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Ming Fang 6/12/2009. DL Overview. 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.
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Ming Fang 6/12/2009 DL Overview
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
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
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’ • 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