Halting Problem Ontology

KIRETH RESEARCH - SEMANTIC LIBRARY
AI CORE -> ONTOLOGIES -> MATHEMATICAL ONTOLOGIES -> MATHEMATICAL LOGIC ONTOLOGIES
COMPUTABILITY THEORY -> HALTING PROBLEM ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory/halting-problem-ontology.html

FLOW_ALGEBRA HaltingProblemOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN HaltingProblemOntology := {
    HaltingProblem,
    ComputabilityTheory,
    Undecidability,
    DecisionProblem,
    UndecidableProblem,
    SemiDecidableProblem,
    RecognisableProblem,
    TuringMachine,
    UniversalTuringMachine,
    MachineEncoding,
    ProgramEncoding,
    InputEncoding,
    Computation,
    Configuration,
    Halting,
    Divergence,
    Acceptance,
    Rejection,
    Decider,
    Recogniser,
    Algorithm,
    Diagonalisation,
    SelfReference,
    Contradiction,
    Reduction,
    ManyOneReduction,
    UniversalSimulation,
    ComputableFunction,
    PartialComputableFunction,
    RiceTheorem
}

DEFINE ENTITY_TYPES := {
    HALTING_PROBLEM,
    DECISION_PROBLEM,
    UNDECIDABLE_PROBLEM,
    MACHINE,
    MACHINE_ENCODING,
    INPUT,
    COMPUTATION,
    CONFIGURATION,
    ALGORITHM,
    DECIDER,
    RECOGNISER,
    REDUCTION,
    PROOF_METHOD,
    COMPUTABILITY_PROPERTY,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    HAS_INPUT,
    ENCODES,
    SIMULATES,
    HALTS_ON,
    DIVERGES_ON,
    ACCEPTS,
    REJECTS,
    DECIDES,
    RECOGNISES,
    REDUCES_TO,
    MANY_ONE_REDUCES_TO,
    CONTRADICTS,
    WITNESSES,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    DEPENDS_ON
}

SET HaltingProblem.type = HALTING_PROBLEM
SET HaltingProblem.parent = UndecidableProblem
SET HaltingProblem.context = ComputabilityTheory
SET HaltingProblem.role =
    "determine whether an arbitrary encoded program or Turing machine eventually halts when executed on a specified input"

RULE halting_instance:
    TuringMachine(M)
    AND Input(w)
    -> HaltingInstance(encode(M,w))

RULE halting_language:
    HALT = {
        encode(M,w)
        | HALTS_ON(M,w)
    }

RULE positive_instance:
    HALTS_ON(M,w)
    -> encode(M,w) MEMBER_OF HALT

RULE negative_instance:
    DIVERGES_ON(M,w)
    -> encode(M,w) NOT_MEMBER_OF HALT

RULE recognisability:
    UniversalTuringMachine(U)
    AND input=encode(M,w)
    AND U simulates M on w
    AND IF M halts THEN U accepts
    -> RecognisableProblem(HaltingProblem)

RULE semi_decidability:
    RecognisableProblem(HaltingProblem)
    -> SemiDecidableProblem(HaltingProblem)

RULE assume_halting_decider:
    Assume EXISTS Algorithm(H) [
        FOR_EVERY encode(M,w) [
            HALTS_ON(H,encode(M,w))
            AND H(M,w)=YES IFF HALTS_ON(M,w)
            AND H(M,w)=NO IFF DIVERGES_ON(M,w)
        ]
    ]
    -> CandidateHaltingDecider(H)

RULE diagonal_machine:
    CandidateHaltingDecider(H)
    -> ConstructMachine(D) {
        input: encode(X),
        behaviour:
            IF H(X,X)=YES
            THEN loop_forever
            ELSE halt
    }

RULE diagonal_self_application:
    ConstructedMachine(D)
    -> Evaluate D on encode(D)

RULE diagonal_case_1:
    H(D,D)=YES
    -> prediction:
        "D halts on D"
    -> construction:
        "D loops forever"
    -> CONTRADICTION

RULE diagonal_case_2:
    H(D,D)=NO
    -> prediction:
        "D does not halt on D"
    -> construction:
        "D halts"
    -> CONTRADICTION

RULE halting_undecidability:
    CandidateHaltingDecider(H)
    AND diagonal_self_application_produces_contradiction
    -> NOT EXISTS TotalCorrectHaltingDecider
    -> UndecidableProblem(HaltingProblem)

