# Question 16207: The Empty Word Is Accepted Exactly When the Start State Is Accepting [Return to catalogue page 82](https://ocme.evemisslab.com/en/questions/page/82/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r07-finite-automaton-foundations/#q-20261014-advance-20261101-r07-finite-automaton-0057) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0031-16207 · q-20261014-advance-20261101-r07-finite-automaton-0057 ## Original question and givens Use a complete deterministic finite automaton with states=Fin(3)={0,…,2}, alphabet=Fin(2)={0,…,1}, total row-state/column-symbol transition=[[0, 2], [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, 1, 1, 0, 0]. Give (word,state_trace,final_state,accepted?). - 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, 2], [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, 1, 1, 0, 0]. Give (word,state_trace,final_state,accepted?). ## Definition and scope 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. - [mko-finite-automaton](https://ocme.evemisslab.com/en/mko/mko-finite-automaton/) ## Step-by-step derivation S001 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore same complete states Fin(3);alphabet Fin(2);row-state column-symbol table=[[0, 2], [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. Step identifiers and dependenciesRule: dfa_complete_typed_carriers_start_and_accepting_subset · Uses: G001, S001 S003 · impliesBefore original transition cell state=0;symbol=0;complete original row=[0, 2] 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S002 S004 · impliesBefore original transition cell state=0;symbol=1;complete original row=[0, 2] After single same-carrier next state=2 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S003 S005 · impliesBefore 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S004 S006 · impliesBefore 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S005 S007 · impliesBefore 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S006 S008 · impliesBefore 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. Step identifiers and dependenciesRule: dfa_every_state_symbol_total_deterministic_cell · Uses: G001, S007 S009 · impliesBefore original ordered word=[0, 1, 1, 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. Step identifiers and dependenciesRule: dfa_initial_state_before_first_symbol_including_empty_word · Uses: G001, S008 S010 · impliesBefore word position=0;original symbol=0;current state=0;original state row=[0, 2] 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. Step identifiers and dependenciesRule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S009 S011 · impliesBefore word position=1;original symbol=1;current state=0;original state row=[0, 2] After lookup column=1;next state=2;prefix trace=[0, 0, 2] 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. Step identifiers and dependenciesRule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S010 S012 · impliesBefore word position=2;original symbol=1;current state=2;original state row=[0, 0] After lookup column=1;next state=0;prefix trace=[0, 0, 2, 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. Step identifiers and dependenciesRule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S011 S013 · impliesBefore word position=3;original symbol=0;current state=0;original state row=[0, 2] After lookup column=0;next state=0;prefix trace=[0, 0, 2, 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. Step identifiers and dependenciesRule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S012 S014 · impliesBefore word position=4;original symbol=0;current state=0;original state row=[0, 2] After lookup column=0;next state=0;prefix trace=[0, 0, 2, 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. Step identifiers and dependenciesRule: dfa_each_original_symbol_one_ordered_table_lookup · Uses: G001, S013 S015 · impliesBefore 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. Step identifiers and dependenciesRule: dfa_acceptance_final_state_not_intermediate_or_length · Uses: G001, S014 ## Reverse checks - 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. ## Result ([0, 1, 1, 0, 0], [0, 0, 2, 0, 0, 0], 0, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 8b7698b955c4d6ed026f0139003fff04a562bf39af1f3db1656b9b1a4b84deed - candidate record SHA-256: f2adeaaca9df1f18caa1d8f6cbb568cc219cc3722d814f39ea52fc743553900e - semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a - question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r07-finite-automaton-0057.json)