Question 23938: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Original question and givens

Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. 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=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. 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=[s12, m12, f12, e112, e212, d12],alphabet=[a12, b12],start=s12;ORIGINAL transition order=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]

final set normalized UTF16=[f12];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL start:stage=0;seed set=[s12];previous set=[s12]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s12, m12, Yes], [1, s12, d12, Yes], [4, f12, e112, No], [5, e112, e212, No]];next set=[d12, m12, s12];stable=No

Subset DFA ORIGINAL start:stage=1;seed set=[s12];previous set=[d12, m12, s12]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s12, m12, Yes], [1, s12, d12, Yes], [4, f12, e112, No], [5, e112, e212, No]];next set=[d12, m12, s12];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

(s12, [d12, m12, s12], 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.