RULE no_finite_observation_decision:
    Computation(M,w)
    AND observed_for_finite_number_of_steps
    AND not_halted_yet
    -> DO_NOT_INFER DIVERGES_ON(M,w)

RULE individual_instance_guard:
    UndecidableProblem(HaltingProblem)
    -> DOES_NOT_IMPLY
       every_specific_halting_instance_is_unanswerable

RULE reduction_source:
    UndecidableProblem(HaltingProblem)
    AND MANY_ONE_REDUCES_TO(HaltingProblem,B)
    -> UndecidableProblem(B)

RULE decidability_contradiction:
    MANY_ONE_REDUCES_TO(HaltingProblem,B)
    AND Assume DecidableProblem(B)
    -> DecidableProblem(HaltingProblem)
    -> CONTRADICTION

DEFINE HALTING_PROBLEM_MODEL := {
    input:
        "encoding of machine M and input w",
    question:
        "does M eventually halt when run on w?",
    yes_condition:
        "M reaches a halting configuration",
    no_condition:
        "M executes forever",
    classification:
        UndecidableProblem
}

DEFINE HALT_SET_MODEL := {
    set_name:
        HALT,
    representation:
        "{encode(M,w) | M halts on w}",
    recognisable:
        TRUE,
    decidable:
        FALSE
}

DEFINE DIAGONAL_ARGUMENT_MODEL := {
    assumption:
        "a total correct halting decider H exists",
    construction:
        "machine D reverses H's prediction when applied to a machine's own encoding",
    self_application:
        "run D on encode(D)",
    result:
        "both possible predictions contradict D's defined behaviour",
    conclusion:
        "H cannot exist"
}

DEFINE RECOGNISER_MODEL := {
    input:
        "encode(M,w)",
    procedure:
        "simulate M on w",
    if_M_halts:
        "eventually observe the halt and accept",
    if_M_diverges:
        "simulation continues forever",
    consequence:
        "HALT is recognisable but not decidable"
}

DEFINE CRITICAL_DISTINCTIONS := {
    halting_vs_acceptance: {
        halting:
            "computation terminates in any halting state",
        acceptance:
            "computation terminates specifically with an accepting outcome"
    },
    nonhalting_vs_not_yet_halted: {
        nonhalting:
            "infinite computation",
        not_yet_halted:
            "finite observation with no halt seen so far",
        equivalence: FALSE
    },
    undecidable_problem_vs_instance: {
        undecidable_problem:
            "no single algorithm decides every instance",
        specific_instance:
            "may still be provably halting or nonhalting"
    },
    recognisable_vs_decidable: {
        recognisable:
            "positive halting instances can eventually be confirmed",
        decidable:
            "both halting and nonhalting instances would need guaranteed finite classification"
    }
}

DEFINE EXAMPLE_1 := {
    machine:
        "M immediately enters a halt state",
    input:
        "w",
    classification:
        PositiveHaltingInstance
}

DEFINE EXAMPLE_2 := {
    machine:
        "M enters a simple infinite loop",
    input:
        "w",
    classification:
        NegativeHaltingInstance
}

DEFINE EXAMPLE_3 := {
    hypothetical_algorithm:
        "H(M,w)",
    requirement:
        "always halt and correctly answer whether M halts on w",
    result:
        "impossible in general by the diagonal argument"
}

DEFINE INSTANCE_ANALYSIS_PIPELINE :=
    EncodedMachineInput
    -> DecodeMachine(M)
    -> DecodeInput(w)
    -> AnalyseMachineStructure
    -> SimulateWhenUseful
    -> SearchForFormalHaltingProof
    -> SearchForFormalDivergenceProof
    -> IF halt_observed
       THEN PositiveInstance
    -> IF divergence_proven
       THEN NegativeInstance
    -> ELSE UnknownForThisAnalysis
    -> DO_NOT_USE finite_nonhalting_observation_as_proof

DEFINE RECOGNITION_PIPELINE :=
    encode(M,w)
    -> UniversalSimulation(M,w)
    -> IF M halts
       THEN DetectHaltingConfiguration
       -> Accept
    -> ELSE ContinueSimulation
    -> EstablishRecognisabilityOfHALT

DEFINE UNDECIDABILITY_PROOF_PIPELINE :=
    AssumeTotalCorrectHaltingDecider(H)
    -> ConstructDiagonalMachine(D)
    -> FeedDItsOwnEncoding
    -> EvaluatePredictionH(D,D)
    -> ReversePredictedBehaviour
    -> DeriveContradictionInBothCases
    -> RejectAssumption
    -> ConcludeHaltingProblemUndecidable

