# Question 23972: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States [Return to catalogue page 120](https://ocme.evemisslab.com/en/questions/page/120/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations/#q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0097) 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. R0046-23972 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0097 ## Original question and givens Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary). - G001 — Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary). ## 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-subset-construction-nfa-to-dfa](https://ocme.evemisslab.com/en/mko/mko-subset-construction-nfa-to-dfa/) ## Step-by-step derivation S001 · impliesBefore source operation=subsetdfa-full-word-trace-invariant;original snapshot and mathematics retained After explicit ordered source output fields=[input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary] Conditions: none Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL states=[s21, m21, f21, e121, e221, d21],alphabet=[a21, b21],start=s21;ORIGINAL transition order=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]] After final set normalized UTF16=[f21];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model Conditions: none This model differs from the earlier NFA node:transition triples are unique and finals are sorted.Subsets must be actual carrier subsets;do not import old duplicate/final-order conventions. Step identifiers and dependenciesRule: subsetdfa_qualified_NFA_signature_DUPLICATE_edges_REJECTED_here · Uses: G001, S001 S003 · impliesBefore Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s21];previous set=[s21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, Yes], [1, s21, d21, Yes], [4, f21, e121, No], [5, e121, e221, No]];next set=[d21, m21, s21];stable=No Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S002 S004 · impliesBefore Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s21];previous set=[d21, m21, s21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, Yes], [1, s21, d21, Yes], [4, f21, e121, No], [5, e121, e221, No]];next set=[d21, m21, s21];stable=Yes Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S003 S005 · impliesBefore Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d21, m21, s21];symbol=a21;ALL matching ORIGINAL edge rows=[[2, m21, m21]] After direct symbol targets=[m21];NO EPS preclosure inserted into S;now close ONLY these targets Conditions: none An arbitrary supplied S need not be EPS-closed;preclosing it would change this operation.Apply move(S,a) and then close its targets.Empty targets yield the legal empty dead state. Step identifiers and dependenciesRule: subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · Uses: G001, S004 S006 · impliesBefore Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m21];previous set=[m21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, No], [5, e121, e221, No]];next set=[m21];stable=Yes Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S005 S007 · impliesBefore Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m21];symbol=b21;ALL matching ORIGINAL edge rows=[[3, m21, f21]] After direct symbol targets=[f21];NO EPS preclosure inserted into S;now close ONLY these targets Conditions: none An arbitrary supplied S need not be EPS-closed;preclosing it would change this operation.Apply move(S,a) and then close its targets.Empty targets yield the legal empty dead state. Step identifiers and dependenciesRule: subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · Uses: G001, S006 S008 · impliesBefore Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[f21];previous set=[f21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, No]];next set=[e121, f21];stable=No Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S007 S009 · impliesBefore Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=1;seed set=[f21];previous set=[e121, f21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];stable=No Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S008 S010 · impliesBefore Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=2;seed set=[f21];previous set=[e121, e221, f21] After ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];stable=Yes Conditions: none Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input. Step identifiers and dependenciesRule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S009 S011 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=0;previous configs=[[s21, 0]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]]];next configs=[[s21, 0], [m21, 0], [d21, 0]];stable=No Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S010 S012 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=1;previous configs=[[s21, 0], [m21, 0], [d21, 0]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1]];stable=No Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S011 S013 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=2;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2]];stable=No Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S012 S014 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=3;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2]];stable=No Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S013 S015 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=4;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]], [[e121, 2], 5, [e221, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]];stable=No Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S014 S016 · impliesBefore ORIGINAL word=[a21, b21];direct NFA configuration stage=5;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]] After ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]], [[e121, 2], 5, [e221, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]];stable=Yes Conditions: none Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation. Step identifiers and dependenciesRule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S015 S017 · impliesBefore ORIGINAL word=[a21, b21];ALL DFA prefix subsets=[[d21, m21, s21], [m21], [e121, e221, f21]];independent direct NFA prefix subsets=[[d21, m21, s21], [m21], [e121, e221, f21]] After prefix_invariant=Yes;accepted=Yes;base is EPS closure of start;for ANY prefix,append one symbol move then finite EPS suffix to obtain EXACTLY all next reachable states Conditions: none General equivalence uses arbitrary-prefix induction with both inclusions:every DFA move has an NFA path witness,and any NFA path for the next prefix decomposes into prior prefix,symbol edge and EPS suffix.Intersection with finals then corresponds to a full accepting run.The fixed No only withholds general equivalence from finite word tests alone;it negates neither this invariant/acceptance nor the theorem. Step identifiers and dependenciesRule: subsetdfa_GENERAL_arbitrary_prefix_two_inclusions_acceptance_rule · Uses: G001, S016 ## Reverse checks - 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 ([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant) 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: 2dce1f6ae419945d3fb54e870c36e059efd2034ed2f45132184df6a4ca9e0f4c - candidate record SHA-256: fae6727e0cf6fdd9e84fa5f6794859731d203835f8ad8a22f954f58384ed1260 - semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1 - question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0097.json)