Question 23945: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Original question and givens

Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. 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. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

• G001 — Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. 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. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).

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-epsilon-start;original snapshot and mathematics retained

explicit ordered source output fields=[NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate]

ORIGINAL states=[s19, m19, f19, e119, e219, d19],alphabet=[a19, b19],start=s19;ORIGINAL transition order=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]

final set normalized UTF16=[f19];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL start:stage=0;seed set=[s19];previous set=[s19]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s19, m19, Yes], [1, s19, d19, Yes], [4, f19, e119, No], [5, e119, e219, No]];next set=[d19, m19, s19];stable=No

Subset DFA ORIGINAL start:stage=1;seed set=[s19];previous set=[d19, m19, s19]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s19, m19, Yes], [1, s19, d19, Yes], [4, f19, e119, No], [5, e119, e219, No]];next set=[d19, m19, s19];stable=Yes

Reverse checks

• 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

(s19, [d19, m19, s19], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

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.