Turing Machine Ontology

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

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

FLOW_ALGEBRA TuringMachineOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN TuringMachineOntology := {
    TuringMachine,
    ComputabilityTheory,
    ComputationModel,
    DeterministicTuringMachine,
    NondeterministicTuringMachine,
    UniversalTuringMachine,
    MultiTapeTuringMachine,
    State,
    StartState,
    AcceptState,
    RejectState,
    HaltingState,
    Tape,
    TapeCell,
    TapeAlphabet,
    InputAlphabet,
    BlankSymbol,
    ReadWriteHead,
    TransitionFunction,
    TransitionRule,
    Configuration,
    InitialConfiguration,
    SuccessorConfiguration,
    Computation,
    ComputationPath,
    Input,
    Output,
    Encoding,
    MachineDescription,
    Acceptance,
    Rejection,
    Halting,
    Divergence,
    Language,
    RecognisableLanguage,
    DecidableLanguage,
    ComputableFunction,
    PartialComputableFunction,
    UniversalSimulation,
    HaltingProblem
}

DEFINE ENTITY_TYPES := {
    TURING_MACHINE,
    COMPUTATION_MODEL,
    STATE,
    SYMBOL,
    ALPHABET,
    TAPE,
    TAPE_CELL,
    READ_WRITE_HEAD,
    TRANSITION_FUNCTION,
    TRANSITION_RULE,
    CONFIGURATION,
    COMPUTATION,
    INPUT,
    OUTPUT,
    ENCODING,
    LANGUAGE,
    FUNCTION,
    COMPUTABILITY_PROPERTY,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    HAS_STATE,
    HAS_START_STATE,
    HAS_ACCEPT_STATE,
    HAS_REJECT_STATE,
    HAS_TAPE,
    HAS_ALPHABET,
    READS,
    WRITES,
    MOVES,
    TRANSITIONS_TO,
    HAS_CONFIGURATION,
    HAS_INPUT,
    HAS_OUTPUT,
    ACCEPTS,
    REJECTS,
    HALTS_ON,
    DIVERGES_ON,
    COMPUTES,
    RECOGNISES,
    DECIDES,
    SIMULATES,
    ENCODES,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    DEPENDS_ON
}

SET TuringMachine.type = TURING_MACHINE
SET TuringMachine.parent = ComputationModel
SET TuringMachine.context = ComputabilityTheory
SET TuringMachine.role =
    "represent an abstract machine that computes by repeatedly reading and writing tape symbols, changing finite control state and moving a read-write head according to transition rules"

DEFINE TURING_MACHINE_SCHEMA := {
    machine:
        "M = (Q, Sigma, Gamma, delta, q_start, q_accept, q_reject)",
    Q:
        "finite set of states",
    Sigma:
        "input alphabet excluding blank symbol",
    Gamma:
        "tape alphabet containing Sigma and blank symbol",
    delta:
        "transition function",
    q_start:
        StartState,
    q_accept:
        AcceptState,
    q_reject:
        RejectState
}

RULE alphabet_relation:
    InputAlphabet(Sigma)
    AND TapeAlphabet(Gamma)
    -> Sigma SUBSET_OF Gamma

RULE blank_not_input:
    BlankSymbol(B)
    AND InputAlphabet(Sigma)
    -> B NOT_MEMBER_OF Sigma

RULE blank_in_tape_alphabet:
    BlankSymbol(B)
    AND TapeAlphabet(Gamma)
    -> B MEMBER_OF Gamma

RULE deterministic_transition:
    DeterministicTuringMachine(M)
    AND CurrentState(q)
    AND READS(head,a)
    AND delta(q,a)=(qPrime,b,d)
    -> WRITES(head,b)
    AND machine_state := qPrime
    AND MOVES(head,d)
    AND d MEMBER_OF {LEFT,RIGHT}

RULE transition_uniqueness:
    DeterministicTuringMachine(M)
    AND State(q)
    AND TapeSymbol(a)
    -> at_most_one_transition_for(q,a)

RULE configuration:
    TuringMachine(M)
    -> Configuration(C)
       contains {
           current_state,
           tape_contents,
           head_position
       }

RULE initial_configuration:
    TuringMachine(M)
    AND Input(w)
    -> InitialConfiguration(C0)
       where {
           state=q_start,
           tape_prefix=w,
           remaining_tape=blank,
           head_position=initial_input_cell
       }

RULE successor_configuration:
    Configuration(C)
    AND applicable_transition_exists(C)
    -> SuccessorConfiguration(CPrime)
    AND TRANSITIONS_TO(C,CPrime)

RULE computation:
    TuringMachine(M)
    AND Input(w)
    -> Computation(M,w)
       = ordered_sequence(
           C0,C1,C2,...
         )
       where C0=InitialConfiguration(M,w)

RULE acceptance:
    Computation(M,w)
    AND reaches_state(q_accept)
    -> ACCEPTS(M,w)
    AND HALTS_ON(M,w)

RULE rejection:
    Computation(M,w)
    AND reaches_state(q_reject)
    -> REJECTS(M,w)
    AND HALTS_ON(M,w)

RULE divergence:
    Computation(M,w)
    AND infinite_sequence_without_halting_state
    -> DIVERGES_ON(M,w)

RULE recognises_language:
    TuringMachine(M)
    AND Language(L)
    AND FOR_EVERY input w [
        w MEMBER_OF L -> ACCEPTS(M,w)
    ]
    AND FOR_EVERY input w [
        w NOT_MEMBER_OF L
        -> M may_reject_or_diverge_on w
    ]
    -> RECOGNISES(M,L)

RULE decides_language:
    TuringMachine(M)
    AND Language(L)
    AND FOR_EVERY input w [
        HALTS_ON(M,w)
        AND (
            w MEMBER_OF L -> ACCEPTS(M,w)
        )
        AND (
            w NOT_MEMBER_OF L -> REJECTS(M,w)
        )
    ]
    -> DECIDES(M,L)

RULE computes_partial_function:
    TuringMachine(M)
    AND FOR_EVERY input x [
        IF HALTS_ON(M,x)
        THEN output(M,x)=f(x)
    ]
    -> COMPUTES(M,f)
    AND PartialComputableFunction(f)

RULE computes_total_function:
    COMPUTES(M,f)
    AND FOR_EVERY input x [
        HALTS_ON(M,x)
    ]
    -> ComputableFunction(f)
    AND TotalComputableFunction(f)

RULE universal_machine:
    UniversalTuringMachine(U)
    AND EncodedMachine(M)
    AND EncodedInput(x)
    -> SIMULATES(U,M_on_x)

RULE universal_simulation_equivalence:
    UniversalTuringMachine(U)
    -> behaviour(U,encode(M,x))
       corresponds_to behaviour(M,x)

RULE nondeterministic_acceptance:
    NondeterministicTuringMachine(N)
    AND Input(w)
    AND EXISTS ComputationPath(p) [
        p reaches_accept_state
    ]
    -> ACCEPTS(N,w)

RULE model_equivalence:
    MultiTapeTuringMachine(Multi)
    -> EXISTS SingleTapeTuringMachine(Single) [
        Single simulates Multi
        AND recognises_same_language
    ]

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

DEFINE TRANSITION_MODEL := {
    deterministic_form:
        "delta(q,a) = (qPrime,b,d)",
    inputs: {
        current_state: q,
        scanned_symbol: a
    },
    outputs: {
        next_state: qPrime,
        symbol_to_write: b,
        movement: d
    },
    movement_domain: {
        LEFT,
        RIGHT
    }
}

DEFINE CONFIGURATION_MODEL := {
    state:
        "current finite-control state",
    tape:
        "finite nonblank content embedded in an otherwise blank unbounded tape",
    head_position:
        "currently scanned tape cell",
    successor:
        "obtained by exactly one valid transition"
}

DEFINE COMPUTATION_MODEL := {
    start:
        InitialConfiguration,
    transition:
        "repeated application of delta",
    outcomes: {
        Accept,
        Reject,
        Diverge
    },
    deterministic:
        "one computation path",
    nondeterministic:
        "possibly branching computation tree"
}

DEFINE LANGUAGE_MODEL := {
    recognised_language:
        "L(M) = {w | M accepts w}",
    decidable_language:
        "M halts on every input and accepts exactly members of L",
    recognisable_language:
        "M accepts exactly members of L but may diverge on nonmembers"
}

DEFINE CRITICAL_DISTINCTIONS := {
    acceptance_vs_halting: {
        acceptance:
            "machine halts in an accepting state",
        halting:
            "machine terminates in any designated halting outcome",
        implication:
            "acceptance implies halting, but halting may also be rejection"
    },
    rejection_vs_divergence: {
        rejection:
            "explicit finite halting outcome",
        divergence:
            "computation continues indefinitely"
    },
    recogniser_vs_decider: {
        recogniser:
            "may diverge on negative instances",
        decider:
            "halts on every input"
    },
    machine_vs_configuration: {
        machine:
            "fixed transition system",
        configuration:
            "instantaneous computational state of that machine on a particular run"
    }
}

DEFINE EXAMPLE_1 := {
    task:
        "recognise binary strings containing at least one 1",
    strategy: {
        "scan input from left to right",
        "if 1 is found, accept",
        "if blank is reached without 1, reject"
    },
    classification:
        Decider
}

DEFINE EXAMPLE_2 := {
    machine:
        UniversalTuringMachine,
    input:
        "encoding of machine M together with input w",
    behaviour:
        "simulate the computation of M on w"
}

DEFINE EXAMPLE_3 := {
    problem:
        HaltingProblem,
    encoded_instance:
        "encode(M,w)",
    question:
        "does M halt on w?",
    general_decider_exists:
        FALSE
}

DEFINE MACHINE_PARSE_PIPELINE :=
    MachineDescription
    -> ParseStateSet
    -> ParseInputAlphabet
    -> ParseTapeAlphabet
    -> ParseTransitionFunction
    -> ParseStartAcceptRejectStates
    -> ValidateAlphabetConstraints
    -> ValidateTransitionArity
    -> ConstructTypedTuringMachine

DEFINE EXECUTION_PIPELINE :=
    TuringMachine
    -> Input
    -> ConstructInitialConfiguration
    -> ReadCurrentSymbol
    -> LookupTransition
    -> WriteSymbol
    -> ChangeState
    -> MoveHead
    -> ConstructSuccessorConfiguration
    -> RepeatUntilAcceptRejectOrDivergence
    -> ReturnComputationOutcome

DEFINE LANGUAGE_CLASSIFICATION_PIPELINE :=
    TuringMachine(M)
    -> AnalyseAcceptedInputs
    -> DetermineWhetherAllInputsHalt
    -> IF all_inputs_halt
       THEN Decider
    -> ELSE IF members_are_accepted
       THEN Recogniser
    -> ReturnLanguageClass

DEFINE SIMULATION_PIPELINE :=
    UniversalMachine(U)
    -> ParseEncodedMachine(M)
    -> ParseEncodedInput(w)
    -> RepresentSimulatedConfiguration
    -> RepeatedlySimulateTransitionOfM
    -> MirrorAcceptRejectOrDivergenceBehaviour
    -> ReturnSimulationResult

DEFINE TURING_MACHINE_GRAPH := {
    node_types: {
        MachineNode,
        StateNode,
        SymbolNode,
        ConfigurationNode,
        TransitionNode,
        InputNode,
        OutputNode
    },
    edge_types: {
        HAS_STATE,
        READS,
        WRITES,
        MOVES,
        TRANSITIONS_TO,
        ACCEPTS,
        REJECTS,
        HALTS_ON,
        DIVERGES_ON,
        SIMULATES
    },
    configuration_sequence_ordered: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    TuringMachine IS_COMPONENT_OF ComputabilityTheory,
    TuringMachine IS_SUBTYPE_OF ComputationModel,
    DeterministicTuringMachine IS_SUBTYPE_OF TuringMachine,
    NondeterministicTuringMachine IS_SUBTYPE_OF TuringMachine,
    UniversalTuringMachine IS_SUBTYPE_OF TuringMachine,
    MultiTapeTuringMachine IS_SUBTYPE_OF TuringMachine,
    TuringMachine HAS_STATE State,
    TuringMachine HAS_TAPE Tape,
    TuringMachine HAS_TRANSITION_FUNCTION TransitionFunction,
    Computation CONTAINS Configuration,
    UniversalTuringMachine SIMULATES EncodedTuringMachine,
    TuringMachine RECOGNISES RecognisableLanguage,
    TuringMachine DECIDES DecidableLanguage,
    TuringMachine COMPUTES ComputableFunction
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "Turing machine" <=> TuringMachine,
    "TM" <=> TuringMachine WHEN computability_context=TRUE,
    "universal Turing machine" <=> UniversalTuringMachine,
    "machine configuration" <=> Configuration,
    "instantaneous description" <=> Configuration WHEN Turing_machine_context=TRUE,
    "transition function" <=> TransitionFunction,
    "Turing recogniser" <=> Recogniser,
    "Turing decider" <=> Decider
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation, execution and analysis of Turing machines",
    preserve_machine_identity: TRUE,
    preserve_state_identity: TRUE,
    preserve_alphabet_identity: TRUE,
    preserve_tape_contents: TRUE,
    preserve_head_position: TRUE,
    preserve_transition_function: TRUE,
    preserve_configuration_order: TRUE,
    distinguish_accept_reject_diverge: TRUE,
    distinguish_recogniser_from_decider: TRUE,
    support_deterministic_execution: TRUE,
    support_nondeterministic_branching: TRUE,
    support_universal_simulation: TRUE,
    support_function_computation: TRUE,
    reasoning_mode: "symbolic + state-transition + algorithmic + computability-aware"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    machine_schema_mapping: 1.00,
    transition_mapping: 0.99,
    configuration_mapping: 0.99,
    deterministic_execution: 0.99,
    recogniser_decider_distinction: 1.00,
    universal_simulation_mapping: 0.99,
    arbitrary_machine_halting_prediction: undecidable_in_general
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        finite_state_control,
        tape_and_head,
        transition_function,
        configuration,
        accept_reject_halting,
        divergence,
        recognised_language,
        universal_simulation
    },
    extensible_layers: {
        nondeterministic_Turing_machines,
        multi_tape_Turing_machines,
        multi_track_Turing_machines,
        oracle_Turing_machines,
        probabilistic_Turing_machines,
        alternating_Turing_machines,
        machine_encodings,
        complexity_aware_machine_models
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + TURING_MACHINE_SCHEMA
    + RULES
    + TRANSITION_MODEL
    + CONFIGURATION_MODEL
    + COMPUTATION_MODEL
    + LANGUAGE_MODEL
    + CRITICAL_DISTINCTIONS
    + MACHINE_PARSE_PIPELINE
    + EXECUTION_PIPELINE
    + LANGUAGE_CLASSIFICATION_PIPELINE
    + SIMULATION_PIPELINE
    + TURING_MACHINE_GRAPH
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: TuringMachineOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/computability-theory/turing-machine-ontology.html",
    parent_domain: ComputabilityTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: TuringMachine,
    transition_reasoning_enabled: TRUE,
    configuration_reasoning_enabled: TRUE,
    execution_reasoning_enabled: TRUE,
    language_classification_enabled: TRUE,
    universal_simulation_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    database_safe_unicode: TRUE,
    flow_complete: TRUE
}

END_FLOW