Question 23962: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

Original question and givens

Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. 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=[a11, a11, a11, b11]. 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=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. 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=[a11, a11, a11, b11]. 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=[s11, m11, f11, e111, e211, d11],alphabet=[a11, b11],start=s11;ORIGINAL transition order=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]

final set normalized UTF16=[f11];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s11];previous set=[s11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, Yes], [1, s11, d11, Yes], [4, f11, e111, No], [5, e111, e211, No]];next set=[d11, m11, s11];stable=No

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s11];previous set=[d11, m11, s11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, Yes], [1, s11, d11, Yes], [4, f11, e111, No], [5, e111, e211, No]];next set=[d11, m11, s11];stable=Yes

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d11, m11, s11];symbol=a11;ALL matching ORIGINAL edge rows=[[2, m11, m11]]

direct symbol targets=[m11];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m11];previous set=[m11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, No], [5, e111, e211, No]];next set=[m11];stable=Yes

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m11];symbol=a11;ALL matching ORIGINAL edge rows=[[2, m11, m11]]

direct symbol targets=[m11];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m11];previous set=[m11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, No], [5, e111, e211, No]];next set=[m11];stable=Yes

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m11];symbol=a11;ALL matching ORIGINAL edge rows=[[2, m11, m11]]

direct symbol targets=[m11];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[m11];previous set=[m11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, No], [5, e111, e211, No]];next set=[m11];stable=Yes

Subset DFA ORIGINAL prefix step3:EXACT ORIGINAL supplied S=[m11];symbol=b11;ALL matching ORIGINAL edge rows=[[3, m11, f11]]

direct symbol targets=[f11];NO EPS preclosure inserted into S;now close ONLY these targets

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=0;seed set=[f11];previous set=[f11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, Yes], [5, e111, e211, No]];next set=[e111, f11];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=1;seed set=[f11];previous set=[e111, f11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, Yes], [5, e111, e211, Yes]];next set=[e111, e211, f11];stable=No

Subset DFA ORIGINAL prefix step3 post-symbol EPS closure:stage=2;seed set=[f11];previous set=[e111, e211, f11]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s11, m11, No], [1, s11, d11, No], [4, f11, e111, Yes], [5, e111, e211, Yes]];next set=[e111, e211, f11];stable=Yes

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=0;previous configs=[[s11, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]]];next configs=[[s11, 0], [m11, 0], [d11, 0]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=1;previous configs=[[s11, 0], [m11, 0], [d11, 0]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=2;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=3;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]], [[m11, 2], 2, [m11, 3]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=4;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]], [[m11, 2], 2, [m11, 3]], [[m11, 3], 3, [f11, 4]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=5;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]], [[m11, 2], 2, [m11, 3]], [[m11, 3], 3, [f11, 4]], [[f11, 4], 4, [e111, 4]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4], [e111, 4]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=6;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4], [e111, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]], [[m11, 2], 2, [m11, 3]], [[m11, 3], 3, [f11, 4]], [[f11, 4], 4, [e111, 4]], [[e111, 4], 5, [e211, 4]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4], [e111, 4], [e211, 4]];stable=No

ORIGINAL word=[a11, a11, a11, b11];direct NFA configuration stage=7;previous configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4], [e111, 4], [e211, 4]]

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s11, 0], 0, [m11, 0]], [[s11, 0], 1, [d11, 0]], [[m11, 0], 2, [m11, 1]], [[m11, 1], 2, [m11, 2]], [[m11, 2], 2, [m11, 3]], [[m11, 3], 3, [f11, 4]], [[f11, 4], 4, [e111, 4]], [[e111, 4], 5, [e211, 4]]];next configs=[[s11, 0], [m11, 0], [d11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4], [e111, 4], [e211, 4]];stable=Yes

ORIGINAL word=[a11, a11, a11, b11];ALL DFA prefix subsets=[[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]];independent direct NFA prefix subsets=[[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]]

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

([a11, a11, a11, b11], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], 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.