Question 23953: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Original question and givens

Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. 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=[a2, a2, b2]. 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=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. 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=[a2, a2, b2]. 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=[s2, m2, f2, e12, e22, d2],alphabet=[a2, b2],start=s2;ORIGINAL transition order=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]

final set normalized UTF16=[f2];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s2];previous set=[s2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, Yes], [1, s2, d2, Yes], [4, f2, e12, No], [5, e12, e22, No]];next set=[d2, m2, s2];stable=No

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s2];previous set=[d2, m2, s2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, Yes], [1, s2, d2, Yes], [4, f2, e12, No], [5, e12, e22, No]];next set=[d2, m2, s2];stable=Yes

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d2, m2, s2];symbol=a2;ALL matching ORIGINAL edge rows=[[2, m2, m2]]

direct symbol targets=[m2];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m2];previous set=[m2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, No], [1, s2, d2, No], [4, f2, e12, No], [5, e12, e22, No]];next set=[m2];stable=Yes

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m2];symbol=a2;ALL matching ORIGINAL edge rows=[[2, m2, m2]]

direct symbol targets=[m2];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m2];previous set=[m2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, No], [1, s2, d2, No], [4, f2, e12, No], [5, e12, e22, No]];next set=[m2];stable=Yes

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m2];symbol=b2;ALL matching ORIGINAL edge rows=[[3, m2, f2]]

direct symbol targets=[f2];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[f2];previous set=[f2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, No], [1, s2, d2, No], [4, f2, e12, Yes], [5, e12, e22, No]];next set=[e12, f2];stable=No

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=1;seed set=[f2];previous set=[e12, f2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, No], [1, s2, d2, No], [4, f2, e12, Yes], [5, e12, e22, Yes]];next set=[e12, e22, f2];stable=No

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=2;seed set=[f2];previous set=[e12, e22, f2]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, No], [1, s2, d2, No], [4, f2, e12, Yes], [5, e12, e22, Yes]];next set=[e12, e22, f2];stable=Yes

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=0;previous configs=[[s2, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]]];next configs=[[s2, 0], [m2, 0], [d2, 0]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=1;previous configs=[[s2, 0], [m2, 0], [d2, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=2;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=3;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=4;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=5;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]], [[e12, 3], 5, [e22, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]];stable=No

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=6;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]], [[e12, 3], 5, [e22, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]];stable=Yes

ORIGINAL word=[a2, a2, b2];ALL DFA prefix subsets=[[d2, m2, s2], [m2], [m2], [e12, e22, f2]];independent direct NFA prefix subsets=[[d2, m2, s2], [m2], [m2], [e12, e22, f2]]

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

• 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

([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], 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.