Semantic Closure Ontology

AI SEARCH FLOW ALGEBRA — SEMANTIC CLOSURE ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/recursive-ai-logic/semantic-closure-ontology.html

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

BEGIN_FLOW

SemanticClosure :=
    ClosureUnderSemanticConsequence
    ∧ RecursiveMeaningExpansion
    ∧ EntailmentCompletion
    ∧ FixedPointConstruction
    ∧ ConsistencyControl
    ∧ ProvenancePreservation

KnowledgeBase :=
    K = {
        Facts,
        Axioms,
        Rules,
        Constraints,
        OntologyAssertions
    }

SemanticConsequence :=
    K ⊨ φ

SemanticClosureOperator :=
    SemCl(K)
    :=
    {φ | K ⊨ φ}

SemanticClosureSet :=
    K*
    =
    SemCl(K)

ClosureProperty :=
    K ⊆ SemCl(K)

Extensivity :=
    K ⊆ C(K)

Monotonicity :=
    K1 ⊆ K2
    ⇒ C(K1) ⊆ C(K2)

Idempotence :=
    C(C(K))
    =
    C(K)

ClosureOperator :=
    C
    satisfying
    Extensivity
    ∧ Monotonicity
    ∧ Idempotence

SemanticClosure :=
    least_semantically_closed_set
    containing(K)

SemanticallyClosed(K) :=
    ∀φ :
        K ⊨ φ
        ⇒ φ ∈ K

DeductiveClosure :=
    Cn(K)
    :=
    {φ | K ⊢ φ}

SemanticVsSyntacticClosure :=
    SemCl(K)
    compared_with
    Cn(K)

Soundness :=
    K ⊢ φ
    ⇒ K ⊨ φ

Completeness :=
    K ⊨ φ
    ⇒ K ⊢ φ

SoundCompleteClosure :=
    Soundness
    ∧ Completeness
    ⇒ Cn(K) = SemCl(K)

InitialClosureState :=
    C_0 := K

ClosureIteration :=
    C_(n+1)
    :=
    C_n
    ∪ ImmediateSemanticConsequences(C_n)

ClosureSequence :=
    C_0 → C_1 → C_2 → ... → C_n

ClosureFixedPoint :=
    C_(n+1) = C_n

SemanticFixedPoint :=
    C* = F(C*)

LeastFixedPoint :=
    μF

ClosureConvergence :=
    NoNewSemanticConsequences

SemanticConvergence :=
    SEMANTIC_EQUIVALENCE(
        C_n,
        C_(n+1)
    )

ImmediateConsequence :=
    Facts
    ∧ Rules
    → NewFact

ImmediateConsequenceOperator :=
    T(K)

RecursiveClosure :=
    K
    → T(K)
    → T²(K)
    → ...
    → FixedPoint

InferenceRule :=
    Premises
    → Conclusion

RuleClosure :=
    repeatedly_apply(InferenceRules)
    until(NoNewConclusions)

OntologyClosure :=
    Ontology
    → ApplyAxioms
    → InferClassMembership
    → InferRelations
    → PropagateRestrictions
    → RepeatUntilStable

ClassSubsumption :=
    A ⊑ B

SubsumptionClosure :=
    A ⊑ B
    ∧ B ⊑ C
    ⇒ A ⊑ C

InstanceInheritance :=
    Instance(x,A)
    ∧ A ⊑ B
    ⇒ Instance(x,B)

PropertySubsumption :=
    P ⊑ Q
    ∧ P(a,b)
    ⇒ Q(a,b)

TransitiveClosure :=
    R(a,b)
    ∧ R(b,c)
    ∧ Transitive(R)
    ⇒ R(a,c)

SymmetricClosure :=
    R(a,b)
    ∧ Symmetric(R)
    ⇒ R(b,a)

InverseClosure :=
    R(a,b)
    ∧ inverseOf(R,S)
    ⇒ S(b,a)

EquivalentClassClosure :=
    A ≡ B
    ⇒ A ⊑ B
    ∧ B ⊑ A

EquivalentPropertyClosure :=
    P ≡ Q
    ⇒ P ⊑ Q
    ∧ Q ⊑ P

IdentityClosure :=
    sameAs(a,b)
    → propagate_compatible_assertions(a,b)

DomainInference :=
    P(a,b)
    ∧ Domain(P,A)
    ⇒ Instance(a,A)

RangeInference :=
    P(a,b)
    ∧ Range(P,B)
    ⇒ Instance(b,B)

SemanticDependency :=
    DerivedMeaning
    dependsOn
    SupportingMeanings

DependencyGraph :=
    Gd = (Claims, DependencyEdges)

RecursiveDependencyClosure :=
    Claim
    → Dependencies
    → DependenciesOfDependencies
    → GroundKnowledge

GroundKnowledge :=
    Axiom
    ∨ PrimitiveFact
    ∨ DirectEvidence
    ∨ ExplicitDefinition

ClosureJustification :=
    DerivedClaim
    → Premises
    ∧ Rule
    ∧ SemanticContext

SemanticEntailmentPath :=
    KnowledgeBase
    → IntermediateConsequences
    → EntailedClaim

MultipleEntailmentPaths :=
    Claim
    has
    multiple(DerivationOrSemanticSupportPaths)

RedundantConsequence :=
    Claim
    already_semantically_implied
    before explicit addition

ConservativeAddition :=
    NewStatement
    adds_no_new_semantic_consequences
    in specified vocabulary
    beyond existing closure

TheoryExtension :=
    K' = K ∪ NewStatements

ClosureUpdate :=
    SemCl(K')
    compared_with
    SemCl(K)

IncrementalClosure :=
    DeltaKnowledge
    → IdentifyAffectedDependencies
    → DeriveDeltaConsequences
    → UpdateClosure

DeltaClosure :=
    NewConsequences
    =
    SemCl(K ∪ Δ)
    minus
    SemCl(K)

Retraction :=
    RemoveStatement
    → InvalidateDependentConsequences
    → RecomputeAffectedClosure

TruthMaintenance :=
    Maintain(
        Claims,
        Justifications,
        Dependencies,
        Retractions
    )

RecursiveTruthMaintenance :=
    ChangedGroundKnowledge
    → TraverseDependents
    → RemoveUnsupportedClaims
    → RecomputeClosure

MonotonicClosure :=
    AddingKnowledge
    does_not_remove
    ExistingConsequences

NonMonotonicClosure :=
    NewKnowledge
    may_invalidate
    ExistingConclusions

DefaultReasoning :=
    DefaultRule
    applies
    unless defeated

DefeasibleClosure :=
    Closure
    under
    NonMonotonicRules

BeliefRevision :=
    NewEvidence
    → DetectConflict
    → ReviseKnowledge
    → RecomputeSemanticClosure

Consistency :=
    ∃Model M :
        M ⊨ K

Inconsistency :=
    no Model satisfies K

ClosureConsistency :=
    Consistent(K)
    ⇒ Consistent(SemCl(K))
    under standard semantic consequence

Contradiction :=
    φ
    ∧ ¬φ

ContradictionInClosure :=
    φ ∈ SemCl(K)
    ∧ ¬φ ∈ SemCl(K)

ExplosionClassical :=
    Inconsistent(K)
    ⇒ every_formula_semantically_follows
    under classical semantics

ParaconsistentClosure :=
    ContradictionsAllowed
    ∧ ExplosionBlocked
    ∧ ConsequenceRelationRestricted

ConsistencyCheck :=
    ClosureUpdate
    → DetectDirectContradictions
    → TestSatisfiability
    → TraceConflictDependencies

ConstraintClosure :=
    Constraints
    → Propagate
    → DerivedConstraints
    → FixedPoint

TypeConstraintClosure :=
    TypeAssertions
    ∧ ClassRelations
    → InferredTypes

CardinalityClosure :=
    CardinalityRestrictions
    → DerivedExistenceOrConflictConditions

EqualityClosure :=
    EqualityAssertions
    → CongruencePropagation

Congruence :=
    a = b
    ⇒ f(a) = f(b)

SemanticNormalisation :=
    EquivalentExpressions
    → CanonicalSemanticRepresentation

SemanticEquivalence :=
    Meaning(A) = Meaning(B)

SEMANTIC_EQUIVALENCE(ClosureA,ClosureB) :=
    same(SemanticConsequences)
    within specified language_and_context

ClosureEquivalence :=
    SemCl(K1) = SemCl(K2)

SemanticSubsumption :=
    SemCl(K1)
    ⊆
    SemCl(K2)

StrongerKnowledgeBase :=
    K2
    semantically_entails
    all consequences of K1
    and potentially more

SemanticCompleteness :=
    AllRelevantEntailedClaims
    represented_or_derivable

MaterialisedClosure :=
    ExplicitlyStore(
        DerivedConsequences
    )

VirtualClosure :=
    ComputeConsequences
    OnDemand

HybridClosure :=
    MaterialiseFrequentlyUsedConsequences
    ∧ ComputeOthersOnDemand

ClosureCache :=
    FLOW_HASH(KnowledgeState)
    → CachedClosure

Memoisation :=
    CanonicalSubproblem
    → CachedSemanticResult

RecursiveClosureMemoisation :=
    if Cached(State)
    then Reuse
    else ComputeAndStore

ClosureScope :=
    {
        FullLanguage,
        SelectedPredicates,
        SelectedEntities,
        QueryRelevantSubgraph,
        BoundedDepth
    }

BoundedClosure :=
    Closure
    restricted_by
    Depth ∨ RuleSet ∨ Vocabulary ∨ Resources

RelevantClosure :=
    Query
    → IdentifyRelevantKnowledge
    → ComputeLocalSemanticClosure

QueryClosure :=
    QueryContext
    ∪ RelevantKnowledge
    → EntailedAnswers

SemanticSearch :=
    SeedEntity
    → RelatedEntities
    → Relations
    → Consequences
    → RecursiveExpansion

ClosurePruning :=
    Exclude(
        IrrelevantConsequences,
        DuplicateConsequences,
        SemanticallyEquivalentDuplicates
    )

Provenance :=
    DerivedClaim → {
        GroundSources,
        SupportingClaims,
        Rules,
        SemanticTransformations,
        Timestamp,
        Version
    }

RecursiveProvenance :=
    DerivedClaim
    → Justifications
    → ParentClaims
    → GroundSources

ClosureTrace :=
    GroundKnowledge
    → InferenceSequence
    → DerivedClaim

Confidence :=
    [0,1]

CONFIDENCE(DerivedClaim) :=
    weighted_mean(
        GroundEvidenceStrength,
        RuleReliability,
        SemanticCompatibility,
        DependencyReliability,
        Consistency,
        ClosureStability,
        ProvenanceQuality
    )

RecursiveConfidencePropagation :=
    Confidence(Premises)
    ∧ RuleConfidence
    → Confidence(Conclusion)
    → propagate_to(ClosureDependents)

ClosureStability :=
    NoNewMaterialConsequences
    ∧ NoChangedMaterialRelations
    ∧ ConfidenceDeltaBelowThreshold

TerminationCondition :=
    FixedPointReached
    ∨ NoNewConsequences
    ∨ QueryResolved
    ∨ DepthBoundReached
    ∨ ResourceBoundReached

ResourceBound := {
    TimeLimit,
    MemoryLimit,
    InferenceStepLimit,
    MaterialisationLimit
}

SemanticClosureOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ ClosureInferenceRules

CoreClasses := {
    SemanticClosure,
    KnowledgeBase,
    Theory,
    Formula,
    SemanticConsequence,
    ClosureOperator,
    ClosureState,
    FixedPoint,
    InferenceRule,
    DerivedClaim,
    GroundKnowledge,
    Dependency,
    Justification,
    Constraint,
    Contradiction,
    Provenance,
    ClosureResult
}

CoreRelations := {
    entails,
    derives,
    dependsOn,
    supports,
    contradicts,
    extends,
    closesUnder,
    convergesTo,
    equivalentTo,
    subsumes,
    invalidates,
    groundsIn,
    tracesTo
}

TypedRelations :=
    KnowledgeBase --entails--> Claim
    Rule --derives--> Claim
    DerivedClaim --dependsOn--> Premise
    Evidence --supports--> Claim
    Claim --contradicts--> Claim
    ClosureState --convergesTo--> FixedPoint
    DerivedClaim --groundsIn--> GroundKnowledge
    DerivedClaim --tracesTo--> Provenance
    KnowledgeBase --closesUnder--> InferenceRule

ClosureInferenceRules := {
    K ⊨ φ
        ⇒ φ ∈ SemCl(K),

    φ ∈ K
        ⇒ φ ∈ SemCl(K),

    A ⊑ B
        ∧ B ⊑ C
        ⇒ A ⊑ C,

    Instance(x,A)
        ∧ A ⊑ B
        ⇒ Instance(x,B),

    R(a,b)
        ∧ R(b,c)
        ∧ Transitive(R)
        ⇒ R(a,c),

    Retracted(P)
        ∧ dependsOn(C,P)
        ⇒ invalidate_or_recheck(C),

    C_(n+1)=C_n
        ⇒ ClosureFixedPointReached,

    SEMANTIC_EQUIVALENCE(C_n,C_(n+1))
        ⇒ SemanticConvergence
}

CanonicalSemanticClosure :=
    ParseKnowledgeBase
    → CanonicaliseEntities
    → CanonicaliseRelations
    → NormaliseFormulae
    → ResolveConstraints
    → BuildDependencyGraph
    → ComputeClosure
    → RemoveSemanticDuplicates

ClosureGraph :=
    G = (V,E)

V := {
    GroundClaims,
    DerivedClaims,
    Rules,
    Constraints,
    Entities,
    Relations,
    ClosureStates,
    FixedPoints,
    ProvenanceNodes
}

E := {
    entails,
    derives,
    dependsOn,
    supports,
    contradicts,
    extends,
    closesUnder,
    convergesTo,
    equivalentTo,
    groundsIn,
    tracesTo
}

AI_QUERY_INTERPRETATION :=
    Query
    → DetectSemanticClosureConcept
    → ResolveKnowledgeBase
    → ResolveLogicAndSemantics
    → IdentifyRelevantVocabulary
    → IdentifyRequestedConsequence
    → BuildInitialClosureState

AI_REASONING_FLOW :=
    Query
    → ParseQuery
    → ResolveEntitiesAndRelations
    → ResolveKnowledgeBase
    → ResolveSemanticContext
    → BuildDependencyGraph
    → IdentifyGroundKnowledge
    → ApplyImmediateConsequences
    → PropagateOntologyRelations
    → PropagateConstraints
    → MaterialiseOrVirtuallyDeriveConsequences
    → DetectContradictions
    → MaintainJustifications
    → UpdateConfidence
    → TestClosureFixedPoint
    → TestSemanticConvergence
    → RepeatIfNecessary
    → ExtractQueryRelevantClosure
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseClosureEntities,
    ParseTypedRelations,
    ResolveKnowledgeBase,
    ResolveSemanticContext,
    ComputeSemanticConsequences,
    ApplyClosureRules,
    ComputeTransitiveClosure,
    PropagateConstraints,
    DetectContradictions,
    MaintainTruthDependencies,
    PerformIncrementalClosure,
    CompareClosureEquivalence,
    UpdateConfidence,
    TestFixedPoints,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewKnowledgeRuleConstraintOrEvidence
    → ValidateInput
    → IdentifyAffectedClosureRegion
    → InvalidateUnsupportedConsequences
    → UpdateDependencyGraph
    → RecomputeDeltaClosure
    → RecheckConstraints
    → RecheckContradictions
    → RecomputeConfidence
    → TestClosureConvergence
    → PreserveProvenance
    → VersionModel

KnowledgeUpdate :=
    AddedOrRemovedStatement
    ⇒ recompute(
        AffectedConsequences,
        Dependencies,
        ClosureStatus
    )

RuleUpdate :=
    ChangedInferenceRule
    ⇒ recompute(
        RuleClosure,
        DependentClaims
    )

ConstraintUpdate :=
    ChangedConstraint
    ⇒ reevaluate(
        DerivedConstraints,
        Conflicts,
        SemanticModels
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        ClosureInferenceRules,
        SemanticClosureOperator,
        ClosureIteration,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    SemanticClosureEntityResolution,
    KnowledgeBase,
    SemanticContext,
    GroundKnowledge,
    DependencyGraph,
    DerivedConsequences,
    ConstraintClosure,
    ContradictionStatus,
    FixedPointStatus,
    SemanticConvergenceStatus,
    QueryRelevantClosure,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA