Church-Turing Thesis Ontology

KIRETH RESEARCH - SEMANTIC LIBRARY
AI CORE -> ONTOLOGIES -> MATHEMATICAL ONTOLOGIES -> MATHEMATICAL LOGIC ONTOLOGIES
COMPUTABILITY THEORY -> CHURCH-TURING THESIS ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory/church-turing-thesis-ontology.html

FLOW_ALGEBRA ChurchTuringThesisOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN ChurchTuringThesisOntology := {
    ChurchTuringThesis,
    ComputabilityTheory,
    EffectiveCalculability,
    EffectiveProcedure,
    Algorithm,
    ComputableFunction,
    TuringComputableFunction,
    RecursiveFunction,
    PartialRecursiveFunction,
    LambdaDefinableFunction,
    LambdaCalculus,
    TuringMachine,
    RegisterMachine,
    PostSystem,
    GeneralRecursiveFunction,
    FormalComputationModel,
    MechanicalProcedure,
    FiniteInstructionProcedure,
    HumanComputor,
    Simulation,
    ComputationalEquivalence,
    Thesis,
    MathematicalTheorem,
    PhysicalChurchTuringThesis,
    ExtendedChurchTuringThesis,
    Complexity,
    QuantumComputation,
    Hypercomputation
}

DEFINE ENTITY_TYPES := {
    THESIS,
    COMPUTABILITY_PRINCIPLE,
    EFFECTIVE_PROCEDURE,
    ALGORITHM,
    COMPUTABLE_FUNCTION,
    COMPUTATION_MODEL,
    MACHINE,
    FORMALISM,
    EQUIVALENCE_RELATION,
    COMPLEXITY_PRINCIPLE,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    CHARACTERISES,
    COMPUTES,
    SIMULATES,
    EQUIVALENT_IN_COMPUTABILITY_TO,
    FORMALISES,
    REPRESENTS,
    SUPPORTS,
    DISTINGUISHED_FROM,
    IMPLIES,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    DEPENDS_ON
}

SET ChurchTuringThesis.type = THESIS
SET ChurchTuringThesis.context = ComputabilityTheory
SET ChurchTuringThesis.role =
    "identify effectively calculable functions with the functions computable by a Turing-equivalent formal model"

RULE core_thesis:
    EffectivelyCalculableFunction(f)
    -> TuringComputableFunction(f)
       UNDER ChurchTuringThesis

RULE reverse_operational_direction:
    TuringComputableFunction(f)
    -> EffectivelyCalculableFunction(f)
       UNDER standard_interpretation_of_effective_procedure

RULE recursive_equivalence:
    TuringComputableFunction(f)
    IFF
    PartialRecursiveFunction(f)
       UNDER standard_number_theoretic_formulations

RULE lambda_equivalence:
    TuringComputableFunction(f)
    IFF
    LambdaDefinableFunction(f)
       UNDER standard_encodings

RULE register_machine_equivalence:
    TuringComputableFunction(f)
    IFF
    RegisterMachineComputable(f)
       UNDER standard_encodings

RULE model_equivalence:
    ComputationModel(A)
    AND ComputationModel(B)
    AND A simulates B
    AND B simulates A
    -> EQUIVALENT_IN_COMPUTABILITY_TO(A,B)

RULE convergence_evidence:
    IndependentFormalModels(M1,...,Mn)
    AND FOR_EVERY i,j [
        EQUIVALENT_IN_COMPUTABILITY_TO(Mi,Mj)
    ]
    -> SUPPORTS(ConvergenceEvidence,ChurchTuringThesis)

RULE thesis_not_formal_theorem:
    ChurchTuringThesis
    -> DISTINGUISHED_FROM(ChurchTuringThesis,MathematicalTheorem)

RULE effective_not_complexity_bounded:
    EffectiveProcedure(P)
    -> DOES_NOT_IMPLY
       efficient_resource_bound(P)

RULE extended_thesis_distinction:
    ExtendedChurchTuringThesis
    -> concerns efficient_simulation_and_complexity
    AND DISTINGUISHED_FROM(
        ExtendedChurchTuringThesis,
        ChurchTuringThesis
    )

RULE physical_thesis_distinction:
    PhysicalChurchTuringThesis
    -> concerns physically_realizable_computation
    AND DISTINGUISHED_FROM(
        PhysicalChurchTuringThesis,
        ChurchTuringThesis
    )

RULE quantum_guard:
    QuantumAlgorithm(Q)
    -> DOES_NOT_BY_ITSELF_REFUTE
       ChurchTuringThesis

RULE hypercomputation_guard:
    ProposedHypercomputer(H)
    AND claims_non_Turing_computable_output
    -> requires_nonstandard_physical_or_mathematical_assumptions
    AND DOES_NOT_AUTOMATICALLY_REFUTE
       mathematical_ChurchTuringThesis

DEFINE CORE_THESIS_MODEL := {
    informal_side:
        EffectiveCalculability,
    formal_side:
        TuringComputability,
    canonical_statement:
        "every effectively calculable function is Turing-computable",
    status:
        Thesis,
    reason_not_formal_theorem:
        "effective calculability is an informal pre-formal notion rather than a single independently formal mathematical predicate"
}

DEFINE COMPUTATIONAL_EQUIVALENCE_MODEL := {
    TuringMachine:
        TuringComputableFunction,
    LambdaCalculus:
        LambdaDefinableFunction,
    GeneralRecursiveFunctions:
        RecursiveFunction,
    RegisterMachines:
        RegisterMachineComputableFunction,
    convergence:
        "standard formalisms developed from different starting points yield the same class of computable functions"
}

DEFINE EFFECTIVE_PROCEDURE_MODEL := {
    characteristics: {
        finite_description,
        discrete_steps,
        mechanically_executable_rules,
        no_unbounded_creative_oracle_at_each_step,
        finite_local_information_per_step
    },
    note:
        "historical formulations vary; these properties describe the intended effective-procedure concept rather than a unique formal definition"
}

DEFINE THESIS_VARIANTS := {
    ChurchTuringThesis: {
        focus:
            "effective calculability and computability"
    },
    ExtendedChurchTuringThesis: {
        focus:
            "efficient simulation and computational complexity",
        status:
            "stronger and distinct claim"
    },
    PhysicalChurchTuringThesis: {
        focus:
            "computational capabilities of physically realizable systems",
        status:
            "distinct physical claim"
    }
}

DEFINE CRITICAL_DISTINCTIONS := {
    thesis_vs_theorem: {
        thesis:
            "connects an informal notion to a formal mathematical class",
        theorem:
            "proved entirely from formal definitions and axioms",
        Church_Turing_status:
            Thesis
    },
    computability_vs_complexity: {
        computability:
            "whether an effective computation exists",
        complexity:
            "resources required by computation",
        relation:
            "the original thesis concerns computability, not efficient computation"
    },
    original_vs_extended_thesis: {
        original:
            "effective calculability equals Turing computability",
        extended:
            "reasonable computation models can efficiently simulate one another under specified complexity assumptions"
    },
    mathematical_vs_physical_claim: {
        mathematical:
            "formal characterisation of effective procedures",
        physical:
            "claims about what physical systems can compute"
    }
}

DEFINE EXAMPLE_1 := {
    formalism_1:
        TuringMachine,
    formalism_2:
        LambdaCalculus,
    relation:
        "same computable function class under standard encodings",
    significance:
        "independent convergence supports the thesis"
}

DEFINE EXAMPLE_2 := {
    function:
        "natural-number addition",
    effectively_calculable:
        TRUE,
    Turing_computable:
        TRUE,
    recursive:
        TRUE
}

DEFINE EXAMPLE_3 := {
    claim:
        "a quantum computer may solve some computable problems more efficiently",
    implication_for_original_thesis:
        "no contradiction, because efficiency and computability are distinct"
}

DEFINE THESIS_ANALYSIS_PIPELINE :=
    CandidateProcedureOrFunction
    -> DetermineWhetherClaimConcernsEffectiveCalculability
    -> FormaliseInKnownComputationModel
    -> CompareWithTuringComputability
    -> CompareWithRecursiveAndLambdaModels
    -> PreserveThesisRatherThanTheoremStatus
    -> ReturnComputabilityCorrespondence

DEFINE MODEL_EQUIVALENCE_PIPELINE :=
    ComputationModel(A)
    -> ComputationModel(B)
    -> ConstructSimulationAByB
    -> ConstructSimulationBByA
    -> VerifyInputOutputBehaviourPreservation
    -> EstablishComputationalEquivalence
    -> AddToConvergenceEvidence

DEFINE VARIANT_CLASSIFICATION_PIPELINE :=
    ChurchTuringRelatedClaim
    -> DetermineWhetherClaimConcerns {
        computability,
        efficiency,
        physical_realizability
    }
    -> IF computability
       THEN OriginalChurchTuringThesis
    -> IF efficiency
       THEN ExtendedChurchTuringThesis
    -> IF physical_realizability
       THEN PhysicalChurchTuringThesis
    -> ReturnQualifiedVariant

DEFINE CLAIM_VALIDATION_PIPELINE :=
    InputClaim
    -> IdentifyFormalAndInformalTerms
    -> CheckWhetherClaimTreatsThesisAsTheorem
    -> CorrectCategoryIfNeeded
    -> CheckComputabilityComplexityDistinction
    -> CheckPhysicalMathematicalDistinction
    -> CheckModelEquivalenceEvidence
    -> ReturnSemanticallyQualifiedClaim

DEFINE CHURCH_TURING_GRAPH := {
    central_node:
        ChurchTuringThesis,
    informal_node:
        EffectiveCalculability,
    formal_nodes: {
        TuringComputability,
        RecursiveFunctions,
        LambdaCalculus,
        RegisterMachines
    },
    edges: {
        CHARACTERISES,
        FORMALISES,
        EQUIVALENT_IN_COMPUTABILITY_TO,
        SUPPORTS,
        DISTINGUISHED_FROM
    },
    convergence_structure: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    ChurchTuringThesis IS_COMPONENT_OF ComputabilityTheory,
    ChurchTuringThesis CHARACTERISES EffectiveCalculability,
    TuringMachine FORMALISES Computation,
    RecursiveFunction EQUIVALENT_IN_COMPUTABILITY_TO TuringComputableFunction,
    LambdaCalculus EQUIVALENT_IN_COMPUTABILITY_TO TuringMachine,
    RegisterMachine EQUIVALENT_IN_COMPUTABILITY_TO TuringMachine,
    ComputationalEquivalence SUPPORTS ChurchTuringThesis,
    ChurchTuringThesis IS_DISTINGUISHED_FROM MathematicalTheorem,
    ExtendedChurchTuringThesis IS_DISTINGUISHED_FROM ChurchTuringThesis,
    PhysicalChurchTuringThesis IS_DISTINGUISHED_FROM ChurchTuringThesis
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "Church-Turing thesis" <=> ChurchTuringThesis,
    "Church Turing thesis" <=> ChurchTuringThesis,
    "Church's thesis" <=> ChurchTuringThesis WHEN computability_context=TRUE,
    "Turing thesis" <=> ChurchTuringThesis WHEN historical_context=TRUE,
    "effective calculability" <=> EffectiveCalculability,
    "extended Church-Turing thesis" <=> ExtendedChurchTuringThesis,
    "physical Church-Turing thesis" <=> PhysicalChurchTuringThesis
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation of the Church-Turing thesis and equivalence of classical computation models",
    preserve_thesis_status: TRUE,
    distinguish_thesis_from_theorem: TRUE,
    preserve_informal_effective_calculability_side: TRUE,
    preserve_formal_Turing_computability_side: TRUE,
    distinguish_computability_from_complexity: TRUE,
    distinguish_original_from_extended_thesis: TRUE,
    distinguish_mathematical_from_physical_claims: TRUE,
    support_computation_model_equivalence: TRUE,
    support_cross_model_simulation_reasoning: TRUE,
    treat_quantum_speedup_as_complexity_issue_unless_computability_changes: TRUE,
    reasoning_mode: "symbolic + conceptual + equivalence-based + computability-aware"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    core_thesis_mapping: 1.00,
    thesis_theorem_distinction: 1.00,
    Turing_recursive_equivalence: 0.99,
    Turing_lambda_equivalence: 0.99,
    original_extended_distinction: 0.99,
    physical_thesis_mapping: context_dependent,
    hypercomputation_claim_assessment: assumption_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        effective_calculability,
        Turing_computability,
        thesis_status,
        classical_model_convergence,
        computability_complexity_distinction,
        original_extended_distinction
    },
    extensible_layers: {
        historical_formulations,
        Gandy_principles,
        physical_computation,
        quantum_computation,
        analog_computation,
        hypercomputation,
        interactive_computation,
        higher_type_computability
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + CORE_THESIS_MODEL
    + COMPUTATIONAL_EQUIVALENCE_MODEL
    + EFFECTIVE_PROCEDURE_MODEL
    + THESIS_VARIANTS
    + CRITICAL_DISTINCTIONS
    + THESIS_ANALYSIS_PIPELINE
    + MODEL_EQUIVALENCE_PIPELINE
    + VARIANT_CLASSIFICATION_PIPELINE
    + CLAIM_VALIDATION_PIPELINE
    + CHURCH_TURING_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: ChurchTuringThesisOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory/church-turing-thesis-ontology.html",
    parent_domain: ComputabilityTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: ChurchTuringThesis,
    thesis_status_preserved: TRUE,
    computation_model_equivalence_enabled: TRUE,
    variant_classification_enabled: TRUE,
    computability_complexity_distinction_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW