logic: Roles

F.O.Logic

Two place predicate

AxAy R(x,y) S(y,x)

AxAy R(x,y) S(x,y)

AxAy R(x,y) => S(x,y)

AxAyAz R(x,y) & R(y,z) => R(x,z)

Description Logic

A role R = {(x.y) | R(x,y) }

R S- inverse role

S- = {(x,y) | R(y,x)}

R S

R S

R is transitive

The Semantic Web

Concept Expressions

Wffs are expressions whose value is true or false

In DL, concept expressions denote a set of individuals for which the value of a wffs is TRUE

C = {x | C(x) }

P Q = {x | P(x) & Q(x)}

P U Q = {x | P(x) V Q(x)}

P U ¬Q = {x | P(x) V ¬Q(x)}

The Semantic Web

Concept Expressions with Roles

some values from:

E R.P = {x | Ey R(x,y) & P(x)}

General form “E Role.Class”

Examples:

E Father.Male =

“the set of fathers who have sons”

Student E Married.Student =

“the set of students who also are married to students”

The Semantic Web

Concept Expressions with Roles

All values from:

A R.P = {x | Ay R(x,y) => P(y)}

General form “A Role.Class”

Examples:

Parents A Parentof.Doctor

“the set of parents whose children are (all) doctors”

Students A Examinedby.Easy

“the set of students who had easy exams“

The Semantic Web

Summary

DL is built around classes –

It is a logic with no variables or functions, and

a restricted number of expressions compared to FOL

The Semantic Web

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