Semantic Interpretation Ontology

AI SEARCH FLOW ALGEBRA — SEMANTIC INTERPRETATION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/logical-semantics/semantic-interpretation-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Mathematical Logic"
SUBDOMAIN := "Logical Semantics"
ONTOLOGY := "Semantic Interpretation Ontology"
PRIMARY_ENTITY := SemanticInterpretation
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

SemanticInterpretation :=
    MappingFromSyntaxToMeaning
    ∧ AssignmentOfSemanticObjects
    ∧ ContextSensitiveEvaluation
    ∧ ModelRelativeMeaning

InterpretationPurpose :=
    connect(FormalLanguage, SemanticDomain)
    ∧ assign(MeaningToNonLogicalSymbols)
    ∧ enable(TruthEvaluation)
    ∧ support(SemanticConsequence)
    ∧ enable(ModelComparison)

FormalLanguage :=
    LogicalSymbols
    ∪ NonLogicalSymbols
    ∪ Variables
    ∪ Terms
    ∪ Formulae

SemanticDomain :=
    MathematicalStructure
    ∧ contains(ObjectsOfInterpretation)

Interpretation :=
    I = (D, ·^I)

where:
    D := NonEmptyDomain
    ·^I := InterpretationFunction

InterpretationFunction :=
    NonLogicalSymbol
    → SemanticObject

ConstantInterpretation :=
    c^I ∈ D

FunctionInterpretation :=
    f^I : D^n → D

PredicateInterpretation :=
    P^I ⊆ D^n

RelationInterpretation :=
    R^I ⊆ D^n

EqualityInterpretation :=
    (=)^I
    =
    IdentityRelationOn(D)

PropositionalInterpretation :=
    v : PropositionVariables → {True,False}

VariableAssignment :=
    g : Variables → D

UpdatedAssignment :=
    g[x↦d]

TermInterpretation :=
    [[t]]_(I,g)

VariableTerm :=
    [[x]]_(I,g)
    =
    g(x)

ConstantTerm :=
    [[c]]_(I,g)
    =
    c^I

FunctionTerm :=
    [[f(t1,...,tn)]]_(I,g)
    =
    f^I(
        [[t1]]_(I,g),
        ...,
        [[tn]]_(I,g)
    )

FormulaInterpretation :=
    [[φ]]_(I,g)
    ∈ TruthDomain

AtomicPredicateSemantics :=
    I,g ⊨ P(t1,...,tn)
    iff
    (
        [[t1]]_(I,g),
        ...,
        [[tn]]_(I,g)
    )
    ∈ P^I

EqualitySemantics :=
    I,g ⊨ t1=t2
    iff
    [[t1]]_(I,g)
    =
    [[t2]]_(I,g)

NegationSemantics :=
    I,g ⊨ ¬φ
    iff
    I,g ⊭ φ

ConjunctionSemantics :=
    I,g ⊨ φ∧ψ
    iff
    I,g ⊨ φ
    ∧ I,g ⊨ ψ

DisjunctionSemantics :=
    I,g ⊨ φ∨ψ
    iff
    I,g ⊨ φ
    ∨ I,g ⊨ ψ

ImplicationSemantics :=
    I,g ⊨ φ→ψ
    iff
    I,g ⊭ φ
    ∨ I,g ⊨ ψ

BiconditionalSemantics :=
    I,g ⊨ φ↔ψ
    iff
    (
        I,g ⊨ φ
        iff
        I,g ⊨ ψ
    )

UniversalQuantifierSemantics :=
    I,g ⊨ ∀x φ
    iff
    ∀d ∈ D :
        I,g[x↦d] ⊨ φ

ExistentialQuantifierSemantics :=
    I,g ⊨ ∃x φ
    iff
    ∃d ∈ D :
        I,g[x↦d] ⊨ φ

Satisfaction :=
    I,g ⊨ φ

NonSatisfaction :=
    I,g ⊭ φ

Sentence :=
    Formula
    ∧ no_free_variables

SentenceTruth :=
    I ⊨ φ

Model :=
    Interpretation
    ∧ satisfies(TheoryOrFormula)

