The Semantic Web – WEEK 10: Introduction to Description Logics

1. 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]

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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