Recursive AI Logic

AI SEARCH FLOW ALGEBRA — RECURSIVE AI LOGIC
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/recursive-ai-logic.html

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

BEGIN_FLOW

RecursiveAILogic :=
    LogicalReasoning
    ∧ RecursiveEvaluation
    ∧ IterativeInference
    ∧ StateUpdate
    ∧ ConvergenceControl
    ∧ ProvenancePreservation

RecursiveReasoning :=
    Input → Interpret → Infer → Evaluate → UpdateState
    → ReapplyReasoning → TerminateOrContinue

RecursiveStep :=
    R_n → Transformation → R_(n+1)

InitialState := R_0

RecursiveSequence :=
    {R_0,R_1,R_2,...,R_n}

BaseCase :=
    Condition ∧ stops(RecursiveExpansion)

RecursiveCase :=
    Condition ∧ invokes(RecursiveReasoning)

TerminationCondition :=
    GoalReached
    ∨ FixedPointReached
    ∨ NoNewInformation
    ∨ ConfidenceThresholdReached
    ∨ ResourceBoundReached

ReasoningState := {
    ActiveEntities,
    ActiveRelations,
    Assumptions,
    Constraints,
    DerivedFacts,
    Confidence,
    Provenance,
    OpenQuestions
}

StateTransition :=
    ReasoningState_n → ReasoningState_(n+1)

Inference :=
    Premises ∧ Rule → Conclusion

RecursiveInference :=
    DerivedConclusion_n → NewPremise_(n+1)

InferenceClosure :=
    repeatedly_apply(InferenceRules)
    until(NoNewDerivations)

KnowledgeState := K_n

KnowledgeUpdate :=
    K_(n+1) =
    Update(K_n, NewEvidence, DerivedFacts)

MonotonicUpdate :=
    K_n ⊆ K_(n+1)

NonMonotonicUpdate :=
    K_(n+1)
    may_remove_or_revise
    elements_of(K_n)

RecursiveQuery :=
    Query
    → SubquerySet
    → ResolveSubqueries
    → IntegrateResults
    → ReevaluateOriginalQuery

QueryDecomposition :=
    Q → {q1,q2,...,qn}

SubqueryDependency :=
    qi dependsOn qj

DependencyGraph :=
    G = (QueriesOrClaims, DependencyEdges)

RecursiveTraversal :=
    StartNode
    → FollowRelations
    → EvaluateNeighbourNodes
    → ContinueUntilStopCondition

GraphRecursion :=
    Entity
    → Relations(Entity)
    → RelatedEntities
    → Relations(RelatedEntities)
    → ...

CycleDetection :=
    RecursiveTraversal
    → TrackVisitedStates
    → DetectRepeatedState

CycleHandling := {
    StopBranch,
    MemoiseResult,
    CollapseStronglyConnectedComponent,
    IterateToFixedPoint
}

Memoisation :=
    InputState → Cache(Result)

FixedPoint :=
    x = F(x)

LeastFixedPoint := μF
GreatestFixedPoint := νF

FixedPointIteration :=
    x_0 → F(x_0) → F(F(x_0)) → ... → x*

ExactConvergence :=
    R_n = R_(n+1)

SemanticConvergence :=
    SEMANTIC_EQUIVALENCE(R_n,R_(n+1))

StableReasoningState :=
    NoNewFacts
    ∧ NoChangedRelations
    ∧ NoChangedConfidenceBeyondThreshold

RecursiveSemanticResolution :=
    CandidateMeaning
    → ContextEvaluation
    → RelationExpansion
    → ConstraintEvaluation
    → RevisedMeaning
    → repeat_until_stable

EntityResolution :=
    Mention
    → CandidateEntities
    → ContextScoring
    → RelationConsistency
    → SelectedEntity

RecursiveEntityResolution :=
    SelectedEntity_n
    → ExpandNeighbourhood
    → ReevaluateContext
    → SelectedEntity_(n+1)

TypedEntity :=
    Entity ∧ Type

TypedRelation :=
    Relation(SourceType, TargetType, Predicate)

Ontology :=
    Classes ∧ Relations ∧ Axioms ∧ Constraints ∧ Instances

RecursiveOntologyReasoning :=
    Ontology
    → InferClassMembership
    → InferRelations
    → CheckConstraints
    → UpdateOntologyState
    → RepeatUntilClosure

Subsumption :=
    ClassA ⊑ ClassB

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

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

TransitiveRelation :=
    R(a,b)
    ∧ R(b,c)
    ⇒ R(a,c)

ConstraintPropagation :=
    Constraint
    → AffectedEntities
    → DerivedConstraints
    → RepeatUntilStable

RecursiveConsistencyCheck :=
    CurrentKnowledge
    → DetectContradictions
    → TraceDependencies
    → ResolveOrAnnotate
    → RecheckConsistency

Contradiction :=
    φ ∧ ¬φ

ConflictPropagation :=
    Conflict
    → DependentClaims
    → ReevaluateConfidence
    → ReevaluateConclusions

