Question 23966: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Original question and givens

Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. 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=[a15, a15, a15, b15]. 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=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. 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=[a15, a15, a15, b15]. 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=[s15, m15, f15, e115, e215, d15],alphabet=[a15, b15],start=s15;ORIGINAL transition order=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]

final set normalized UTF16=[f15];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s15];previous set=[s15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, Yes], [1, s15, d15, Yes], [4, f15, e115, No], [5, e115, e215, No]];next set=[d15, m15, s15];stable=No

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s15];previous set=[d15, m15, s15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, Yes], [1, s15, d15, Yes], [4, f15, e115, No], [5, e115, e215, No]];next set=[d15, m15, s15];stable=Yes

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d15, m15, s15];symbol=a15;ALL matching ORIGINAL edge rows=[[2, m15, m15]]

direct symbol targets=[m15];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m15];previous set=[m15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, No], [5, e115, e215, No]];next set=[m15];stable=Yes

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m15];symbol=a15;ALL matching ORIGINAL edge rows=[[2, m15, m15]]

direct symbol targets=[m15];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m15];previous set=[m15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, No], [5, e115, e215, No]];next set=[m15];stable=Yes

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m15];symbol=a15;ALL matching ORIGINAL edge rows=[[2, m15, m15]]

direct symbol targets=[m15];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[m15];previous set=[m15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, No], [5, e115, e215, No]];next set=[m15];stable=Yes

Subset DFA ORIGINAL prefix step3:EXACT ORIGINAL supplied S=[m15];symbol=b15;ALL matching ORIGINAL edge rows=[[3, m15, f15]]

direct symbol targets=[f15];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=0;seed set=[f15];previous set=[f15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, No]];next set=[e115, f15];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=1;seed set=[f15];previous set=[e115, f15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, Yes]];next set=[e115, e215, f15];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=2;seed set=[f15];previous set=[e115, e215, f15]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s15, m15, No], [1, s15, d15, No], [4, f15, e115, Yes], [5, e115, e215, Yes]];next set=[e115, e215, f15];stable=Yes

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=0;previous configs=[[s15, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]]];next configs=[[s15, 0], [m15, 0], [d15, 0]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=1;previous configs=[[s15, 0], [m15, 0], [d15, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=2;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=3;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]], [[m15, 2], 2, [m15, 3]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=4;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]], [[m15, 2], 2, [m15, 3]], [[m15, 3], 3, [f15, 4]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=5;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]], [[m15, 2], 2, [m15, 3]], [[m15, 3], 3, [f15, 4]], [[f15, 4], 4, [e115, 4]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4], [e115, 4]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=6;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4], [e115, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]], [[m15, 2], 2, [m15, 3]], [[m15, 3], 3, [f15, 4]], [[f15, 4], 4, [e115, 4]], [[e115, 4], 5, [e215, 4]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4], [e115, 4], [e215, 4]];stable=No

ORIGINAL word=[a15, a15, a15, b15];direct NFA configuration stage=7;previous configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4], [e115, 4], [e215, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[m15, 0], 2, [m15, 1]], [[m15, 1], 2, [m15, 2]], [[m15, 2], 2, [m15, 3]], [[m15, 3], 3, [f15, 4]], [[f15, 4], 4, [e115, 4]], [[e115, 4], 5, [e215, 4]]];next configs=[[s15, 0], [m15, 0], [d15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4], [e115, 4], [e215, 4]];stable=Yes

ORIGINAL word=[a15, a15, a15, b15];ALL DFA prefix subsets=[[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]];independent direct NFA prefix subsets=[[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]]

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

([a15, a15, a15, b15], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], 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.