Automata Theory Ontology

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

1. PURPOSE
Represent formal machines, languages, transitions, acceptance, computation and equivalence as typed entities and verifiable relations. Distinguish recognisers from generators and finite-state from unbounded-memory computation.

2. TYPED ENTITIES
Alphabet := finite nonempty set of symbols
Word(Sigma) := finite sequence over Alphabet Sigma, including epsilon
Language(Sigma) := subset of Sigma*
State := discrete machine configuration identifier
Transition := typed state/symbol/stack/tape update
Automaton := tuple of states, alphabets, transition relation, initial configuration and acceptance condition
DFA := (Q, Sigma, delta: Q x Sigma -> Q, q0 in Q, F subset Q)
NFA := (Q, Sigma, Delta: Q x (Sigma union {epsilon}) -> P(Q), q0, F)
PDA := (Q, Sigma, Gamma, Delta, q0, Z0, F), with explicit stack semantics
TuringMachine := (Q, Sigma, Gamma, delta, q0, q_accept, q_reject), with tape and head conventions
Grammar := (Nonterminals, Terminals, Productions, StartSymbol)
Configuration := machine state together with relevant unread input, stack or tape
Run := finite or infinite sequence of configurations linked by transitions

3. CORE RELATIONSHIPS
HAS_STATE(Automaton, State)
HAS_ALPHABET(Automaton, Alphabet)
HAS_INITIAL_STATE(Automaton, State)
HAS_ACCEPTING_STATE(Automaton, State)
TRANSITIONS_TO(Configuration, Configuration)
RECOGNISES(Automaton, Language)
GENERATES(Grammar, Language)
ACCEPTS(Automaton, Word)
REJECTS(Automaton, Word)
EQUIVALENT_LANGUAGE(Automaton, Automaton)
SIMULATES(Machine, Machine)
DETERMINISES_TO(NFA, DFA)
MINIMISES_TO(DFA, DFA)

4. FORMAL OPERATIONS
EXTENDED_DFA_TRANSITION(delta*, q, epsilon) = q
EXTENDED_DFA_TRANSITION(delta*, q, wa) = delta(delta*(q,w), a)
DFA_ACCEPTS(M,w) iff delta*(q0,w) in F
NFA_ACCEPTS(M,w) iff some permitted run on w ends in F
LANGUAGE(M) = {w in Sigma* | ACCEPTS(M,w)}
COMPLEMENT_DFA(M) := swap accepting and nonaccepting states only when the DFA is complete
PRODUCT_DFA(M1,M2) := synchronous product for common alphabet, with acceptance condition chosen for union/intersection/difference
CONCATENATE_LANGUAGES(L1,L2) = {uv | u in L1 and v in L2}
KLEENE_STAR(L) = union over n >= 0 of L^n, where L^0 = {epsilon}

5. INVARIANTS AND THEOREMS
DFA_DETERMINISM: for every q in Q and a in Sigma exactly one successor delta(q,a) exists.
NFA_NONDETERMINISM: zero or more successor states may be available; epsilon transitions may be present.
FINITE_AUTOMATA_EQUIVALENCE: DFA, NFA and regular expressions define exactly the regular languages.
DETERMINISATION: every finite NFA has an equivalent DFA (subset construction).
MINIMISATION: a complete DFA has a unique minimal equivalent DFA up to state renaming.
REGULAR_CLOSURE: regular languages are closed under union, intersection, complement, concatenation and Kleene star.
PDA_CLASS: nondeterministic PDAs recognise context-free languages under standard acceptance conventions.
TURING_RECOGNISABILITY: a recogniser may fail to halt on rejected inputs; a decider halts on all inputs.
CHURCH_TURING_CONTEXT: the Church–Turing thesis is a foundational thesis, not a formally proven theorem.

6. INFERENCE RULES
IF DFA_ACCEPTS(M,w) THEN w in LANGUAGE(M).
IF M1 and M2 are equivalent DFAs THEN LANGUAGE(M1) = LANGUAGE(M2).
IF L is regular THEN COMPLEMENT(L) is regular (relative to a specified Sigma*).
IF M is a complete DFA THEN COMPLEMENT_DFA(M) recognises Sigma* minus LANGUAGE(M).
IF a grammar is context-free THEN an equivalent nondeterministic PDA exists.
IF a Turing machine halts and accepts w THEN w belongs to its recognised language.
DO_NOT_INFER: finite test agreement does not establish language equivalence.
DO_NOT_INFER: a PDA language is necessarily regular.
DO_NOT_INFER: a recogniser always halts.

7. VALIDATION
CHECK finite Q and Sigma for finite automata.
CHECK transition input/output types and declared alphabets.
CHECK q0 in Q and F subset Q.
CHECK DFA transition totality before complement.
CHECK epsilon closure during NFA evaluation and determinisation.
CHECK product constructions use compatible alphabets.
CHECK PDA stack alphabet and stack action semantics.
CHECK Turing machine acceptance, rejection and nontermination are distinct.
CHECK language-equivalence claims via proof, decision procedure where applicable, or verified construction; testing alone is insufficient.

8. SEMANTIC_EQUIVALENCE
DFA ~ NFA iff they recognise the same language.
RegularExpression ~ DFA iff denoted language equals recognised language.
State-renamed machines are structurally isomorphic when transitions and acceptance are preserved.
Language equivalence does not imply identical transition structure or resource use.
CONFIDENCE: formal_definition=1.0 when all assumptions are satisfied; empirical_claim=unverified until proven.

9. FLOW_HASH
FLOW_HASH_METHOD: SHA-256 of canonical UTF-8 serialisation of ontology version, typed entities, operations, constraints and inference rules.
FLOW_HASH_VALUE: computed by ingestion pipeline; never fabricate a digest.

10. UPDATE_MODEL
UPDATE_MODEL: versioned, append provenance, record semantic changes, revalidate affected definitions and proofs, recompute FLOW_HASH after canonicalisation.
BREAKING_CHANGE: changes to acceptance semantics, transition typing, alphabet interpretation or computational model.

11. AI_CONSUMPTION
AI_CONSUMPTION:
  representation: typed ontology and executable-style rules
  reasoning: language membership, automata transformation, closure and equivalence
  verification: type checks, invariants, constructive proofs, counterexamples
  uncertainty: separate formal theorems from unproved empirical claims
  provenance: preserve SOURCE_URL, version and authorship
  safety: do not claim decidability of arbitrary Turing-machine properties

END FLOW ALGEBRA