Question 23946: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Original question and givens

Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. 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=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. 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=[s20, m20, f20, e120, e220, d20],alphabet=[a20, b20],start=s20;ORIGINAL transition order=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]

final set normalized UTF16=[f20];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL start:stage=0;seed set=[s20];previous set=[s20]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s20, m20, Yes], [1, s20, d20, Yes], [4, f20, e120, No], [5, e120, e220, No]];next set=[d20, m20, s20];stable=No

Subset DFA ORIGINAL start:stage=1;seed set=[s20];previous set=[d20, m20, s20]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s20, m20, Yes], [1, s20, d20, Yes], [4, f20, e120, No], [5, e120, e220, No]];next set=[d20, m20, s20];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

(s20, [d20, m20, s20], 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.