Question 23926: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Original question and givens

Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. 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=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. 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=[s0, m0, f0, e10, e20, d0],alphabet=[a0, b0],start=s0;ORIGINAL transition order=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]

final set normalized UTF16=[f0];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL start:stage=0;seed set=[s0];previous set=[s0]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s0, m0, Yes], [1, s0, d0, Yes], [4, f0, e10, No], [5, e10, e20, No]];next set=[d0, m0, s0];stable=No

Subset DFA ORIGINAL start:stage=1;seed set=[s0];previous set=[d0, m0, s0]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s0, m0, Yes], [1, s0, d0, Yes], [4, f0, e10, No], [5, e10, e20, No]];next set=[d0, m0, s0];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

(s0, [d0, m0, s0], 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.