Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. 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. Given subset S=[m1, d1]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).
G001 — Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. 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. Given subset S=[m1, d1]. Give (subset, NFA_final_states, intersection, accepting_subset, 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.
Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.
final set normalized UTF16=[f1];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model
Conditions: none
This model differs from the earlier NFA node:transition triples are unique and finals are sorted.Subsets must be actual carrier subsets;do not import old duplicate/final-order conventions.
EXACT ORIGINAL supplied S=[d1, m1];normalized SAME-machine final set=[f1]
After
actual intersection=[];accepting_subset=No;NO EPS closure inserted
Conditions: none
This is the definition of an accepting DFA subset,not NFA empty-word acceptance from an arbitrary unclosed S.The empty subset is a legal rejecting dead state.
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
([d1, m1], [f1], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)
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.