Probability Tree Ontology

FULL EXECUTABLE FLOW ALGEBRA — PROBABILITY TREE ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/probability-rules/probability-tree-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Probability Rules"
ONTOLOGY := "Probability Tree Ontology"
PRIMARY_ENTITY := ProbabilityTree
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

ProbabilityTree :=
    RootedBranchingProbabilityStructure
    ∧ SequentialEvents
    ∧ ConditionalProbabilities
    ∧ PathProbabilityRules
    ∧ OutcomeAggregation

ProbabilitySpace :=
    (Ω,F,P)

Tree :=
    (V,E)

Root :=
    InitialState
    ∈ V

Node :=
    StateOrPartialOutcome
    ∈ V

Branch :=
    DirectedEdge
    representing
    PossibleNextEventOrOutcome

Leaf :=
    TerminalNode
    representing
    CompleteOutcomeOrTerminalState

InternalNode :=
    Node
    with outgoing branches

Depth :=
    NumberOfBranches
    from Root
    to Node

Stage :=
    SetOfNodes
    at same sequential step
    or modelled decision layer

BranchLabel :=
    EventOrOutcome
    associated_with
    Branch

BranchProbability :=
    P(NextEvent|CurrentPath)

OutgoingBranchNormalisation :=
    For each nonterminal node N:
        Σ_{e outgoing from N}
        P(e|N)
        =
        1

BranchNonNegativity :=
    0≤P(e|N)≤1

Path :=
    Root
    → Node_1
    → ...
    → Leaf

PathEvent :=
    IntersectionOfBranchEventsAlongPath

PathProbability :=
    ProductOfConditionalBranchProbabilities

PathMultiplicationRule :=
    Path(e1,...,en)
    ⇒
    P(Path)
    =
    P(e1)
    Π_{i=2..n}
    P(e_i|e1∩...∩e_(i-1))

TwoStagePath :=
    A
    → B

TwoStagePathProbability :=
    P(A∩B)
    =
    P(A)P(B|A)

ThreeStagePathProbability :=
    P(A∩B∩C)
    =
    P(A)P(B|A)P(C|A∩B)

IndependentBranchSimplification :=
    Independent(B,A)
    ⇒
    P(B|A)=P(B)

DependentBranch :=
    P(B|A)
    ≠
    P(B)

ConditionalBranch :=
    BranchProbability
    depends_on
    preceding path

UnconditionalFirstBranch :=
    FirstStageBranch
    labelled_by
    P(A)

CompleteTree :=
    AllRelevantOutcomes
    represented_by
    RootToLeafPaths

MutuallyExclusiveSiblingBranches :=
    BranchesFromSameNode
    represent
    mutually exclusive alternatives

CollectivelyExhaustiveSiblingBranches :=
    UnionOfSiblingBranchEvents
    =
    CurrentNodeEvent

SiblingPartition :=
    OutgoingBranches(Node)
    partition
    NodeEvent

LeafPartition :=
    CompleteRootToLeafEvents
    partition
    SampleSpace
    when tree is complete and non-overlapping

LeafProbability :=
    PathProbability(LeafPath)

TreeNormalisation :=
    Σ_{leaf L}
    P(L)
    =
    1
    for complete valid tree

EventProbabilityFromLeaves :=
    P(A)
    =
    Σ_{leaf L satisfying A}
    P(L)

PathAdditionRule :=
    MutuallyExclusivePathsToTarget
    ⇒
    P(Target)
    =
    Sum(PathProbabilities)

ProbabilityTreeEvaluation :=
    MultiplyAlongBranches
    ∧ AddAcrossMutuallyExclusivePaths

TreeQuery :=
    TargetEvent
    → IdentifyCompatibleLeaves
    → ComputeLeafProbabilities
    → SumCompatibleLeaves

PartialPath :=
    Root
    → IntermediateNode

PartialPathProbability :=
    ProductOfBranchProbabilities
    along partial path

NodeProbability :=
    SumOfProbabilities
    of all paths reaching node

UniquePathTreeNode :=
    If node has unique root path:
        P(Node)=PathProbability

MergedStateGraph :=
    If multiple paths reach same state:
        NodeProbability
        =
        Sum(PathProbabilities)

TreeVsGeneralGraph :=
    Tree
    has unique parent per non-root node
    whereas
    probability graph may merge states

SequentialExperiment :=
    Stage_1
    → Stage_2
    → ...
    → Stage_n

TrialSequence :=
    Outcomes
    generated
    sequentially

WithReplacementTree :=
    Sampling
    where composition resets
    after each draw

WithoutReplacementTree :=
    Sampling
    where later branch probabilities
    change after earlier draws

IndependentTrialTree :=
    SameBranchProbabilities
    repeated
    across equivalent nodes
    under independence

DependentTrialTree :=
    BranchProbabilities
    vary_with
    prior outcomes

BernoulliTree :=
    EachStage
    has branches
    Success and Failure

BernoulliBranchProbabilities :=
    P(Success)=p
    ∧ P(Failure)=1-p

BinomialTree :=
    RepeatedIndependentBernoulliStages

BinomialPathProbability :=
    For path with k successes in n trials:
        p^k(1-p)^(n-k)

BinomialOutcomeAggregation :=
    P(K=k)
    =
    C(n,k)p^k(1-p)^(n-k)

UrnTree :=
    SequentialDraws
    represented_by
    branches for item categories

WithoutReplacementUrnUpdate :=
    DrawnItem
    → UpdateComposition
    → UpdateNextBranchProbabilities

CardTree :=
    SequentialCardDraws
    with or without replacement

DiceTree :=
    SequentialDiceOutcomes

CoinTree :=
    SequentialCoinOutcomes

DiagnosticTree :=
    ConditionStatus
    → TestOutcome

DiagnosticBranches := {
    Condition → Positive,
    Condition → Negative,
    NoCondition → Positive,
    NoCondition → Negative
}

DiagnosticPathProbability :=
    PriorConditionProbability
    × ConditionalTestProbability

PositiveTestProbability :=
    Sum(
        ConditionAndPositivePath,
        NoConditionAndPositivePath
    )

BayesFromTree :=
    P(Condition|Positive)
    =
    P(Condition∩Positive)
    /
    P(Positive)

BayesTreeDenominator :=
    P(Positive)
    =
    Sum(AllPositiveLeafPaths)

LawOfTotalProbabilityTree :=
    PartitionBranches B_i
    → Event A
    ⇒
    P(A)
    =
    Σ_i P(B_i)P(A|B_i)

BayesTree :=
    PriorBranches
    → EvidenceBranches
    → PosteriorByNormalisedPathWeights

PriorBranch :=
    P(H_i)

LikelihoodBranch :=
    P(E|H_i)

JointEvidencePath :=
    P(H_i)P(E|H_i)

PosteriorFromTree :=
    P(H_i|E)
    =
    JointEvidencePath_i
    /
    Sum_j JointEvidencePath_j

DecisionTreeRelation :=
    ProbabilityTree
    may combine
    chance nodes
    with decision nodes

ChanceNode :=
    Node
    whose outgoing branches
    have probabilities

DecisionNode :=
    Node
    whose outgoing branches
    represent actions
    rather_than random outcomes

UtilityLeaf :=
    Leaf
    with associated
    payoff_or_utility

ExpectedUtility :=
    Σ_leaf
    P(leaf|decision_policy)
    × Utility(leaf)

DecisionPolicy :=
    Mapping
    from decision nodes
    to chosen actions

ProbabilityTreeConstraint :=
    StructuralOrProbabilityRequirement
    on tree

TreeStructuralConstraints := {
    OneRoot,
    DirectedAcyclicStructure,
    UniqueParentForNonRootTreeNodes,
    ValidBranchTargets,
    CompletePathDefinitions
}

TreeProbabilityConstraints := {
    BranchNonNegativity,
    OutgoingBranchNormalisation,
    ConditionalConsistency,
    LeafNormalisation
}

BranchConsistency :=
    BranchProbability
    equals
    relevant conditional probability

