# Question 17212: Construct a Complete Prefix-Trie DFA for an Exact Finite Language [Return to catalogue page 87](https://ocme.evemisslab.com/en/questions/page/87/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r07-regular-language-foundations/#q-20261014-advance-20261102-r07-regular-language-0062) 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. R0033-17212 · q-20261014-advance-20261102-r07-regular-language-0062 ## Original question and givens Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 1, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[0]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language). - G001 — Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 1, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[0]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language). ## 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-regular-language](https://ocme.evemisslab.com/en/mko/mko-regular-language/) ## Step-by-step derivation S001 · impliesBefore DECLARED complete nonempty duplicate-free word FAMILY=[[0], [1, 1, 1, 0, 1, 0]];alphabet Fin(2) After all prefixes including empty sorted by length then source JSON order=[[], [0], [1], [1, 1], [1, 1, 1], [1, 1, 1, 0], [1, 1, 1, 0, 1], [1, 1, 1, 0, 1, 0]];dead state appended index=8 Conditions: none This family is the entire finite language,not test samples.It may contain the empty word but the family is nonempty.Sort prefixes by length and the source JSON order;symbols0..9 make same-length ordering compatible.Do not inherit another family's30-state cap. Step identifiers and dependenciesRule: regular_complete_finite_language_prefix_states_not_sampled_tests · Uses: G001 S002 · impliesBefore prefix state=0;represented prefix=[];next symbol=0;extended word=[0] After extension is declared prefix=Yes;next state=1 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S001 S003 · impliesBefore prefix state=0;represented prefix=[];next symbol=1;extended word=[1] After extension is declared prefix=Yes;next state=2 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S002 S004 · impliesBefore prefix state=1;represented prefix=[0];next symbol=0;extended word=[0, 0] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S003 S005 · impliesBefore prefix state=1;represented prefix=[0];next symbol=1;extended word=[0, 1] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S004 S006 · impliesBefore prefix state=2;represented prefix=[1];next symbol=0;extended word=[1, 0] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S005 S007 · impliesBefore prefix state=2;represented prefix=[1];next symbol=1;extended word=[1, 1] After extension is declared prefix=Yes;next state=3 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S006 S008 · impliesBefore prefix state=3;represented prefix=[1, 1];next symbol=0;extended word=[1, 1, 0] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S007 S009 · impliesBefore prefix state=3;represented prefix=[1, 1];next symbol=1;extended word=[1, 1, 1] After extension is declared prefix=Yes;next state=4 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S008 S010 · impliesBefore prefix state=4;represented prefix=[1, 1, 1];next symbol=0;extended word=[1, 1, 1, 0] After extension is declared prefix=Yes;next state=5 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S009 S011 · impliesBefore prefix state=4;represented prefix=[1, 1, 1];next symbol=1;extended word=[1, 1, 1, 1] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S010 S012 · impliesBefore prefix state=5;represented prefix=[1, 1, 1, 0];next symbol=0;extended word=[1, 1, 1, 0, 0] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S011 S013 · impliesBefore prefix state=5;represented prefix=[1, 1, 1, 0];next symbol=1;extended word=[1, 1, 1, 0, 1] After extension is declared prefix=Yes;next state=6 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S012 S014 · impliesBefore prefix state=6;represented prefix=[1, 1, 1, 0, 1];next symbol=0;extended word=[1, 1, 1, 0, 1, 0] After extension is declared prefix=Yes;next state=7 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S013 S015 · impliesBefore prefix state=6;represented prefix=[1, 1, 1, 0, 1];next symbol=1;extended word=[1, 1, 1, 0, 1, 1] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S014 S016 · impliesBefore prefix state=7;represented prefix=[1, 1, 1, 0, 1, 0];next symbol=0;extended word=[1, 1, 1, 0, 1, 0, 0] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S015 S017 · impliesBefore prefix state=7;represented prefix=[1, 1, 1, 0, 1, 0];next symbol=1;extended word=[1, 1, 1, 0, 1, 0, 1] After extension is declared prefix=No;next state=8 Conditions: none Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching. Step identifiers and dependenciesRule: regular_trie_every_prefix_symbol_total_transition · Uses: G001, S016 S018 · impliesBefore dead state=8;all alphabet symbols Fin(2);full supplied words=[[0], [1, 1, 1, 0, 1, 0]] After dead loops row=[8, 8];accepting prefix-state indices=[1, 7];complete table=[[1, 2], [8, 8], [8, 3], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]] Conditions: none Every symbol loops at dead.Accept exactly states equal to full supplied words,not every proper prefix.If the empty word is supplied,start0 is accepting. Step identifiers and dependenciesRule: regular_trie_dead_absorbing_accepts_full_words_only · Uses: G001, S017 S019 · impliesBefore same complete prefix-state table and full word family=[[0], [1, 1, 1, 0, 1, 0]];start state0 represents empty prefix After for EVERY finite input prefix,its state is that exact prefix index if present,otherwise dead;prefix extension closure and dead absorption preserve this by induction;accepted iff input is one FULL supplied word Conditions: none The empty prefix is the base case.Each transition follows exact extension.If a consumed word is not a supplied-word prefix,no later extension can become one,so dead absorption is valid.Induction over arbitrary word length gives acceptance exactly for full listed words,not an inference from a successful probe. Step identifiers and dependenciesRule: regular_trie_all_word_induction_exact_language_not_probe_only · Uses: G001, S018 S020 · impliesBefore finite-language probe:original ordered word=[0];start=0 After trace before consuming symbols=[0] Conditions: none Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide. Step identifiers and dependenciesRule: regular_word_initial_state_empty_word_retained · Uses: G001, S019 S021 · impliesBefore finite-language probe:word position=0;symbol=0;current state=0;complete original row=[1, 2] After next state=1;ordered prefix trace=[0, 1] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S020 S022 · impliesBefore original probe=[0];final state=1;accepting indices=[1, 7] After probe accepted=Yes;separate all-word construction certificate retained Conditions: none The probe is one run.The complete construction and induction above supply the full finite-language certificate. Step identifiers and dependenciesRule: regular_probe_result_separate_universal_construction · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[], [0], [1], [1, 1], [1, 1, 1], [1, 1, 1, 0], [1, 1, 1, 0, 1], [1, 1, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [8, 3], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [0], [0, 1], Yes, explicit_DFA_recognizes_exact_finite_language) 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: fb8687d040bf421a6b971ca1284c1698f43d26a61d57adbefae9c9588b532639 - candidate record SHA-256: 9904bbe6f17cbf40fe724669cb7b04f705e6ff92ad964453cd6b745953171f06 - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r07-regular-language-0062.json)