Turing Machine Ontology

FLOW ALGEBRA
TITLE: Turing Machine Ontology
DOMAIN: Discrete Mathematics / Automata Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/automata-theory/turing-machine-ontology-2.html
VERSION: 1.0
DATE: 2026-10-11
AUTHORS: Mark Higgins — Sky (Recursive AI)
SOURCE_STATUS: Mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent Turing machines as typed mathematical objects with explicit transition semantics, computation traces, acceptance, rejection, halting, and decidability constraints. Separate mathematical assertions from implementation conventions.

2. TYPED ENTITIES
  State := finite identifier
  TapeSymbol := symbol
  InputSymbol <: TapeSymbol
  Direction := {L, R}
  Machine := (Q, Sigma, Gamma, delta, q0, q_accept, q_reject)
  Q: finite set of states
  Sigma: finite input alphabet excluding blank
  Gamma: finite tape alphabet with Sigma subset Gamma
  blank: distinguished element of Gamma not in Sigma
  q0: initial state in Q
  q_accept, q_reject: distinct halting states in Q
  delta: partial function (Q \ {q_accept,q_reject}) x Gamma -> Q x Gamma x Direction
  Configuration := (tape: Z -> Gamma, head: Z, state: Q)
  Trace := finite or infinite sequence of configurations
  Language := subset of Sigma*

3. WELL-FORMEDNESS
  finite(Q) AND finite(Sigma) AND finite(Gamma)
  Sigma subset Gamma AND blank notin Sigma
  q0 in Q AND q_accept in Q AND q_reject in Q
  q_accept != q_reject
  delta is single-valued wherever defined
  tape has finite nonblank support at every finite step
  Convention: bi-infinite tape, deterministic machine, no outgoing halting transitions.
  Alternative conventions (one-way infinite tape, stay-put movement) require explicit translation.

4. CORE OPERATIONS
  INITIALISE(M,w): write input w from cell 0, blank elsewhere; head=0; state=q0.
  STEP(M,C): if delta(q,tape(head))=(q',a',d), write a' at head,
    move head by -1 for L or +1 for R, and enter q'.
  HALT(C): state in {q_accept,q_reject} OR delta undefined (undefined-transition convention must be specified).
  ACCEPT(M,w): exists finite t with STEP^t(INITIALISE(M,w)).state=q_accept.
  REJECT(M,w): exists finite t with STEP^t(INITIALISE(M,w)).state=q_reject.
  DIVERGE(M,w): execution has infinitely many transitions without halting.
  LANGUAGE(M): {w in Sigma* | ACCEPT(M,w)}.

5. FORMAL RELATIONS
  HAS_STATE(M,q); HAS_TAPE_SYMBOL(M,a); STARTS_IN(M,q0)
  TRANSITIONS_TO((q,a),(q',a',d))
  REACHES(C,C') iff C ->* C'
  ACCEPTS(M,w); REJECTS(M,w); LOOPS_ON(M,w)
  RECOGNISES(M,L) iff LANGUAGE(M)=L
  DECIDES(M,L) iff M halts on every input and LANGUAGE(M)=L
  COMPUTES(M,f) iff M realises partial function f under declared input/output encoding.

6. INFERENCE RULES
  TRANSITION: delta(q,a)=(q',b,d) AND READ(C)=a => C -> WRITE_MOVE(C,b,d,q').
  REACHABILITY: C -> C' => C ->* C'.
  REACHABILITY_TRANSITIVE: C ->* C' AND C' ->* C'' => C ->* C''.
  ACCEPTANCE: C0=INITIALISE(M,w) AND C0 ->* C AND STATE(C)=q_accept => ACCEPTS(M,w).
  DECIDER: HALTS_ON_ALL(M) AND LANGUAGE(M)=L => DECIDES(M,L).
  DECIDABLE_IMPLIES_RECOGNISABLE: DECIDES(M,L) => RECOGNISABLE(L).
  COMPLEMENT_DECIDER: DECIDES(M,L) => EXISTS decider for Sigma* \ L by swapping accept/reject states.
  WARNING: recognisability alone does not imply decidability or recognisability of complement.

7. MATHEMATICAL LAWS AND LIMITATIONS
  Deterministic transition yields at most one successor per nonhalting configuration.
  Every decidable language is Turing-recognisable.
  A language is decidable iff both it and its complement are Turing-recognisable.
  There exist Turing-recognisable languages that are undecidable.
  The general halting problem is undecidable.
  Multi-tape and nondeterministic Turing machines recognise the same language class as standard deterministic single-tape machines, though complexity can differ.
  Church-Turing thesis is a thesis about effective computation, not a formal theorem.

8. SEMANTIC_EQUIVALENCE
  Turing machine ~= abstract machine with unbounded tape and finite control [under declared model].
  Recognisable language ~= recursively enumerable language.
  Decidable language ~= recursive language.
  Halting != accepting; rejecting != diverging.
  Machine equivalence by accepted language does not imply equal runtime or transition structure.
  Equivalent models require a proven encoding/simulation, not merely similar names.

9. VALIDATION
  CHECK_TYPES: verify alphabets, states, blank, transition codomain and deterministic uniqueness.
  CHECK_INITIAL: verify input belongs to Sigma* and initial configuration matches encoding.
  CHECK_STEP: compare each configuration pair against delta and tape/head update.
  CHECK_ACCEPT: require finite witnessed accepting trace.
  CHECK_REJECT: require finite witnessed rejecting trace.
  CHECK_DIVERGENCE: finite timeout is UNKNOWN, not proof of divergence.
  CHECK_DECIDER: empirical test coverage is not proof of halting on all inputs.
  CHECK_PROVENANCE: distinguish proved theorem, illustrative example, hypothesis, and simulation result.

10. EXAMPLE
  Let Sigma={0,1}; M decide whether input contains at least one 1.
  Q={scan,accept,reject}; Gamma={0,1,blank}; q0=scan.
  delta(scan,0)=(scan,0,R)
  delta(scan,1)=(accept,1,R)
  delta(scan,blank)=(reject,blank,R)
  Thus ACCEPT(M,"0010")=true; REJECT(M,"000")=true; REJECT(M,"")=true.
  This machine halts on every finite input, so it decides {w in {0,1}* | w contains 1}.

11. CONFIDENCE
  FORMAL_DEFINITIONS: high (standard theory; conventions explicit)
  MATHEMATICAL_THEOREMS: high (established results)
  EXAMPLE: high (trace-checkable)
  LIVE_PAGE_ALIGNMENT: unverified
  Do not convert confidence annotations into probabilities without calibration evidence.

12. FLOW_HASH
  HASH_METHOD: SHA-256 of canonical UTF-8 flow text excluding the FLOW_HASH value itself.
  FLOW_HASH_VALUE: UNCOMPUTED (generate on publication after canonicalisation).
  HASH_SCOPE: versioned content and declared conventions.

13. UPDATE_MODEL
  Trigger: new source revision, corrected formalisation, new theorem citation, or ontology linkage.
  Procedure: compare source -> validate changes -> increment version -> regenerate hash -> archive previous revision.
  Preserve distinctions between mathematical definitions, model conventions, and empirical assertions.

14. AI_CONSUMPTION
  TASKS: machine validation, transition simulation, proof checking, language classification, formal knowledge graph integration.
  INPUTS: machine tuple, encoded input, transition table, declared tape convention.
  OUTPUTS: well-formedness report, bounded execution trace, acceptance/rejection witness, or UNKNOWN.
  SAFETY: never infer nontermination from timeout; never claim general decidability from finite tests.
  LINKED_CONCEPTS: Finite Automaton, Pushdown Automaton, Formal Language, Computability, Decidability, Halting Problem.

END FLOW ALGEBRA