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Introduction to Description Logic

Bangladesh Short-Term Course (Artificial Intelligence)

Plaban Kumar Bhowmick

Assistant Professor

Centre of Excellence in Artificial Intelligence

Indian Institute of Technology Kharagpur

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Reasoning with OWL Knowledge Base

OWL Knowledge base

Formal Description

mapping

Reasoning Algorithm

New Inferences

Answer to an input query

Description Logic

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Description Logic Constructs

Existential Restriction:

 

 

hasDegree

 

type

UG

Universal Restriction:

 

type

EligibleForMasters

 

hasChild

 

type

Female

type

FatherOfGirlChild

 

type

Male

type

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Description Logic Constructs

Negation:

 

 

 

 

 

Negation over Universal Quantification:

An Equivalence:

 

 

Individuals that did not only won AI Coding Challenge

Individuals that have also own challenges other than AI Coding Challenge

 

 

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Description Logic Constructs

Intersection and Union:

 

 

Bottom (owl:Nothing) and Top (owl:Thing) Concept

 

 

 

 

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Description Logic Constructs

Concept Inclusion Axioms

 

Defined Concept:

 

Primitive Concept:

 

 

 

General Class Inclusion (GCI)Axioms

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Description Logic Constructs

Role Inclusion Axioms

 

 

 

 

 

 

Atomic Roles

Inverse Roles

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Description Logic Constructs

Role Constructors

 

 

 

 

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Two Ways to Represent Semantics

  • Model theoretic Semantics
    • Representing DL element as set
    • Membership in Class 🡪 Membership in a set
    • Interpretation of a DL formula 🡪 Set theoretic operations

​

  • First Order Logic (FOL)-based Semantics
    • Representing elements as predicates
    • Interpretation of a formula 🡪 evaluating FOL logic expression
    • DL is a subset of FOL

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Model Theoretic Semantics of DL

 

 

 

Vocabulary

 

 

 

 

 

Interpretation Function

 

 

 

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Model Theoretic Semantics of DL

  •  

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Interpretation in DL

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Interpretation of Class Expressions

 

 

 

 

 

 

 

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Evaluation of Concept Expression

 

 

 

 

 

 

 

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Evaluation of Concept Expression

 

 

 

 

 

 

 

S

S

S

A

A

A

Closed World:

Open World:

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Evaluation of Concept Expression

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Transforming DL Expression into FOL

 

 

Atomic Concepts

Translated into unary predicate in FOL

 

Atomic Roles

Translated into binary predicate in FOL

 

 

 

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Transforming DL Expression into FOL

Concept Intersection

 

 

Universal Quantification

 

 

 

 

 

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Transforming DL Expression into FOL

Existential Quantification