Question 23958: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Original question and givens

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. 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=[a7, a7, a7, b7]. 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=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. 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=[a7, a7, a7, b7]. 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

Step-by-step derivation

source operation=subsetdfa-full-word-trace-invariant;original snapshot and mathematics retained

explicit ordered source output fields=[input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary]

ORIGINAL states=[s7, m7, f7, e17, e27, d7],alphabet=[a7, b7],start=s7;ORIGINAL transition order=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]

final set normalized UTF16=[f7];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s7];previous set=[s7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, Yes], [1, s7, d7, Yes], [4, f7, e17, No], [5, e17, e27, No]];next set=[d7, m7, s7];stable=No

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s7];previous set=[d7, m7, s7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, Yes], [1, s7, d7, Yes], [4, f7, e17, No], [5, e17, e27, No]];next set=[d7, m7, s7];stable=Yes

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d7, m7, s7];symbol=a7;ALL matching ORIGINAL edge rows=[[2, m7, m7]]

direct symbol targets=[m7];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m7];previous set=[m7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, No], [5, e17, e27, No]];next set=[m7];stable=Yes

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m7];symbol=a7;ALL matching ORIGINAL edge rows=[[2, m7, m7]]

direct symbol targets=[m7];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m7];previous set=[m7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, No], [5, e17, e27, No]];next set=[m7];stable=Yes

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m7];symbol=a7;ALL matching ORIGINAL edge rows=[[2, m7, m7]]

direct symbol targets=[m7];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[m7];previous set=[m7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, No], [5, e17, e27, No]];next set=[m7];stable=Yes

Subset DFA ORIGINAL prefix step3:EXACT ORIGINAL supplied S=[m7];symbol=b7;ALL matching ORIGINAL edge rows=[[3, m7, f7]]

direct symbol targets=[f7];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=0;seed set=[f7];previous set=[f7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, Yes], [5, e17, e27, No]];next set=[e17, f7];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=1;seed set=[f7];previous set=[e17, f7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, Yes], [5, e17, e27, Yes]];next set=[e17, e27, f7];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=2;seed set=[f7];previous set=[e17, e27, f7]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, No], [1, s7, d7, No], [4, f7, e17, Yes], [5, e17, e27, Yes]];next set=[e17, e27, f7];stable=Yes

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=0;previous configs=[[s7, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]]];next configs=[[s7, 0], [m7, 0], [d7, 0]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=1;previous configs=[[s7, 0], [m7, 0], [d7, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=2;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=3;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]], [[m7, 2], 2, [m7, 3]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=4;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]], [[m7, 2], 2, [m7, 3]], [[m7, 3], 3, [f7, 4]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=5;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]], [[m7, 2], 2, [m7, 3]], [[m7, 3], 3, [f7, 4]], [[f7, 4], 4, [e17, 4]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4], [e17, 4]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=6;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4], [e17, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]], [[m7, 2], 2, [m7, 3]], [[m7, 3], 3, [f7, 4]], [[f7, 4], 4, [e17, 4]], [[e17, 4], 5, [e27, 4]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4], [e17, 4], [e27, 4]];stable=No

ORIGINAL word=[a7, a7, a7, b7];direct NFA configuration stage=7;previous configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4], [e17, 4], [e27, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[m7, 0], 2, [m7, 1]], [[m7, 1], 2, [m7, 2]], [[m7, 2], 2, [m7, 3]], [[m7, 3], 3, [f7, 4]], [[f7, 4], 4, [e17, 4]], [[e17, 4], 5, [e27, 4]]];next configs=[[s7, 0], [m7, 0], [d7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4], [e17, 4], [e27, 4]];stable=Yes

ORIGINAL word=[a7, a7, a7, b7];ALL DFA prefix subsets=[[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]];independent direct NFA prefix subsets=[[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]]

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

Reverse checks

• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([a7, a7, a7, b7], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], 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.