DL Overview

1. Ming Fang 6/12/2009 DL Overview

2. Outlines • Classical logics • Introduction to DL • Syntax of DL • Semantics of DL • KR in DL • Reasoning in DL • Applications

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

4. 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

5. Two Familiar Logics • Propositional Logic atomic formula + connectives  propositional formula • First-order Logic atomic formula + connectives + existential and universal quantifiers  well formed formulas

6. An Example

7. 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

8. Introduction to DL cont’ • Originated from frames and semantic networks • Provides formal logical extension • Structured logic

9. 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

10. Syntax of DL cont’

11. 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.

12. 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

13. 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

14. Reasoning in DL • TBox

15. Reasoning in DL cont’

16. Reasoning in DL cont’ • ABox

17. 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

18. 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