ModelOfFormula :=
    I ⊨ φ

ModelOfTheory :=
    I ⊨ T
    iff
    ∀φ ∈ T :
        I ⊨ φ

Theory :=
    SetOfSentences

ModelClass :=
    Mod(T)
    =
    {I | I ⊨ T}

Validity :=
    ⊨ φ
    iff
    ∀I :
        I ⊨ φ

Satisfiability :=
    ∃I :
        I ⊨ φ

Unsatisfiability :=
    ¬∃I :
        I ⊨ φ

SemanticConsequence :=
    Γ ⊨ φ

SemanticConsequenceMeaning :=
    ∀I :
        I ⊨ Γ
        ⇒ I ⊨ φ

Counterinterpretation :=
    I
    ∧ I ⊨ Γ
    ∧ I ⊭ φ

Countermodel :=
    Counterinterpretation

SemanticEquivalence :=
    φ ≡ ψ
    iff
    ∀I :
        I ⊨ φ ↔ ψ

InterpretationEquivalence :=
    Interpretations
    agree_on
    relevant(Symbols, Formulae, TruthConditions)

Denotation :=
    Expression
    → SemanticObject

Reference :=
    Term
    → DomainObject

Extension :=
    Predicate
    → SetOfTuples

Intension :=
    Expression
    → Function(ContextOrWorld, Extension)

Context :=
    InformationRequiredToResolveMeaning

ContextualInterpretation :=
    Syntax
    ∧ Context
    → Meaning

PossibleWorld :=
    AlternativeSemanticState

KripkeInterpretation :=
    M = (W,R,V)

WorldRelativeInterpretation :=
    M,w ⊨ φ

AccessibilityRelation :=
    R ⊆ W × W

ModalNecessity :=
    M,w ⊨ □φ
    iff
    ∀v :
        R(w,v)
        ⇒ M,v ⊨ φ

ModalPossibility :=
    M,w ⊨ ◇φ
    iff
    ∃v :
        R(w,v)
        ∧ M,v ⊨ φ

TemporalInterpretation :=
    Formula
    ∧ TimePointOrState
    → TruthValue

Next :=
    M,i ⊨ Xφ
    iff
    M,i+1 ⊨ φ

Eventually :=
    M,i ⊨ Fφ
    iff
    ∃j ≥ i :
        M,j ⊨ φ

Globally :=
    M,i ⊨ Gφ
    iff
    ∀j ≥ i :
        M,j ⊨ φ

Until :=
    M,i ⊨ φ U ψ
    iff
    ∃j ≥ i :
        M,j ⊨ ψ
        ∧
        ∀k :
            i ≤ k < j
            ⇒ M,k ⊨ φ

ManyValuedInterpretation :=
    Formula
    → TruthValueSet

ClassicalTruthDomain :=
    {True,False}

ThreeValuedTruthDomain :=
    {True,False,Unknown}

FuzzyTruthDomain :=
    [0,1]

ProbabilisticInterpretation :=
    Formula
    → ProbabilityValue

AlgebraicInterpretation :=
    LogicalConstructs
    → AlgebraicOperations

BooleanInterpretation :=
    ClassicalPropositionalLogic
    → BooleanAlgebra

HeytingInterpretation :=
    IntuitionisticLogic
    → HeytingAlgebra

InterpretationConstraint :=
    RestrictionOn(
        Domain
        ∨ SymbolAssignment
        ∨ Relation
        ∨ Function
        ∨ TruthValue
    )

AdmissibleInterpretation :=
    Interpretation
    ∧ satisfies(AllInterpretationConstraints)

IntendedInterpretation :=
    Interpretation
    ∧ corresponds_to(IntendedDomainMeaning)

NonStandardInterpretation :=
    ValidFormalInterpretation
    ∧ differs_from(IntendedInterpretation)

AmbiguousInterpretation :=
    Expression
    ∧ multiple_plausible(Interpretations)

Disambiguation :=
    Context
    ∧ TypeInformation
    ∧ OntologyRelations
    ∧ Constraints
    → PreferredInterpretation

InterpretationSelection :=
    CandidateInterpretations
    → ConstraintFiltering
    → SemanticCompatibility
    → ContextRanking
    → SelectedInterpretation

InterpretationComposition :=
    MeaningOfCompoundExpression
    derived_from
    MeaningsOfComponents

CompositionalInterpretation :=
    [[F(e1,...,en)]]
    =
    SemanticFunction_F(
        [[e1]],...,[[en]]
    )

InterpretationEnvironment :=
    VariableBindings
    ∧ ContextBindings
    ∧ DomainBindings

BindingResolution :=
    Identifier
    → BoundEntity

ScopeResolution :=
    VariableOccurrence
    → BinderOrEnvironmentEntry

SubstitutionLemma :=
    Interpretation(
        φ[x←t],
        g
    )
    related_to
    Interpretation(
        φ,
        g[x↦[[t]]_(I,g)]
    )

Isomorphism :=
    BijectionBetweenDomains
    ∧ preserves(Constants, Functions, Relations)

IsomorphicInterpretations :=
    I ≅ J

SemanticInvarianceUnderIsomorphism :=
    I ≅ J
    ⇒
    same_truth_for_corresponding_sentences

Homomorphism :=
    StructurePreservingMapping

Embedding :=
    InjectiveStructurePreservingMapping

ElementaryEmbedding :=
    Embedding
    ∧ preserves(FirstOrderTruth)

Reduct :=
    Interpretation
    restricted_to
    SmallerSignature

Expansion :=
    Interpretation
    extended_with
    AdditionalSymbols

Signature :=
    Constants
    ∪ FunctionSymbols
    ∪ RelationSymbols

InterpretationOfSignature :=
    assign_meaning_to_every_nonlogical_symbol(Signature)

SemanticCompatibility :=
    Interpretation
    ∧ FormulaConstraints
    ∧ OntologyConstraints
    are_jointly_satisfiable

InterpretationConflict :=
    AssignmentsOrConstraints
    yield(Contradiction)

InterpretationConsistency :=
    no_symbol_receives
    incompatible_meanings
    within same semantic context

InterpretationCompleteness :=
    every_required_symbol
    has_defined_interpretation

PartialInterpretation :=
    some_symbols_uninterpreted

TotalInterpretation :=
    every_required_symbol_interpreted

InterpretationExtension :=
    PartialInterpretation
    → TotalInterpretation

SemanticGrounding :=
    Symbol
    → Concept
    → EntityOrRelation
    → FormalMeaning

OntologyInterpretation :=
    OntologyVocabulary
    → SemanticDomain

ClassInterpretation :=
    ClassName
    → SetOfEntities

PropertyInterpretation :=
    PropertyName
    → RelationOrFunction

IndividualInterpretation :=
    IndividualName
    → Entity

AxiomInterpretation :=
    OntologyAxiom
    → ConstraintOnModels

OntologyModel :=
    Interpretation
    ∧ satisfies(AllOntologyAxioms)

SemanticInterpretationOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ InferenceRules

CoreClasses := {
    SemanticInterpretation,
    FormalLanguage,
    Signature,
    Domain,
    InterpretationFunction,
    VariableAssignment,
    Term,
    Formula,
    Theory,
    Model,
    SemanticValue,
    TruthValue,
    Denotation,
    Reference,
    Extension,
    Intension,
    Context,
    PossibleWorld,
    Countermodel
}

CoreRelations := {
    interprets,
    assigns,
    denotes,
    refersTo,
    satisfies,
    falsifies,
    models,
    evaluates,
    constrains,
    extends,
    restricts,
    equivalentTo,
    isomorphicTo,
    grounds,
    dependsOn
}

TypedRelations :=
    Interpretation --interprets--> FormalLanguage
    InterpretationFunction --assigns--> NonLogicalSymbol
    Term --denotes--> DomainElement
    Formula --evaluates--> TruthValue
    Model --satisfies--> Formula
    Model --models--> Theory
    Countermodel --falsifies--> EntailmentClaim
    Context --constrains--> Interpretation
    Symbol --grounds--> OntologyEntity
    Interpretation --extends--> PartialInterpretation

InferenceRules := {
    I ⊨ Γ
        ∧ Γ ⊨ φ
        ⇒ I ⊨ φ,

    I,g ⊨ φ∧ψ
        ⇒ I,g ⊨ φ
        ∧ I,g ⊨ ψ,

    I,g ⊨ ∀x φ
        ⇒
        ∀d∈D :
            I,g[x↦d] ⊨ φ,

    ∃Countermodel I :
        I ⊨ Γ
        ∧ I ⊭ φ
        ⇒ Γ ⊭ φ,

    InterpretationsIsomorphic(I,J)
        ⇒ preserve(
            FirstOrderSentenceTruth
        ),

    TotalInterpretation(I)
        ∧ satisfies(I,T)
        ⇒ ModelOf(I,T),

    SemanticEquivalent(φ,ψ)
        ⇒ Models(φ) = Models(ψ)
}

SEMANTIC_EQUIVALENCE(InterpretationA, InterpretationB) :=
    ∀RelevantExpression e :
        [[e]]_InterpretationA
        =
        [[e]]_InterpretationB

InterpretationSubsumption :=
    Constraints(InterpretationA)
    ⇒ Constraints(InterpretationB)

CanonicalInterpretation :=
    ResolveSignature
    → ResolveDomain
    → ResolveSymbolAssignments
    → ResolveVariableAssignments
    → NormaliseSemanticObjects
    → ValidateConstraints

InterpretationGraph :=
    G = (V,E)

V := {
    Languages,
    Signatures,
    Symbols,
    Domains,
    Interpretations,
    Assignments,
    Terms,
    Formulae,
    Theories,
    Models,
    SemanticValues,
    Contexts,
    Countermodels
}

E := {
    interprets,
    assigns,
    denotes,
    refersTo,
    satisfies,
    falsifies,
    models,
    evaluates,
    constrains,
    extends,
    equivalentTo,
    grounds
}

CONFIDENCE(SemanticInterpretation) :=
    weighted_mean(
        SignatureResolution,
        DomainResolution,
        SymbolAssignmentPrecision,
        ContextCompatibility,
        ModelConsistency,
        SemanticEvidence,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectInterpretationConcept
    → ResolveFormalLanguage
    → ResolveSignature
    → IdentifySymbols
    → IdentifyContext
    → GenerateCandidateInterpretations

AI_REASONING_FLOW :=
    Query
    → ParseExpression
    → ResolveFormalLanguage
    → ResolveSignature
    → IdentifyDomainRequirements
    → ConstructCandidateDomain
    → AssignConstants
    → AssignFunctions
    → AssignPredicatesAndRelations
    → ResolveVariableAssignment
    → EvaluateTerms
    → EvaluateFormulae
    → TestSatisfaction
    → TestSemanticConsequence
    → SearchForCountermodelIfRequired
    → CompareCandidateInterpretations
    → ResolveSemanticEquivalence
    → BuildInterpretationGraph
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseInterpretationEntities,
    ParseTypedRelations,
    ResolveSignature,
    ResolveDomain,
    AssignSemanticObjects,
    ResolveVariableBindings,
    EvaluateTerms,
    EvaluateFormulae,
    TestSatisfaction,
    TestEntailment,
    ConstructCountermodels,
    CompareInterpretations,
    DetectSemanticConflict,
    FollowOntologyGrounding,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewInterpretationEvidence
    → ValidateEvidence
    → IdentifyAffectedSymbols
    → RecomputeAssignments
    → ReevaluateDependentExpressions
    → DetectInterpretationConflict
    → UpdateInterpretationGraph
    → RecomputeModelStatus
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

SignatureUpdate :=
    AddedOrRemovedSymbol
    ⇒ revalidate(
        InterpretationCompleteness,
        DependentFormulae
    )

DomainUpdate :=
    ChangedDomain
    ⇒ reevaluate(
        Constants,
        Functions,
        Predicates,
        QuantifiedFormulae
    )

ContextUpdate :=
    ChangedContext
    ⇒ rerank(CandidateInterpretations)

AxiomUpdate :=
    ChangedTheoryOrOntologyAxiom
    ⇒ recompute(
        AdmissibleModels,
        SemanticConsequences,
        Countermodels
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        InterpretationFunction,
        Satisfaction,
        SemanticConsequence,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    InterpretationEntityResolution,
    FormalLanguage,
    Signature,
    Domain,
    SymbolAssignments,
    VariableAssignments,
    TermDenotations,
    FormulaTruthValues,
    SatisfactionStatus,
    EntailmentStatus,
    CountermodelStatus,
    SemanticEquivalence,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA