Computability Theory

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

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

FLOW_ALGEBRA ComputabilityTheory_v1.0

BEGIN_FLOW

DEFINE DOMAIN ComputabilityTheory := {
    ComputabilityTheory,
    Computation,
    Algorithm,
    EffectiveProcedure,
    ComputableFunction,
    PartialComputableFunction,
    TotalComputableFunction,
    RecursiveFunction,
    PartialRecursiveFunction,
    PrimitiveRecursiveFunction,
    TuringMachine,
    UniversalTuringMachine,
    Configuration,
    State,
    Tape,
    Alphabet,
    TransitionFunction,
    Input,
    Output,
    Halting,
    NonHalting,
    DecisionProblem,
    DecidableProblem,
    UndecidableProblem,
    RecognisableLanguage,
    RecursivelyEnumerableLanguage,
    RecursiveLanguage,
    SemiDecidableProblem,
    HaltingProblem,
    Reduction,
    ManyOneReduction,
    TuringReduction,
    ChurchTuringThesis,
    LambdaCalculus,
    MuRecursiveFunction,
    RegisterMachine,
    ComputableSet,
    EnumerableSet,
    Enumeration,
    Diagonalisation,
    RiceTheorem,
    RecursionTheorem,
    KleeneNormalForm,
    DegreeOfUnsolvability,
    Oracle,
    RelativeComputability
}

DEFINE ENTITY_TYPES := {
    COMPUTABILITY_THEORY,
    COMPUTATION_MODEL,
    ALGORITHM,
    FUNCTION,
    MACHINE,
    MACHINE_STATE,
    CONFIGURATION,
    LANGUAGE,
    SET,
    DECISION_PROBLEM,
    REDUCTION,
    THEOREM,
    COMPUTABILITY_PROPERTY,
    ORACLE,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    COMPUTES,
    ACCEPTS,
    REJECTS,
    HALTS_ON,
    DIVERGES_ON,
    DECIDES,
    RECOGNISES,
    ENUMERATES,
    REDUCES_TO,
    MANY_ONE_REDUCES_TO,
    TURING_REDUCES_TO,
    SIMULATES,
    ENCODES,
    HAS_INPUT,
    HAS_OUTPUT,
    TRANSITIONS_TO,
    IS_COMPUTABLE_BY,
    IS_DECIDABLE_BY,
    IS_RECOGNISABLE_BY,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    DEPENDS_ON
}

SET ComputabilityTheory.type = COMPUTABILITY_THEORY
SET ComputabilityTheory.domain = MathematicalLogic
SET ComputabilityTheory.role =
    "study which functions, sets and decision problems can be computed by effective procedures and identify formal limits of algorithmic computation"

RULE deterministic_turing_transition:
    TuringMachine(M)
    AND Configuration(C)
    AND transition_defined(M,C)
    -> unique_next_configuration(CPrime)
    AND TRANSITIONS_TO(C,CPrime)

RULE turing_computation:
    TuringMachine(M)
    AND Input(x)
    -> Computation(M,x)
       = sequence_of_configurations_from_initial_configuration

RULE halting:
    Computation(M,x)
    AND reaches_halting_state
    -> HALTS_ON(M,x)

RULE divergence:
    Computation(M,x)
    AND never_reaches_halting_state
    -> DIVERGES_ON(M,x)

RULE partial_computable_function:
    TuringMachine(M)
    AND FOR_EVERY input x where M halts [
        output(M,x)=f(x)
    ]
    -> PartialComputableFunction(f)

RULE total_computable_function:
    PartialComputableFunction(f)
    AND FOR_EVERY input x [
        f(x) defined
    ]
    -> TotalComputableFunction(f)

RULE decidable_language:
    Language(L)
    AND EXISTS TuringMachine(M) [
        FOR_EVERY input x [
            M halts_on x
            AND (
                x MEMBER_OF L -> ACCEPTS(M,x)
            )
            AND (
                x NOT_MEMBER_OF L -> REJECTS(M,x)
            )
        ]
    ]
    -> Decidable(L)

RULE recognisable_language:
    Language(L)
    AND EXISTS TuringMachine(M) [
        FOR_EVERY input x [
            x MEMBER_OF L -> ACCEPTS(M,x)
        ]
        AND FOR x NOT_MEMBER_OF L [
            M may_reject_or_diverge
        ]
    ]
    -> Recognisable(L)

RULE decidable_implies_recognisable:
    Decidable(L)
    -> Recognisable(L)

RULE complement_recognisable_decidable:
    Recognisable(L)
    AND Recognisable(Complement(L))
    -> Decidable(L)

RULE enumeration_equivalence:
    Language(L)
    AND L is recursively_enumerable
    -> EXISTS Enumerator(E) [
        ENUMERATES(E,L)
    ]

RULE many_one_reduction:
    DecisionProblem(A)
    AND DecisionProblem(B)
    AND EXISTS TotalComputableFunction(f) [
        FOR_EVERY x [
            x MEMBER_OF A
            IFF
            f(x) MEMBER_OF B
        ]
    ]
    -> MANY_ONE_REDUCES_TO(A,B)

RULE reduction_transfers_undecidability:
    MANY_ONE_REDUCES_TO(A,B)
    AND Undecidable(A)
    -> Undecidable(B)

RULE reduction_transfers_decidability_backward:
    MANY_ONE_REDUCES_TO(A,B)
    AND Decidable(B)
    -> Decidable(A)

RULE turing_reduction:
    Problem(A)
    AND Problem(B)
    AND OracleMachine(M^B) decides A
    -> TURING_REDUCES_TO(A,B)

RULE universal_machine:
    UniversalTuringMachine(U)
    AND EncodedMachine(M)
    AND Input(x)
    -> U simulates M on x

RULE halting_problem_definition:
    HaltingProblem(HALT)
    -> HALT
       = {
           encode(M,x)
           | HALTS_ON(M,x)
       }

RULE halting_problem_undecidable:
    -> Undecidable(HaltingProblem)

RULE diagonalisation_pattern:
    Assume(decider_for_target_problem)
    -> ConstructSelfReferentialInput
    -> DeriveBehaviourContradiction
    -> RejectAssumedDecider

RULE Rice_theorem:
    Property(P)
    AND P is_nontrivial_semantic_property_of_partial_computable_functions
    -> Undecidable(
        {
            encode(M)
            | function_computed_by(M) has_property P
        }
    )

RULE Church_Turing_correspondence:
    EffectiveProcedure(P)
    -> represented_by_TuringComputableProcess(P)
       UNDER ChurchTuringThesis

DEFINE TURING_MACHINE_MODEL := {
    components: {
        finite_state_set,
        input_alphabet,
        tape_alphabet,
        transition_function,
        start_state,
        accept_state,
        reject_state
    },
    configuration:
        "current state, tape contents and head position",
    computation:
        "successive application of the transition function"
}

DEFINE COMPUTABLE_FUNCTION_MODEL := {
    partial:
        "machine returns f(x) whenever f(x) is defined and may diverge otherwise",
    total:
        "machine halts and returns f(x) for every valid input",
    machine_independence:
        "computability is defined by existence of a suitable effective computation model"
}

DEFINE DECIDABILITY_MODEL := {
    decidable:
        "there exists an algorithm that halts on every input and answers the membership or yes/no question correctly",
    recognisable:
        "there exists an algorithm that halts and accepts every positive instance but may fail to halt on negative instances",
    undecidable:
        "no total correct decision algorithm exists in the specified computation model"
}

DEFINE REDUCTION_MODEL := {
    purpose:
        "compare computational difficulty by transforming instances or allowing controlled access to another problem",
    many_one:
        "single computable transformation preserving yes/no membership",
    Turing:
        "computation may query an oracle for the target problem"
}

DEFINE CHURCH_TURING_MODEL := {
    status:
        "thesis rather than formal theorem",
    claim:
        "effectively calculable functions coincide with functions computable by a Turing-equivalent formal model",
    equivalent_formalisms: {
        TuringMachines,
        LambdaCalculus,
        PartialRecursiveFunctions,
        RegisterMachines
    }
}

DEFINE CRITICAL_DISTINCTIONS := {
    computable_vs_efficient: {
        computable:
            "some terminating algorithm exists",
        efficient:
            "resource use is suitably bounded",
        equivalence: FALSE
    },
    decidable_vs_recognisable: {
        decidable:
            "halts on all inputs",
        recognisable:
            "guaranteed to halt on positive instances only"
    },
    partial_vs_total_computable: {
        partial:
            "may be undefined or diverge on some inputs",
        total:
            "defined and terminating on every input"
    },
    computability_vs_complexity: {
        computability:
            "whether computation is possible in principle",
        complexity:
            "resources required to perform computation"
    }
}

DEFINE EXAMPLE_1 := {
    function:
        "addition on natural numbers",
    classification:
        TotalComputableFunction
}

DEFINE EXAMPLE_2 := {
    problem:
        HaltingProblem,
    question:
        "does machine M halt on input x?",
    classification:
        UndecidableProblem
}

DEFINE EXAMPLE_3 := {
    language:
        "set of encodings of Turing machines that halt on blank input",
    classification:
        RecognisableLanguage,
    decidable:
        FALSE
}

DEFINE MACHINE_EXECUTION_PIPELINE :=
    MachineDescription
    -> ParseStatesAlphabetAndTransitionFunction
    -> EncodeInput
    -> CreateInitialConfiguration
    -> RepeatedlyApplyTransition
    -> IF AcceptState
       THEN Accept
    -> IF RejectState
       THEN Reject
    -> IF NoHaltingStateReached
       THEN DivergenceOrUnknownUnderFiniteObservation

DEFINE COMPUTABILITY_CLASSIFICATION_PIPELINE :=
    FunctionOrProblem
    -> IdentifyInputOutputSpecification
    -> SearchForEffectiveProcedure
    -> FormaliseAsComputationModel
    -> AnalyseHaltingBehaviour
    -> DeterminePartialOrTotalComputability
    -> DetermineRecognisability
    -> DetermineDecidability
    -> ReturnComputabilityClass

DEFINE REDUCTION_PIPELINE :=
    SourceProblem(A)
    -> TargetProblem(B)
    -> ConstructComputableTransformationOrOracleProcedure
    -> ProveCorrectnessOfReduction
    -> EstablishReductionRelation
    -> TransferKnownDecidabilityOrUndecidabilityConsequences
    -> ReturnRelativeComputabilityResult

DEFINE UNDECIDABILITY_PIPELINE :=
    TargetProblem
    -> SelectKnownUndecidableProblem
    -> ConstructReduction
    -> VerifyReductionDirection
    -> ApplyUndecidabilityTransfer
    -> OR ConstructDiagonalArgument
    -> DeriveNoDeciderConclusion
    -> ReturnUndecidabilityProof

DEFINE COMPUTABILITY_GRAPH := {
    nodes: {
        FunctionNode,
        ProblemNode,
        LanguageNode,
        MachineNode,
        AlgorithmNode,
        ReductionNode,
        OracleNode
    },
    edges: {
        COMPUTES,
        DECIDES,
        RECOGNISES,
        ENUMERATES,
        REDUCES_TO,
        SIMULATES,
        HALTS_ON,
        DIVERGES_ON
    },
    recursive_simulation_enabled: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    ComputabilityTheory IS_SUBTYPE_OF MathematicalLogic,
    TuringMachine IS_SUBTYPE_OF ComputationModel,
    UniversalTuringMachine SIMULATES TuringMachine,
    Algorithm COMPUTES ComputableFunction,
    DecidableLanguage IS_SUBTYPE_OF RecognisableLanguage,
    PartialComputableFunction IS_SUPERTYPE_OF TotalComputableFunction,
    ManyOneReduction IS_SUBTYPE_OF Reduction,
    TuringReduction IS_SUBTYPE_OF Reduction,
    HaltingProblem IS_INSTANCE_OF UndecidableProblem,
    RiceTheorem ESTABLISHES UndecidabilityForNontrivialSemanticProperties,
    ChurchTuringThesis RELATES EffectiveProcedure TO TuringComputability
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "computability theory" <=> ComputabilityTheory,
    "recursion theory" <=> ComputabilityTheory WHEN classical_mathematical_context=TRUE,
    "computable function" <=> ComputableFunction,
    "recursive function" <=> ComputableFunction WHEN historical_context=TRUE,
    "recursively enumerable" <=> RecognisableLanguage,
    "semi-decidable" <=> RecognisableLanguage,
    "Turing-recognisable" <=> RecognisableLanguage,
    "Turing-decidable" <=> DecidableLanguage,
    "halting problem" <=> HaltingProblem
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation of computability, decision procedures, reductions and undecidability",
    preserve_computation_model_identity: TRUE,
    preserve_machine_identity: TRUE,
    preserve_input_identity: TRUE,
    preserve_halting_status: TRUE,
    distinguish_partial_from_total_computation: TRUE,
    distinguish_decidable_from_recognisable: TRUE,
    distinguish_computability_from_complexity: TRUE,
    preserve_reduction_direction: TRUE,
    validate_reduction_correctness: TRUE,
    support_machine_simulation: TRUE,
    support_undecidability_reasoning: TRUE,
    support_diagonalisation_reasoning: TRUE,
    support_relative_computability: TRUE,
    reasoning_mode: "symbolic + algorithmic + reduction-based + meta-computational"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    Turing_machine_mapping: 0.99,
    computable_function_classification: 0.99,
    decidability_recognisability_distinction: 1.00,
    reduction_direction_mapping: 1.00,
    halting_problem_mapping: 1.00,
    Rice_theorem_mapping: 0.99,
    arbitrary_program_halting_analysis: undecidable_in_general
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        effective_computation,
        Turing_computability,
        partial_total_distinction,
        decidable_recognisable_distinction,
        halting_problem_undecidability,
        reduction_direction,
        Church_Turing_thesis_status
    },
    extensible_layers: {
        primitive_recursive_functions,
        partial_recursive_functions,
        lambda_calculus,
        register_machines,
        computability_degrees,
        oracle_computation,
        arithmetic_hierarchy,
        recursion_theorem,
        descriptive_computability,
        computable_analysis
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + TURING_MACHINE_MODEL
    + COMPUTABLE_FUNCTION_MODEL
    + DECIDABILITY_MODEL
    + REDUCTION_MODEL
    + CHURCH_TURING_MODEL
    + CRITICAL_DISTINCTIONS
    + MACHINE_EXECUTION_PIPELINE
    + COMPUTABILITY_CLASSIFICATION_PIPELINE
    + REDUCTION_PIPELINE
    + UNDECIDABILITY_PIPELINE
    + COMPUTABILITY_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: ComputabilityTheory,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory.html",
    parent_domain: MathematicalLogicOntologies,
    semantic_domain: MathematicalLogic,
    primary_entity: ComputabilityTheory,
    machine_reasoning_enabled: TRUE,
    computability_classification_enabled: TRUE,
    reduction_reasoning_enabled: TRUE,
    undecidability_reasoning_enabled: TRUE,
    relative_computability_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW