Question 23969: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Original question and givens

Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. 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=[a18, a18, b18]. 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=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. 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=[a18, a18, b18]. 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=[s18, m18, f18, e118, e218, d18],alphabet=[a18, b18],start=s18;ORIGINAL transition order=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]

final set normalized UTF16=[f18];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s18];previous set=[s18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, Yes], [1, s18, d18, Yes], [4, f18, e118, No], [5, e118, e218, No]];next set=[d18, m18, s18];stable=No

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s18];previous set=[d18, m18, s18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, Yes], [1, s18, d18, Yes], [4, f18, e118, No], [5, e118, e218, No]];next set=[d18, m18, s18];stable=Yes

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d18, m18, s18];symbol=a18;ALL matching ORIGINAL edge rows=[[2, m18, m18]]

direct symbol targets=[m18];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m18];previous set=[m18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, No], [1, s18, d18, No], [4, f18, e118, No], [5, e118, e218, No]];next set=[m18];stable=Yes

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m18];symbol=a18;ALL matching ORIGINAL edge rows=[[2, m18, m18]]

direct symbol targets=[m18];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m18];previous set=[m18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, No], [1, s18, d18, No], [4, f18, e118, No], [5, e118, e218, No]];next set=[m18];stable=Yes

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m18];symbol=b18;ALL matching ORIGINAL edge rows=[[3, m18, f18]]

direct symbol targets=[f18];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[f18];previous set=[f18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, No], [1, s18, d18, No], [4, f18, e118, Yes], [5, e118, e218, No]];next set=[e118, f18];stable=No

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=1;seed set=[f18];previous set=[e118, f18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, No], [1, s18, d18, No], [4, f18, e118, Yes], [5, e118, e218, Yes]];next set=[e118, e218, f18];stable=No

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=2;seed set=[f18];previous set=[e118, e218, f18]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s18, m18, No], [1, s18, d18, No], [4, f18, e118, Yes], [5, e118, e218, Yes]];next set=[e118, e218, f18];stable=Yes

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=0;previous configs=[[s18, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]]];next configs=[[s18, 0], [m18, 0], [d18, 0]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=1;previous configs=[[s18, 0], [m18, 0], [d18, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=2;previous configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]], [[m18, 1], 2, [m18, 2]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=3;previous configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]], [[m18, 1], 2, [m18, 2]], [[m18, 2], 3, [f18, 3]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=4;previous configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]], [[m18, 1], 2, [m18, 2]], [[m18, 2], 3, [f18, 3]], [[f18, 3], 4, [e118, 3]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3], [e118, 3]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=5;previous configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3], [e118, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]], [[m18, 1], 2, [m18, 2]], [[m18, 2], 3, [f18, 3]], [[f18, 3], 4, [e118, 3]], [[e118, 3], 5, [e218, 3]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3], [e118, 3], [e218, 3]];stable=No

ORIGINAL word=[a18, a18, b18];direct NFA configuration stage=6;previous configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3], [e118, 3], [e218, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s18, 0], 0, [m18, 0]], [[s18, 0], 1, [d18, 0]], [[m18, 0], 2, [m18, 1]], [[m18, 1], 2, [m18, 2]], [[m18, 2], 3, [f18, 3]], [[f18, 3], 4, [e118, 3]], [[e118, 3], 5, [e218, 3]]];next configs=[[s18, 0], [m18, 0], [d18, 0], [m18, 1], [m18, 2], [f18, 3], [e118, 3], [e218, 3]];stable=Yes

ORIGINAL word=[a18, a18, b18];ALL DFA prefix subsets=[[d18, m18, s18], [m18], [m18], [e118, e218, f18]];independent direct NFA prefix subsets=[[d18, m18, s18], [m18], [m18], [e118, e218, f18]]

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

([a18, a18, b18], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], 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.