Fuzzy DL, Fuzzy SWRL, Fuzzy Carin

(report from visit to Athens)

M.Vacura

VŠE Praha(used materials by G.Stoilos, NTU Athens)

Description Logics

Concept and Role Oriented Concepts (Unary): Man, Tall, Human, Brain Roles (Binary): hasChild, hasColor Individuals: John, Object1, Italy, Monday

Concepts

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

Concept Constructors: , ⊔, ⊓, , , , ( Animal ⊓ Rational)

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

Typology of DLs

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

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)

Typology of DLs

Description logics S ALCR+ = 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)

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.

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.

Important DLs

ALC – base DL SHOIN – OWL DL SROIQ – OWL DL 1.1

(Support for datatypes)

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

Uncertainty

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

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

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

Fuzzy DLs

Syntax Extensions A box Fuzzy assertions: DLAssertion {, , >, <} [0,1]

George:Tall 0.7, (Prague, Paris):Far 0.6

Complex concepts

Bob:Tall 0.8 Bob:Athletic 0.6

Bob:(Athletic ⊓ Tall) t(0.6,0.8)

Reasoning

Usually DL Reasoning is done with tableaux algorithms.

Tableaux algorithms can be extended to deal with fuzziness

NTU Athens - Implementation for fKD-SHIN Reasoner FIRE

Future

Fuzzy T box <C D> 0,6

Fuzzy R box <R S> 0,3

Fuzzy SWRL

SWRL

A Semantic Web Rule Language Combining OWL and RuleML (undecidable)

RuleML – Rule Markup Language (www.ruleml.org)

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.

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)

Fuzzy Carin

Fuzzy Carin

Carin combines the description logic ALCNR with Horn Rules.

Fuzzy Carin adds fuzziness to Carin.

(decidable)

Fuzzy Carin

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