DL OVERVIEW

Ming Fang

6/12/2009

Outlines

Classical logics Introduction to DL Syntax of DL Semantics of DL KR in DL Reasoning in DL Applications

Classical Logics

Logics are formal languages for representing information such that conclusions can be drawn.

Important Questions Expressive Power of representation language

able to represent the problem Soundness of entailment procedure

no false conclusions are drawn Completeness of entailment procedure

all correct conclusions are drawn Decidability of entailment problem

there exists a (terminating) algorithm to compute entailment

Complexity

resources needed for computing the solution

Two Familiar Logics

Propositional Logic

atomic formula + connectives propositional formula

First-order Logic

atomic formula + connectives + existential and universal quantifiers well formed formulas

An Example

Introduction to DL

To form a middle ground solution, DL includes some more expressive operations than propositional logic and has decidable or more efficient decision problems than first-order predicate logic

A fragment of FOL Inherits open-world assumption and

non-unique name assumption

Introduction to DL cont’ Originated from frames and semantic

networks Provides formal logical extension Structured logic

Syntax of DL Unary predicates: denote concepts e.g.

student(Ming) Binary predicates: denote roles e.g.

major(Ming, CS) FOL constructors: intersection, union,

negation, universal quantifier, etc. Other constructors: inverse, transitivity, etc. Any (basic) Description Logic is a subset of

L3, i.e. the function-free FOL using only at most three variable names

Syntax of DL cont’

Semantics of DL An atomic concept is interpreted as a set of individuals that is a subset of

the domain. An atomic role is interpreted as a set of pairs of individuals from the domain,

i.e., a binary relation over the domain. In this case, if an individual x is related to y via a role R, then y is called an R-successor of x.

The top concept is interpreted as the whole domain. The bottom concept is interpreted as the empty set. The interpretation of ¬C is the set of all individuals in the domain which does

not belong to the interpretation of C. Intersection of two concepts C and D is interpreted, as set-intersection i.e.,

the set of all individuals in the domain that belongs to both the interpretation of C and the interpretation of D.

The value restriction R.C is interpreted as the set of all individuals in the ∀domain whose R-successors (if any) all belong to the interpretation of C.

The limited existential restriction is interpreted as the set of all individuals in the domain that have at least one R-successor.

KR in DL

A DL KB typically contains two components: TBox and ABox

TBox (terminological box): contains intensional knowledge in the form of a terminology, e.g.

Normally doesn’t change Assumed to be acyclic

KR in DL cont’

ABox (assertional box): contains extensional knowledge that is specific to individuals, e.g.

Subject to occasional or even constant change

The TBox/ABox distinction is not significant

Reasoning in DL

TBox

Reasoning in DL cont’

Reasoning in DL cont’

ABox

Applications

OWL

cornerstone of the semantic web for its use in the design of ontologies

OWL DL and Lite are basted on DL

OWL DLP: intersection of DL and Horn Logic Programs. It’s the largest fragment on which the choice for CWA and UNA doesn’t matter

Applications cont’ Configuration Conceptual Modeling Query Optimization and View Maintenance Natural Language Semantics I3 (Intelligent Integration of Information) Information Access and Intelligent Interfaces Terminologies and Ontologies Software Management Planning