Fuzzy DL Fuzzy SWRL Fuzzy Carin report from

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Fuzzy DL, Fuzzy SWRL, Fuzzy Carin (report from visit to Athens) M. Vacura VŠE

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

Description Logics l Concept and Role Oriented l l l Concepts (Unary): Man, Tall,

Description Logics l Concept and Role Oriented l l l Concepts (Unary): Man, Tall, Human, Brain Roles (Binary): has. Child, has. Color Individuals: John, Object 1, Italy, Monday

Concepts l Concepts: l l l Universal ⊤ Empty ⊥ Atomic/primitive concepts (concept names)

Concepts l Concepts: l l l Universal ⊤ Empty ⊥ Atomic/primitive concepts (concept names) Complex concepts (terms) Concept Constructors: l l , ⊔, ⊓, , ( Animal ⊓ Rational)

Axioms l Concept Axioms – T box (terminology) l l l Role Axioms –

Axioms l Concept Axioms – T box (terminology) l l l Role Axioms – R box l l l Woman Person ⊓ Female Parent Person ⊓ has. Child. Person has. Son has. Child Trans(has. Offspring) Instance Axioms (Assertions) – A box l l Bob: Parent (Bob, Helen): has. Child

Typology of DLs l Constructors of Description logics AL l l Negation: A (A

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

Typology of DLs l Constructors of Description logics ALU l l Constructors of Description

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

Typology of DLs l Description logics S l l ALCR+ = ALC + transitive

Typology of DLs l Description logics S l l ALCR+ = ALC + transitive roles axioms. Trans(has. Offspring) Description logics SH l SH = S + role hiearchy axioms. l has. Son has. Child Description logics SHf l l SHf = SH + role functional axioms. Func(R)

Typology of DLs l l Description logics SHO l SHO = SH + nominal

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

Typology of DLs l Description logics SHOIQ l l SHOIQ = SHOI + qualified

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

Important DLs l ALC – base DL SHOIN – OWL DL SROIQ – OWL

Important DLs l ALC – base DL SHOIN – OWL DL SROIQ – OWL DL 1. 1 l (Support for datatypes) l l

Uncertainty and Applications l Several Applications from Industry and Academic face uncertain imprecision: l

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

Uncertainty l l l Imprecision (Possibility Theory) Vagueness (Fuzzy Set Theory) Randomness (Probability Theory)

Uncertainty l l l Imprecision (Possibility Theory) Vagueness (Fuzzy Set Theory) Randomness (Probability Theory)

Fuzzy Set Theory l An object belongs to a set to a degree between

Fuzzy Set Theory l An object belongs to a set to a degree between 0 and 1. (membership degree). l l Tall(George)=0. 7 A pair of objects belongs to a relation to a degree between 0 and 1. (membership degree). l Far(Prague, Paris)=0. 6

Fuzzy Set Theoretic Operations l Complement: c(x) l l Intersection: t(x, y) l l

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

Fuzzy DLs l Syntax Extensions l l A box Fuzzy assertions: DLAssertion { ,

Fuzzy DLs l Syntax Extensions l l A box Fuzzy assertions: DLAssertion { , , >, <} [0, 1] l l George: Tall 0. 7, (Prague, Paris): Far 0. 6

Complex concepts l Bob: Tall 0. 8 Bob: Athletic 0. 6 l Bob: (Athletic

Complex concepts l Bob: Tall 0. 8 Bob: Athletic 0. 6 l Bob: (Athletic ⊓ Tall) t(0. 6, 0. 8) l

Reasoning l l l Usually DL Reasoning is done with tableaux algorithms. Tableaux algorithms

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

Future l l Fuzzy T box l <C D> 0, 6 Fuzzy R box

Future l l Fuzzy T box l <C D> 0, 6 Fuzzy R box l <R S> 0, 3

Fuzzy SWRL

Fuzzy SWRL

SWRL l A Semantic Web Rule Language Combining OWL and Rule. ML l l

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

Fuzzy SWRL l OWL – A box: l l OWL asserions can include a

Fuzzy SWRL l OWL – A box: l l 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). Rule. ML l atoms can include a “weight” (a truth value between 0 and 1) that represents the “importance” of the atom in a rule.

Fuzzy SWRL l Fuzzy rule assertions: l l l antecedent → consequent parent(? x,

Fuzzy SWRL l Fuzzy rule assertions: l l l antecedent → consequent parent(? x, ? p) ∧ Happy(? p) → Happy(? x) *0. 8, Eyebrows. Raised(? a)*0. 9 ∧ Mouth. Open(? a)*0. 8 → Happy(? a)

Fuzzy Carin

Fuzzy Carin

Fuzzy Carin l Carin combines the description logic ALCNR with Horn Rules. l Fuzzy

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

Fuzzy Carin

Fuzzy Carin

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