1. Fuzzy DL, Fuzzy SWRL, Fuzzy Carin (report from visit to Athens) M.Vacura VŠE Praha (used materials by G.Stoilos, NTU Athens)

2. Description Logics Concept and Role Oriented Concepts (Unary): Man, Tall, Human, Brain Roles (Binary): hasChild, hasColor Individuals: John, Object 1, Italy, Monday

3. Concepts Concepts: Universal ⊤ Empty ⊥ Atomic/primitive concepts (concept names) Complex concepts (terms) Concept Constructors: , ⊔, ⊓, , , ,  ( Animal ⊓ Rational)

4. Axioms Concept Axioms – T box (terminology) Woman  Person ⊓ Female Parent  Person ⊓  hasChild.Person Role Axioms – R box hasSon  hasChild Trans(hasOffspring) Instance Axioms (Assertions) – A box Bob: Parent (Bob,Helen):hasChild

5. Typology of DLs Constructors of Description logics AL Negation:  A(A primitive) Conjunction: (A ⊓ B) Universal quantification:  R.C Limited existential quantification:  R. ⊤

6. Typology of DLs Constructors of Description logics ALU (A ⊔ B)(disjunction) Constructors of Description logics ALE  R.C(full existencial quantification) Constructors of Description logics ALN (  n C), (  n C)(numerical restriction) Constructors of Description logics ALC (  A) (full negation)

7. Typology of DLs Description logics S ALC R+ = ALC + transitive roles axioms. Trans(hasOffspring) Description logics SH SH = S + role hiearchy axioms. hasSon  hasChild Description logics SHf SHf = SH + role functional axioms. Func(R)

8. Typology of DLs Description logics SHO SHO = SH + nominal axioms. C  {a} Description logics SHOI SHO = SH + inverse role axioms. Description logics SHOIN SHOIN = SHOI + numerical restrictions.

9. Typology of DLs Description logics SHOIQ SHOIQ = SHOI + qualified numerical restrictions. Description logics SROIQ SROIQ = SHOIQ + extended role axioms disjoint roles, reflexive and irreflexive roles, negated role assertions (A box), complex role inclusion axioms, local reflexivity axioms.

10. Important DLs ALC – base DL SHOIN – OWL DL SROIQ – OWL DL 1.1 (Support for datatypes)

11. Uncertainty and Applications Several Applications from Industry and Academic face uncertain imprecision: Multimedia Processing (Image Analysis and Annotation) Medical Diagnosis Geospatial Applications Information Retrieval Sensor Readings Decision Making

12. Uncertainty Imprecision (Possibility Theory) Vagueness (Fuzzy Set Theory) Randomness (Probability Theory)

13. Fuzzy Set Theory An object belongs to a set to a degree between 0 and 1. (membership degree). Tall(George)=0.7 A pair of objects belongs to a relation to a degree between 0 and 1. (membership degree). Far(Prague,Paris)=0.6

14. Fuzzy Set Theoretic Operations Complement: c(x) c(x)=1-x Intersection: t(x,y) t(x,y)=min(x,y), t(x,y)=max(0,x+y-1) t-norm Godel, Lukasiewicz Union: u(x,y) u(x,y)=max(x,y), u(x,y)=min(1,x+y) s-norm Godel, Lukasiewicz Implication: J(x,y) J(x,y)=max(1-x,y), J(x,y)=min(1,1-x+y) Kleene-Dienes, Lukasiewicz

15. Fuzzy DLs Syntax Extensions A box Fuzzy assertions: DLAssertion { , , >, <} [0,1] George:Tall  0.7, (Prague, Paris):Far  0.6

16. Complex concepts Bob:Tall  0.8 Bob:Athletic  0.6 Bob:(Athletic ⊓ Tall)  t(0.6,0.8)

17. Reasoning Usually DL Reasoning is done with tableaux algorithms. Tableaux algorithms can be extended to deal with fuzziness NTU Athens - Implementation for f KD -SHIN Reasoner FIRE

18. Future Fuzzy T box  0,6 Fuzzy R box  0,3

19. Fuzzy SWRL

20. SWRL A Semantic Web Rule Language Combining OWL and RuleML (undecidable) RuleML – Rule Markup Language (www.ruleml.org)www.ruleml.org

21. Fuzzy SWRL OWL – A box: OWL asserions can include a specification of the “degree” (a truth value between 0 and 1) of confidence with which we assert that an individual (resp. pair of individuals) is an instance of a given class (resp.property). RuleML atoms can include a “weight” (a truth value between 0 and 1) that represents the “importance” of the atom in a rule.

22. Fuzzy SWRL Fuzzy rule assertions: antecedent → consequent parent(?x, ?p) ∧ Happy(?p) → Happy(?x) *0.8, EyebrowsRaised(?a)*0.9 ∧ MouthOpen(?a)*0.8 → Happy(?a)

23. Fuzzy Carin

24. Carin combines the description logic ALCNR with Horn Rules. Fuzzy Carin adds fuzziness to Carin. (decidable)

25. Fuzzy Carin

26. END