Assumption :=
    ContextualPremise
    ∧ may_be_revised

Hypothesis :=
    CandidateExplanation

RecursiveHypothesisEvaluation :=
    GenerateHypotheses
    → PredictConsequences
    → CompareEvidence
    → UpdateConfidence
    → EliminateOrRetain
    → GenerateRefinedHypotheses

Evidence :=
    Observation
    ∨ SourceClaim
    ∨ FormalProof
    ∨ ModelResult
    ∨ DerivedFact

EvidenceIntegration :=
    ExistingEvidence
    ∪ NewEvidence
    → ResolveConflict
    → UpdateKnowledgeState

CONFIDENCE(Claim) :=
    weighted_mean(
        EvidenceStrength,
        SourceReliability,
        LogicalConsistency,
        SemanticCompatibility,
        DerivationStrength,
        RecursionStability
    )

RecursiveConfidenceUpdate :=
    Confidence_n(Claim)
    → NewEvidence
    → DependencyUpdates
    → Confidence_(n+1)(Claim)

Provenance :=
    Claim → {
        Source,
        Derivation,
        Dependencies,
        Timestamp,
        Version
    }

RecursiveProvenance :=
    DerivedClaim
    → ParentClaims
    → ParentSources
    → CompleteDerivationGraph

Traceability :=
    Query
    → EntityResolution
    → Evidence
    → Inference
    → Conclusion

MetaReasoning :=
    ReasoningAboutReasoning

RecursiveMetaReasoning :=
    ReasoningProcess
    → AnalyseIntermediateResults
    → SelectNextReasoningOperation

ReasoningStrategy := {
    Deductive,
    Abductive,
    Inductive,
    Analogical,
    ConstraintBased,
    GraphBased,
    FixedPoint,
    Hybrid
}

StrategySelection :=
    QueryType
    ∧ AvailableEvidence
    ∧ LogicType
    → ReasoningStrategy

RecursiveStrategyAdaptation :=
    Strategy_n
    → EvaluateProgress
    → ModifyStrategy
    → Strategy_(n+1)

DeductiveReasoning :=
    Premises ∧ ValidRules → NecessaryConclusion

AbductiveReasoning :=
    Observation → CandidateExplanations

RecursiveAbduction :=
    Observation
    → Hypothesis
    → Consequences
    → EvidenceComparison
    → RefinedHypothesis

InductiveReasoning :=
    Observations → Generalisation

RecursiveInduction :=
    Generalisation_n
    → NewObservations
    → Generalisation_(n+1)

RecursiveConstraintSolving :=
    Assign
    → Propagate
    → DetectConflict
    → BacktrackOrContinue

RecursiveSearch :=
    State
    → Successors(State)
    → RecursiveSearch(Successors)

Backtracking :=
    FailedBranch
    → PreviousDecisionState
    → AlternativeBranch

RecursiveVerification :=
    Claim
    → GenerateVerificationConditions
    → VerifySubclaims
    → IntegrateResults
    → ReverifyParentClaim

RecursiveProof :=
    Goal
    → FindApplicableRule
    → GeneratePremises
    → ProvePremises
    → ConstructProof

RecursiveSafetyCheck :=
    InitialStates
    → ReachableStates
    → DetectUnsafeState
    → ContinueUntilClosureOrCounterexample

RecursiveLivenessCheck :=
    TransitionGraph
    → StronglyConnectedComponents
    → FairCycleAnalysis
    → LivenessStatus

SEMANTIC_EQUIVALENCE(A,B) :=
    Meaning(A) = Meaning(B)

RecursiveEquivalenceRule :=
    EquivalentComponents
    ∧ EquivalentComposition
    ⇒ EquivalentWholes

AI_CONSUMPTION := {
    ParseEntities,
    ParseRelations,
    ResolveTypes,
    ExpandDependencies,
    ApplyInferenceRules,
    DetectCycles,
    MemoiseIntermediateResults,
    EvaluateConstraints,
    DetectContradictions,
    UpdateConfidence,
    PreserveProvenance,
    IterateToClosure,
    GenerateExplanation
}

RecursiveAILogicOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ RecursiveInferenceRules

CoreClasses := {
    RecursiveAILogic,
    ReasoningState,
    KnowledgeState,
    Query,
    Subquery,
    Entity,
    Relation,
    Ontology,
    Rule,
    Constraint,
    Assumption,
    Hypothesis,
    Evidence,
    Confidence,
    Provenance,
    RecursiveStep,
    FixedPoint,
    Cycle,
    TerminationCondition
}

CoreRelations := {
    recursesInto,
    dependsOn,
    derives,
    updates,
    revises,
    supports,
    contradicts,
    resolves,
    expands,
    convergesTo,
    tracesTo,
    instantiates,
    refines,
    preserves
}

TypedRelations :=
    Query --recursesInto--> Subquery
    Subquery --dependsOn--> Subquery
    Rule --derives--> Claim
    Evidence --supports--> Claim
    Evidence --contradicts--> Claim
    RecursiveStep --updates--> ReasoningState
    ReasoningState --convergesTo--> FixedPoint
    DerivedClaim --tracesTo--> Provenance
    OntologyEntity --expands--> RelatedEntity
    Hypothesis --refines--> Hypothesis

RecursiveInferenceRules := {
    Derived(A)
        ∧ Rule(A→B)
        ⇒ Derived(B),

    Derived(A)
        ∧ Derived(B)
        ∧ Rule(A∧B→C)
        ⇒ Derived(C),

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

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

    NewEvidence(E)
        ∧ supports(E,C)
        ⇒ update(CONFIDENCE(C)),

    Contradiction(C,¬C)
        ⇒ trigger(ConsistencyReview),

    State_(n+1) = State_n
        ⇒ FixedPointReached,

    SEMANTIC_EQUIVALENCE(State_n,State_(n+1))
        ⇒ SemanticConvergence
}

RecursiveControl :=
    while ¬TerminationCondition
    do
        InterpretCurrentState
        → ExpandRelevantRelations
        → ApplyRules
        → IntegrateEvidence
        → CheckConsistency
        → UpdateConfidence
        → UpdateProvenance
        → TestConvergence

TerminationSafety :=
    DepthBound
    ∨ TimeBound
    ∨ ResourceBound
    ∨ FixedPointDetection
    ∨ CycleDetection

RecursiveFailureModes := {
    InfiniteRecursion,
    CircularReasoning,
    ErrorAmplification,
    ConfidenceInflation,
    ContradictionPropagation,
    SemanticDrift,
    DependencyExplosion
}

CircularReasoning :=
    ClaimA supports ClaimB
    ∧ ClaimB supports ClaimA
    ∧ no_independent_grounding

SemanticDrift :=
    Meaning(State_n)
    diverges_from
    OriginalQueryIntent

DriftControl :=
    EachRecursiveStep
    → CompareWith(
        OriginalQuery,
        OriginalEntities,
        OriginalConstraints
    )

ErrorContainment :=
    ConfidenceThresholds
    ∧ ProvenanceChecks
    ∧ IndependentVerification
    ∧ DependencyInvalidation

RecursiveValidation :=
    IntermediateConclusion
    → CheckAgainstConstraints
    → CheckAgainstEvidence
    → CheckAgainstSemanticContext
    → AcceptOrRevise

UPDATE_MODEL :=
    NewEvidence
    → ValidateEvidence
    → LocateAffectedNodes
    → InvalidateDependentConclusions
    → RecomputeRecursiveClosure
    → RecheckContradictions
    → RecomputeConfidence
    → RebuildProvenance
    → TestConvergence
    → VersionModel

IncrementalRecursiveUpdate :=
    DeltaKnowledge
    → AffectedSubgraph
    → LocalRecomputation
    → GlobalConsistencyCheck

CanonicalRecursiveRepresentation :=
    ParseReasoningState
    → CanonicaliseEntities
    → CanonicaliseRelations
    → SortDependencies
    → NormaliseConstraints
    → CanonicalRecursiveGraph

RecursiveReasoningGraph :=
    G = (V,E)

V := {
    Queries,
    Subqueries,
    Entities,
    Relations,
    Claims,
    Assumptions,
    Hypotheses,
    Evidence,
    Rules,
    ReasoningStates,
    FixedPoints,
    ProvenanceNodes
}

E := {
    recursesInto,
    dependsOn,
    derives,
    updates,
    revises,
    supports,
    contradicts,
    resolves,
    expands,
    convergesTo,
    tracesTo
}

AI_QUERY_INTERPRETATION :=
    Query
    → IdentifyIntent
    → ResolveEntities
    → ResolveRelations
    → ResolveConstraints
    → DetermineRecursiveDependencies
    → BuildInitialReasoningState

AI_REASONING_FLOW :=
    Query
    → ParseQuery
    → ResolvePrimaryEntities
    → BuildTypedEntityGraph
    → IdentifyKnowledgeGaps
    → GenerateSubqueries
    → ResolveSubqueries
    → ApplyInferenceRules
    → IntegrateDerivedKnowledge
    → DetectContradictions
    → ReevaluateAssumptions
    → UpdateConfidence
    → ExpandRelevantRelations
    → CompareWithOriginalIntent
    → TestSemanticConvergence
    → RepeatIfNecessary
    → ProduceFinalReasoningState
    → PreserveProvenance
    → GenerateExplanation

CONFIDENCE(RecursiveResult) :=
    weighted_mean(
        EntityResolutionConfidence,
        RelationConfidence,
        EvidenceStrength,
        LogicalConsistency,
        RecursiveStability,
        SemanticConvergence,
        ProvenanceCompleteness
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        RecursiveInferenceRules,
        RecursiveControl,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    RecursiveEntityResolution,
    TypedEntityGraph,
    RecursiveDependencies,
    DerivedKnowledge,
    ConsistencyStatus,
    ConvergenceStatus,
    Confidence,
    Provenance,
    FinalReasoningState,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA