Fuzzy Set Ontology

FLOW_ALGEBRA FuzzySetOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/fuzzy-logic/fuzzy-set-ontology.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > FuzzyLogic > FuzzySetOntology

BEGIN_FLOW

ENTITY FuzzySet {
    TYPE := GeneralisedSet
    SYMBOL := A
    UNIVERSE := X
    DEFINITION := "A set whose elements have graded membership values, normally represented by a membership function mapping each element of a universe of discourse to a value in [0,1]."
}

ENTITY MembershipFunction {
    SYMBOL := μ_A
    SIGNATURE := μ_A:X->[0,1]
}

ENTITY MembershipDegree {
    TYPE := RealValue
    DOMAIN := [0,1]
}

ENTITY UniverseOfDiscourse {
    TYPE := Domain
    SYMBOL := X
}

RELATIONSHIP FuzzySet --defined_by--> MembershipFunction
RELATIONSHIP FuzzySet --defined_over--> UniverseOfDiscourse
RELATIONSHIP MembershipFunction --returns--> MembershipDegree
RELATIONSHIP FuzzySet --generalises--> CrispSet
RELATIONSHIP FuzzySet --supports--> FuzzyLogic
RELATIONSHIP FuzzySet --supports--> ApproximateReasoning

FORMAL_REPRESENTATION {
    A :=
        {(x, μ_A(x)) | x∈X}
}

MEMBERSHIP_SEMANTICS {
    μ_A(x)=0 := no_membership
    μ_A(x)=1 := full_membership
    0<μ_A(x)<1 := partial_membership
}

CRISP_SET_SPECIAL_CASE {
    IF FOR_ALL x∈X:
        μ_A(x)∈{0,1}
    THEN
        A := CrispSet
}

SEMANTIC_DISTINCTION {
    MembershipDegree != Probability
    MembershipDegree != Frequency
    MembershipDegree != StatisticalConfidence
}

EQUALITY_OF_FUZZY_SETS {
    A=B
    IFF
        FOR_ALL x∈X:
            μ_A(x)=μ_B(x)
}

FUZZY_SUBSET {
    A⊆B
    IFF
        FOR_ALL x∈X:
            μ_A(x)<=μ_B(x)
}

PROPER_FUZZY_SUBSET {
    A⊂B
    IFF
        A⊆B
        AND A!=B
}

EMPTY_FUZZY_SET {
    μ_empty(x)=0
    FOR_ALL x∈X
}

UNIVERSAL_FUZZY_SET {
    μ_X(x)=1
    FOR_ALL x∈X
}

SUPPORT {
    support(A) :=
        {x∈X | μ_A(x)>0}
}

CORE {
    core(A) :=
        {x∈X | μ_A(x)=1}
}

BOUNDARY {
    boundary(A) :=
        {x∈X | 0<μ_A(x)<1}
}

HEIGHT {
    height(A) :=
        sup_x μ_A(x)
}

NORMAL_FUZZY_SET {
    A IS normal
    IFF
        height(A)=1
}

SUBNORMAL_FUZZY_SET {
    A IS subnormal
    IFF
        height(A)<1
}

ALPHA_CUT {
    A_alpha :=
        {x∈X | μ_A(x)>=alpha}
    REQUIRE alpha∈[0,1]
}

STRONG_ALPHA_CUT {
    A_alpha_plus :=
        {x∈X | μ_A(x)>alpha}
}

ALPHA_CUT_NESTING {
    IF alpha1<=alpha2
    THEN
        A_alpha2 ⊆ A_alpha1
}

RECONSTRUCTION_PRINCIPLE {
    FuzzySet
        MAY be represented through
        its family of alpha_cuts
        under suitable fuzzy-set reconstruction semantics
}

FUZZY_COMPLEMENT {
    USING standard_negation:
        μ_notA(x) := 1-μ_A(x)
}

FUZZY_INTERSECTION {
    GENERAL:
        μ_(A∩B)(x) :=
            T(μ_A(x),μ_B(x))

    STANDARD_MIN:
        μ_(A∩B)(x) :=
            min(μ_A(x),μ_B(x))
}

FUZZY_UNION {
    GENERAL:
        μ_(A∪B)(x) :=
            S(μ_A(x),μ_B(x))

    STANDARD_MAX:
        μ_(A∪B)(x) :=
            max(μ_A(x),μ_B(x))
}

ENTITY TNorm {
    TYPE := FuzzyIntersectionOperator
}

ENTITY TConorm {
    TYPE := FuzzyUnionOperator
}

STANDARD_MIN_MAX_IDENTITIES {
    min(x,1)=x
    min(x,0)=0
    max(x,0)=x
    max(x,1)=1
}

STANDARD_MIN_MAX_IDEMPOTENCE {
    min(x,x)=x
    max(x,x)=x
}

STANDARD_MIN_MAX_ABSORPTION {
    min(x,max(x,y))=x
    max(x,min(x,y))=x
}

STANDARD_NEGATION_INVOLUTION {
    1-(1-x)=x
}

STANDARD_DE_MORGAN {
    1-min(x,y)
        =
    max(1-x,1-y)

    1-max(x,y)
        =
    min(1-x,1-y)
}

FUZZY_DIFFERENCE {
    POSSIBLE_DEFINITION :=
        A ∩ complement(B)

    REQUIRE :=
        selected complement and intersection operators
}

CARTESIAN_PRODUCT {
    A×B IS fuzzy_set_on X×Y

    μ_(A×B)(x,y) :=
        T(μ_A(x),μ_B(y))
}

FUZZY_RELATION {
    TYPE := FuzzySetOnCartesianProduct
}

RELATIONSHIP FuzzySet --generalises_to--> FuzzyRelation

FUZZY_CONVEXITY {
    A IS fuzzy_convex
    IFF
        FOR_ALL x,y
        FOR_ALL λ∈[0,1]:
            μ_A(λx+(1-λ)y)
            >=
            min(μ_A(x),μ_A(y))
}

FUZZY_NUMBER {
    TYPE := FuzzySetOnRealLine

    COMMON_REQUIREMENTS := {
        Normality,
        Convexity,
        SuitableContinuity,
        BoundedSupport
    }
}

RELATIONSHIP FuzzySet --specialises_to--> FuzzyNumber

LINGUISTIC_TERM {
    TYPE := LabelledFuzzySet
}

LINGUISTIC_VARIABLE {
    VALUES := collection_of_linguistic_terms
}

EXAMPLE_LINGUISTIC_STRUCTURE {
    VARIABLE := Temperature
    TERMS := {Cold,Warm,Hot}

    EACH term :=
        fuzzy_set_over_temperature_domain
}

FUZZY_PARTITION {
    TYPE := CollectionOfFuzzySets
    PURPOSE :=
        cover or describe a common universe with overlapping graded categories
}

RUSPINI_PARTITION {
    FOR_ALL x∈X:
        SUM_i μ_Ai(x)=1
}

OVERLAP {
    GIVEN A,B

    overlap MAY be measured by
        selected fuzzy intersection/similarity measure

    REQUIRE :=
        measure explicitly declared
}

FUZZY_CARDINALITY {
    FOR finite X:

    sigma_count(A) :=
        SUM_x μ_A(x)
}

RELATIVE_CARDINALITY {
    FOR finite X:
        relative_cardinality(A) :=
            sigma_count(A)/|X|

    REQUIRE |X|>0
}

FUZZY_POWER_SET {
    TYPE := CollectionOfFuzzySubsets
    NOTE :=
        representation depends on chosen fuzzy-set framework
}

FUZZY_SET_TRANSFORMATION {
    GIVEN f:X->Y

    IMAGE_FUZZY_SET B:
        μ_B(y) :=
            sup{
                μ_A(x)
                |
                f(x)=y
            }
}

EXTENSION_PRINCIPLE {
    GIVEN f:X1×...×Xn->Y
    GIVEN fuzzy_sets A1,...,An

    μ_B(y) :=
        sup over f(x1,...,xn)=y
        of T(
            μ_A1(x1),
            ...,
            μ_An(xn)
        )

    REQUIRE selected t_norm T
}

FUZZY_ARITHMETIC {
    FuzzyNumbers
        -> ExtensionPrinciple OR AlphaCutArithmetic
        -> FuzzyResult
}

LINGUISTIC_HEDGE {
    TYPE := MembershipTransformation
}

CONCENTRATION {
    μ_veryA(x) :=
        μ_A(x)^p
    COMMONLY p>1
}

DILATION {
    μ_more_or_less_A(x) :=
        μ_A(x)^p
    COMMONLY 0<p<1
}

NORMALISATION {
    GIVEN subnormal A
    AND height(A)>0

    μ_normalised_A(x) :=
        μ_A(x)/height(A)
}

FUZZY_SIMILARITY {
    INPUT := A,B
    OUTPUT := degree_in_[0,1]
    REQUIRE selected_similarity_measure
}

FUZZY_DISTANCE {
    INPUT := A,B
    OUTPUT := nonnegative_value
    REQUIRE selected_distance_metric
}

HAMMING_STYLE_DISTANCE {
    FOR finite X:
        d_H(A,B) :=
            SUM_x abs(μ_A(x)-μ_B(x))
}

EUCLIDEAN_STYLE_DISTANCE {
    FOR finite X:
        d_E(A,B) :=
            sqrt(
                SUM_x(
                    μ_A(x)-μ_B(x)
                )^2
            )
}

TYPE_1_FUZZY_SET {
    MEMBERSHIP :=
        μ_A(x)∈[0,1]
}

TYPE_2_FUZZY_SET {
    MEMBERSHIP :=
        μ_A(x,u)∈[0,1]

    INTERPRETATION :=
        membership degree itself has fuzzy uncertainty
}

INTERVAL_TYPE_2_FUZZY_SET {
    PRIMARY_MEMBERSHIP :=
        interval [
            lower_mu_A(x),
            upper_mu_A(x)
        ]
}

FOOTPRINT_OF_UNCERTAINTY {
    ABBREVIATION := FOU
    TYPE := RegionOfPossiblePrimaryMembershipValues
}

RELATIONSHIP FuzzySet --extends_to--> Type2FuzzySet

INTUITIONISTIC_FUZZY_SET {
    COMPONENTS := {
        membership_degree μ_A(x),
        nonmembership_degree ν_A(x)
    }

    REQUIRE :=
        0 <= μ_A(x)+ν_A(x) <= 1
}

HESITATION_DEGREE {
    π_A(x) :=
        1-μ_A(x)-ν_A(x)
}

RELATIONSHIP FuzzySet --relates_to--> IntuitionisticFuzzySet

FUZZY_ROUGH_SET_RELATIONSHIP {
    FuzzySet --may_combine_with--> RoughSetTheory
}

FUZZY_CLASSIFICATION {
    OBJECT x
        -> membership_degrees_across_fuzzy_classes
        -> graded_class_representation
}

FUZZY_CLUSTERING_RELATIONSHIP {
    ClusterMembership
        MAY be represented_as
        fuzzy_membership_degrees

    REQUIRE :=
        clustering algorithm semantics
}

FUZZIFICATION {
    crisp_value x
        -> evaluate membership_functions
        -> memberships in fuzzy_sets
}

FUZZY_RULE_ANTECEDENT {
    "x IS A"
        -> truth_degree μ_A(x)
}

APPLICATION_RELATIONSHIPS {
    FuzzySet --supports--> FuzzyLogic
    FuzzySet --supports--> FuzzyControl
    FuzzySet --supports--> Classification
    FuzzySet --supports--> Clustering
    FuzzySet --supports--> DecisionSupport
    FuzzySet --supports--> PatternRecognition
    FuzzySet --supports--> ApproximateReasoning
    FuzzySet --supports--> ArtificialIntelligence
    FuzzySet --supports--> DataFusion
}

ONTOLOGY_LINKS {
    PARENT := FuzzyLogicOntology

    RELATED := {
        MembershipFunctionOntology,
        FuzzyLogicOntology,
        TNormOntology,
        TConormOntology,
        FuzzyRelationOntology,
        FuzzyNumberOntology,
        LinguisticVariableOntology,
        Type2FuzzySetOntology,
        IntuitionisticFuzzySetOntology,
        CrispSetOntology,
        SetTheoryOntology
    }
}

SEMANTIC_EQUIVALENCE {
    FuzzySet ≈ GradedSet
        WHEN context=fuzzy_set_theory

    MembershipDegree ≈ DegreeOfMembership
    μ_A(x) ≈ MembershipOf(x,A)

    Support(A) ≈ PositiveMembershipRegion
    Core(A) ≈ FullMembershipRegion
}

CONSTRAINTS {
    C1 :=
        standard Type-1 fuzzy membership values MUST remain in [0,1]

    C2 :=
        membership degree MUST NOT be interpreted as probability by default

    C3 :=
        fuzzy-set equality MUST compare membership functions pointwise

    C4 :=
        fuzzy subsethood MUST preserve the selected pointwise order definition

    C5 :=
        union, intersection and complement MUST identify selected fuzzy operators where nonstandard choices are possible

    C6 :=
        normality and convexity MUST NOT be assumed without validation

    C7 :=
        alpha-cut thresholds MUST remain within the declared membership domain

    C8 :=
        Type-1, Type-2 and intuitionistic fuzzy sets MUST remain semantically distinct

    C9 :=
        fuzzy arithmetic MUST preserve the declared extension semantics

    C10 :=
        similarity and distance claims MUST identify their measures

    C11 :=
        linguistic labels MUST be grounded in fuzzy-set semantics

    C12 :=
        transformations MUST preserve membership meaning and declared domain
}

REASONING_CHAIN Query Q {
    IDENTIFY target_fuzzy_set
    IDENTIFY universe_of_discourse
    IDENTIFY membership_function
    IDENTIFY fuzzy_set_type

    IF membership_requested
        THEN evaluate μ_A(x)

    IF equality_requested
        THEN compare membership_functions pointwise

    IF subset_requested
        THEN test μ_A(x)<=μ_B(x)

    IF support_requested
        THEN solve μ_A(x)>0

    IF core_requested
        THEN solve μ_A(x)=1

    IF alpha_cut_requested
        THEN solve μ_A(x)>=alpha

    IF normality_requested
        THEN compute height(A)

    IF convexity_requested
        THEN test fuzzy_convexity

    IF union_requested
        THEN apply selected_t_conorm

    IF intersection_requested
        THEN apply selected_t_norm

    IF complement_requested
        THEN apply selected_negation

    IF image_or_arithmetic_requested
        THEN apply extension_principle_or_alpha_cut_method

    IF Type2_context
        THEN identify secondary_membership_semantics

    IF intuitionistic_context
        THEN identify membership_nonmembership_hesitation

    VALIDATE constraints
    RETURN result WITH derivation
}

CONFIDENCE {
    CORE_FUZZY_SET_DEFINITION := 1.00
    TYPE1_MEMBERSHIP_DOMAIN := 1.00
    SUPPORT_CORE_ALPHA_CUT := 1.00
    STANDARD_MIN_MAX_OPERATIONS := 1.00
    EXTENSION_PRINCIPLE := 0.99

    COMPUTED_RESULT :=
        IF membership_functions_and_operators_complete THEN 1.00
        ELSE REQUIRE missing_information
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        FuzzySetMembershipMapping,
        MembershipDegreeDomain,
        PointwiseEquality,
        PointwiseSubsetConcept
    }

    EXTENSIBLE := {
        FuzzyOperators,
        FuzzySetFamilies,
        SimilarityMeasures,
        DistanceMeasures,
        Type2Extensions,
        IntuitionisticExtensions,
        LinguisticStructures,
        ApplicationDomains,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        IDENTIFY fuzzy_set_framework
        VALIDATE universe
        VALIDATE membership_range
        VALIDATE operators
        CHECK set_property_claims
        CHECK semantic_consistency
        ADD relationship IF non_contradictory
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    SYMBOL_RESOLUTION := TRUE
    RELATIONSHIP_TRAVERSAL := TRUE
    FUZZY_SET_REASONING := TRUE
    MEMBERSHIP_EVALUATION := TRUE
    ALPHA_CUT_REASONING := TRUE
    SET_OPERATION_REASONING := TRUE
    FUZZY_ARITHMETIC_REASONING := TRUE
    TYPE2_REASONING := TRUE
    INTUITIONISTIC_FUZZY_REASONING := TRUE
    VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
    TARGET := TARGET_URL
    SUBJECT := "Fuzzy Set Ontology"
    VERSION := "1.0"
    CANONICAL_SIGNATURE :=
        HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
    PRIMARY_ENTITY := FuzzySet

    INTERPRETATION :=
        "A fuzzy set generalises ordinary set membership by assigning each element a degree of membership, normally in [0,1]. Its membership function determines support, core, alpha cuts and graded set operations, while extensions such as fuzzy numbers, Type-2 sets and intuitionistic fuzzy sets provide richer representations of vagueness and uncertainty."
}

END_FLOW