Question 23931: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

Original question and givens

Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. 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=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. 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=[s5, m5, f5, e15, e25, d5],alphabet=[a5, b5],start=s5;ORIGINAL transition order=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]

final set normalized UTF16=[f5];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

Subset DFA ORIGINAL start:stage=0;seed set=[s5];previous set=[s5]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s5, m5, Yes], [1, s5, d5, Yes], [4, f5, e15, No], [5, e15, e25, No]];next set=[d5, m5, s5];stable=No

Subset DFA ORIGINAL start:stage=1;seed set=[s5];previous set=[d5, m5, s5]

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s5, m5, Yes], [1, s5, d5, Yes], [4, f5, e15, No], [5, e15, e25, No]];next set=[d5, m5, s5];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

(s5, [d5, m5, s5], 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.