DEFINE REDUCTION_PIPELINE :=
    HaltingProblem
    -> TargetProblem(B)
    -> ConstructComputableMapping
    -> ProveHaltingYesCorrespondsToBYes
    -> ProveHaltingNoCorrespondsToBNo
    -> Establish HaltingProblem REDUCES_TO B
    -> TransferUndecidabilityToB

DEFINE HALTING_GRAPH := {
    node_types: {
        MachineNode,
        InputNode,
        EncodingNode,
        ComputationNode,
        HaltingStateNode,
        DivergenceNode,
        DeciderNode,
        ContradictionNode
    },
    edge_types: {
        ENCODES,
        HAS_INPUT,
        HALTS_ON,
        DIVERGES_ON,
        SIMULATES,
        DECIDES,
        CONTRADICTS,
        REDUCES_TO
    },
    self_reference_enabled: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    HaltingProblem IS_COMPONENT_OF ComputabilityTheory,
    HaltingProblem IS_INSTANCE_OF UndecidableProblem,
    HaltingProblem IS_INSTANCE_OF RecognisableProblem,
    HaltingProblem IS_INSTANCE_OF SemiDecidableProblem,
    UniversalTuringMachine RECOGNISES HaltingProblem,
    Diagonalisation PROVES UndecidabilityOfHaltingProblem,
    SelfReference IS_USED_BY Diagonalisation,
    HaltingProblem IS_SOURCE_FOR UndecidabilityReductions,
    MachineEncoding ENABLES UniversalSimulation,
    Computation HAS_OUTCOME Halting OR Divergence
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "halting problem" <=> HaltingProblem,
    "Turing machine halting problem" <=> HaltingProblem,
    "HALT" <=> HaltingProblem WHEN computability_context=TRUE,
    "machine halts on input" <=> HALTS_ON,
    "machine diverges" <=> DIVERGES_ON,
    "non-terminating computation" <=> Divergence,
    "halting decider" <=> DeciderForHaltingProblem
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation of the halting problem, its recognisability and its undecidability proof",
    preserve_machine_identity: TRUE,
    preserve_input_identity: TRUE,
    preserve_machine_encoding: TRUE,
    preserve_halting_divergence_distinction: TRUE,
    distinguish_halting_from_acceptance: TRUE,
    distinguish_nonhalting_from_not_yet_halted: TRUE,
    distinguish_problem_level_undecidability_from_instance_analysis: TRUE,
    preserve_self_reference_structure: TRUE,
    support_universal_simulation: TRUE,
    support_diagonalisation_reasoning: TRUE,
    preserve_reduction_direction: TRUE,
    support_halting_based_reductions: TRUE,
    reasoning_mode: "symbolic + computational + self-referential + contradiction-aware"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    halting_problem_definition: 1.00,
    recognisability_mapping: 1.00,
    undecidability_mapping: 1.00,
    diagonal_argument_structure: 0.99,
    reduction_source_mapping: 0.99,
    finite_observation_guard: 1.00,
    arbitrary_instance_halting_prediction: undecidable_in_general
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        encoded_machine_input,
        halt_vs_diverge,
        recognisable_not_decidable,
        diagonal_self_reference,
        no_total_correct_halting_decider,
        finite_observation_guard
    },
    extensible_layers: {
        blank_tape_halting_problem,
        mortality_problems,
        program_termination,
        Rice_theorem_reductions,
        oracle_halting_problems,
        relative_halting_problems,
        arithmetic_hierarchy,
        formal_termination_provers
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + HALTING_PROBLEM_MODEL
    + HALT_SET_MODEL
    + DIAGONAL_ARGUMENT_MODEL
    + RECOGNISER_MODEL
    + CRITICAL_DISTINCTIONS
    + INSTANCE_ANALYSIS_PIPELINE
    + RECOGNITION_PIPELINE
    + UNDECIDABILITY_PROOF_PIPELINE
    + REDUCTION_PIPELINE
    + HALTING_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: HaltingProblemOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory/halting-problem-ontology.html",
    parent_domain: ComputabilityTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: HaltingProblem,
    recognisability_reasoning_enabled: TRUE,
    diagonalisation_reasoning_enabled: TRUE,
    self_reference_reasoning_enabled: TRUE,
    reduction_reasoning_enabled: TRUE,
    instance_analysis_guard_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW