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R0031-16251 · q-20261014-advance-20261101-r07-finite-automaton-0076
Use a complete deterministic finite automaton with states=Fin(3)={0,…,2}, alphabet=Fin(2)={0,…,1}, total row-state/column-symbol transition=[[0, 0], [0, 0], [0, 0]], unique start=0, and increasing accepting subset=[0]. Every transition output lies in this same state set. There are no epsilon transitions or nondeterministic branches. Read a finite word in symbol order; the empty word leaves the automaton at its start state. Word=[0, 0, 0, 0, 0]. Give (word,state_trace,accepted?,proves_language_equivalence?,one_trace_does_not_prove_language_equivalence_or_regularity).
G001 — Use a complete deterministic finite automaton with states=Fin(3)={0,…,2}, alphabet=Fin(2)={0,…,1}, total row-state/column-symbol transition=[[0, 0], [0, 0], [0, 0]], unique start=0, and increasing accepting subset=[0]. Every transition output lies in this same state set. There are no epsilon transitions or nondeterministic branches. Read a finite word in symbol order; the empty word leaves the automaton at its start state. Word=[0, 0, 0, 0, 0]. Give (word,state_trace,accepted?,proves_language_equivalence?,one_trace_does_not_prove_language_equivalence_or_regularity).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
DFA total state-symbol cell count := mul(3, 2)
After
DFA total state-symbol cell count = 6
Conditions: none
Compute exactly from original field roles and established intermediates,retaining denominators,signs,indices and the complete carrier.
Rule: exact_rational_arithmetic · Uses: G001
Before
same complete states Fin(3);alphabet Fin(2);row-state column-symbol table=[[0, 0], [0, 0], [0, 0]]
After
single original start=0;canonical accepting subset=[0];required transition cells=6
Conditions: none
Fix states,alphabet,one start state and accepting subset.Rows index states and columns index symbols;every output must remain in the same state set.Check the whole table,not just cells visited by this word.
Rule: dfa_complete_typed_carriers_start_and_accepting_subset · Uses: G001, S001
Before
original transition cell state=0;symbol=0;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S002
Before
original transition cell state=0;symbol=1;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S003
Before
original transition cell state=1;symbol=0;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S004
Before
original transition cell state=1;symbol=1;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S005
Before
original transition cell state=2;symbol=0;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S006
Before
original transition cell state=2;symbol=1;complete original row=[0, 0]
After
single same-carrier next state=0
Conditions: none
Each(state,symbol)has exactly one valid output,with no missing cell,branch or epsilon transition.This complete typed table supplies total deterministic DFA qualification.
Rule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S007
Before
original ordered word=[0, 0, 0, 0, 0];original start=0
After
initial trace before consuming any symbol=[0]
Conditions: none
Place the initial state in the trace before reading symbols.The empty word consumes nothing and performs no invented transition;its final state is the start.
Rule: dfa_initial_state_before_first_symbol_including_empty_word · Uses: G001, S008
Before
word position=0;original symbol=0;current state=0;original state row=[0, 0]
After
lookup column=0;next state=0;prefix trace=[0, 0]
Conditions: none
Read the original symbol order with one column lookup per symbol.Do not reverse the word or swap state/symbol roles.Each consumed symbol adds one state,so trace length is word length plus1.
Rule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S009
Before
word position=1;original symbol=0;current state=0;original state row=[0, 0]
After
lookup column=0;next state=0;prefix trace=[0, 0, 0]
Conditions: none
Read the original symbol order with one column lookup per symbol.Do not reverse the word or swap state/symbol roles.Each consumed symbol adds one state,so trace length is word length plus1.
Rule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S010
Before
word position=2;original symbol=0;current state=0;original state row=[0, 0]
After
lookup column=0;next state=0;prefix trace=[0, 0, 0, 0]
Conditions: none
Read the original symbol order with one column lookup per symbol.Do not reverse the word or swap state/symbol roles.Each consumed symbol adds one state,so trace length is word length plus1.
Rule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S011
Before
word position=3;original symbol=0;current state=0;original state row=[0, 0]
After
lookup column=0;next state=0;prefix trace=[0, 0, 0, 0, 0]
Conditions: none
Read the original symbol order with one column lookup per symbol.Do not reverse the word or swap state/symbol roles.Each consumed symbol adds one state,so trace length is word length plus1.
Rule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S012
Before
word position=4;original symbol=0;current state=0;original state row=[0, 0]
After
lookup column=0;next state=0;prefix trace=[0, 0, 0, 0, 0, 0]
Conditions: none
Read the original symbol order with one column lookup per symbol.Do not reverse the word or swap state/symbol roles.Each consumed symbol adds one state,so trace length is word length plus1.
Rule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S013
Before
complete original word consumed;final state=0;same accepting subset=[0]
After
accepted=Yes
Conditions: none
Acceptance tests the final state,not an intermediate accepting visit or the word length.For an empty word,test the start state directly.
Rule: dfa_acceptance_final_state_not_intermediate_or_length · Uses: G001, S014
Before
this one word=[0, 0, 0, 0, 0];full trace=[0, 0, 0, 0, 0, 0];accepted=Yes
After
proves equivalence to unspecified language or another automaton=No;one_trace_does_not_prove_language_equivalence_or_regularity
Conditions: none
One trace establishes this word's result,not equivalence with an unspecified language or another automaton.The No does not deny that a complete DFA recognizes a regular language by definition.Full-language constructions and equivalence have separate obligations.
Rule: dfa_one_word_not_full_language_equivalence_or_irregularity · Uses: G001, S015
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Recompute DFA total state-symbol cell count with exact rational arithmetic, not rounded decimals.([0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], Yes, No, one_trace_does_not_prove_language_equivalence_or_regularity)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ff2094181ba4508e29a8832c79bc2ea7eff9373520032da80e9bf551771c33dbcandidate record SHA-256: 1ddbffd4fbf5ae29a84fe17b60cf9b35ee54cc4e6eb674166c9c745c57ec386asemantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369aquestion-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bbLean: NOT_RUN · human review: NOT_PERFORMED