PathConsistency :=
    PathProbability
    equals
    product of branch probabilities

LeafConsistency :=
    LeafProbability
    equals
    path probability

TreeProbabilityConflict :=
    ViolationOf(
        BranchBounds
        ∨ OutgoingNormalisation
        ∨ ConditionalConsistency
        ∨ PathMultiplication
        ∨ LeafNormalisation
    )

IncompleteTree :=
    MissingPossibleBranchesOrLeaves
    relative_to modelled experiment

OverlappingSiblingBranches :=
    SiblingEvents
    not mutually exclusive

NonExhaustiveSiblingBranches :=
    SiblingEvents
    fail to cover
    parent event

TreeValidation :=
    ResolveRoot
    → TraverseNodes
    → ValidateSiblingPartitions
    → ValidateBranchProbabilities
    → ComputePathProbabilities
    → ValidateLeafNormalisation

RecursiveTreeTraversal :=
    Evaluate(Node N, PathProbability p)
    :=
    if Leaf(N)
    then Record(N,p)
    else
        for each Branch e from N:
            Evaluate(
                Target(e),
                p×P(e|N)
            )

ForwardProbabilityPropagation :=
    RootProbability=1
    → MultiplyAlongBranches
    → AssignLeafProbabilities

BackwardEventAggregation :=
    TargetLeaves
    → SumProbabilities
    → EventProbability

DynamicTreeUpdate :=
    NewEvidenceOrBranchProbability
    → UpdateAffectedBranch
    → RecomputeDescendantPaths
    → RecomputeAggregatedEvents

EvidenceConditioningTree :=
    Observe(Event E)
    → RemoveOrZeroIncompatibleLeaves
    → RenormaliseCompatibleLeafProbabilities

PosteriorTree :=
    ConditionedProbabilityTree
    after evidence

PrunedTree :=
    Tree
    with impossible_or_irrelevant branches removed

TreePruning :=
    ConstraintOrEvidence
    → RemoveIncompatibleBranches
    → PreserveRemainingProbabilityStructure
    → RenormaliseIfConditioning

ProbabilityTreeState := {
    Root,
    Nodes,
    Branches,
    Stages,
    BranchProbabilities,
    Paths,
    LeafProbabilities,
    TargetEvents,
    Evidence,
    Confidence,
    Provenance
}

TreeEvidence :=
    ExperimentDefinition
    ∨ ConditionalProbability
    ∨ JointProbability
    ∨ ObservedOutcome
    ∨ ModelAssumption

TreeProvenance :=
    NodeOrBranch
    → {
        Source,
        ProbabilitySource,
        ConditionalContext,
        ParentPath,
        Version
    }

SEMANTIC_EQUIVALENCE(TreeA,TreeB) :=
    same(LeafOutcomeSemantics)
    ∧ same(RootToLeafProbabilityMeasure)
    under specified query scope

TreeCompression :=
    EquivalentSubtrees
    → SharedRepresentation
    while preserving probabilities

TreeExpansion :=
    CoarseBranch
    → FinerConditionalBranches

CanonicalProbabilityTree :=
    ResolveRoot
    → CanonicaliseBranchLabels
    → CanonicaliseConditionalContexts
    → SortSiblingBranches
    → ComputePathProbabilities
    → CanonicaliseLeaves

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

CoreClasses := {
    ProbabilityTree,
    Root,
    Node,
    Branch,
    Leaf,
    Stage,
    Path,
    Event,
    ConditionalProbability,
    PathProbability,
    SequentialExperiment,
    ChanceNode,
    DecisionNode,
    Evidence,
    PosteriorTree,
    TreeConstraint
}

CoreRelations := {
    hasRoot,
    hasNode,
    hasBranch,
    branchesTo,
    hasProbability,
    conditionedOn,
    partOfPath,
    terminatesAt,
    representsEvent,
    precedes,
    dependsOn,
    supports,
    contradicts,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    ProbabilityTree --hasRoot--> Root
    ProbabilityTree --hasNode--> Node
    Node --hasBranch--> Branch
    Branch --branchesTo--> Node
    Branch --hasProbability--> ProbabilityValue
    BranchProbability --conditionedOn--> PathContext
    Branch --partOfPath--> Path
    Path --terminatesAt--> Leaf
    Leaf --representsEvent--> Event
    ProbabilityTree --tracesTo--> Provenance

InferenceRules := {
    Branch(e)
        ⇒ 0≤P(e)≤1,

    SiblingPartition(Node,{e_i})
        ⇒ Σ_iP(e_i|Node)=1,

    Path(e1,...,en)
        ⇒ P(Path)=Π_iP(e_i|PreviousPath),

    MutuallyExclusivePaths({Path_i})
        ∧ all_paths_imply(EventA)
        ⇒ P(A)=Σ_iP(Path_i),

    CompleteTree(T)
        ⇒ Σ_leaf P(leaf)=1,

    IndependentNextEvent(B,A)
        ⇒ P(B|A)=P(B),

    Evidence(E)
        ⇒ ConditionTreeOn(E),

    InvalidSiblingPartition
        ⇒ InvalidProbabilityTree
}

CONFIDENCE(ProbabilityTreeResult) :=
    weighted_mean(
        TreeStructureResolution,
        BranchEventResolution,
        ConditionalProbabilityQuality,
        SiblingPartitionValidity,
        PathCalculationValidity,
        LeafNormalisation,
        EvidenceQuality,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectProbabilityTreeIntent
    → IdentifySequentialExperiment
    → IdentifyStagesAndBranches
    → ResolveConditionalProbabilities
    → IdentifyTargetOutcomeOrEvent
    → BuildInitialTree

AI_REASONING_FLOW :=
    Query
    → ParseSequentialProbabilityProblem
    → ResolveProbabilitySpace
    → ResolveRootState
    → IdentifyStages
    → ResolveBranchEvents
    → ResolveConditionalBranchProbabilities
    → ValidateSiblingPartitions
    → ConstructRootToLeafPaths
    → MultiplyProbabilitiesAlongPaths
    → IdentifyTargetCompatibleLeaves
    → AddProbabilitiesAcrossExclusivePaths
    → ApplyEvidenceConditioningIfRequired
    → ValidateLeafNormalisation
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseProbabilityTreeEntities,
    ParseTypedRelations,
    ResolveSequentialExperiments,
    ResolveTreeNodes,
    ResolveBranches,
    ResolveConditionalProbabilities,
    ValidateSiblingPartitions,
    ComputePathProbabilities,
    AggregateLeafProbabilities,
    EvaluateDependentBranches,
    EvaluateIndependentBranches,
    ApplyLawOfTotalProbability,
    ApplyBayesTheoremFromTree,
    ConditionTreesOnEvidence,
    ValidateProbabilityTrees,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewBranchProbabilityEvidenceOrStructure
    → ValidateInput
    → IdentifyAffectedNodesAndPaths
    → UpdateTree
    → RevalidateSiblingPartitions
    → RecomputeDescendantPathProbabilities
    → RecomputeTargetEventProbabilities
    → RevalidateLeafNormalisation
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

BranchUpdate :=
    ChangedBranchProbability
    ⇒ recompute(
        DescendantPaths,
        LeafProbabilities,
        AggregatedEvents
    )

StructureUpdate :=
    AddedRemovedOrChangedBranch
    ⇒ revalidate(
        TreeStructure,
        SiblingPartitions,
        Completeness
    )

EvidenceUpdate :=
    NewObservedEvidence
    ⇒ condition(
        ProbabilityTree,
        CompatiblePaths
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        ProbabilityTreeEvaluation,
        PathMultiplicationRule,
        PathAdditionRule,
        TreeValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ProbabilityTreeEntityResolution,
    ProbabilitySpace,
    Root,
    TreeStructure,
    Stages,
    BranchEvents,
    ConditionalBranchProbabilities,
    Paths,
    PathProbabilities,
    LeafProbabilities,
    TargetEventProbability,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA