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Question 23930: The DFA Start State Is the Epsilon Closure of the NFA Start Singleton

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0046-23930 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0005

Original question and givens

Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. 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=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. 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.

Step-by-step derivation

S001 · implies

Before

source operation=subsetdfa-epsilon-start;original snapshot and mathematics retained

After

explicit ordered source output fields=[NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate]

Conditions: none

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.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL states=[s4, m4, f4, e14, e24, d4],alphabet=[a4, b4],start=s4;ORIGINAL transition order=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]

After

final set normalized UTF16=[f4];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.

Step identifiers and dependencies

Rule: subsetdfa_qualified_NFA_signature_DUPLICATE_edges_REJECTED_here · Uses: G001, S001

S003 · implies

Before

Subset DFA ORIGINAL start:stage=0;seed set=[s4];previous set=[s4]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s4, m4, Yes], [1, s4, d4, Yes], [4, f4, e14, No], [5, e14, e24, No]];next set=[d4, m4, s4];stable=No

Conditions: none

Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input.

Step identifiers and dependencies

Rule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S002

S004 · implies

Before

Subset DFA ORIGINAL start:stage=1;seed set=[s4];previous set=[d4, m4, s4]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s4, m4, Yes], [1, s4, d4, Yes], [4, f4, e14, No], [5, e14, e24, No]];next set=[d4, m4, s4];stable=Yes

Conditions: none

Each addition has a finite EPS predecessor path;every EPS-closed seed superset contains each stage.Finiteness gives the complete least closure,without consuming input.

Step identifiers and dependencies

Rule: subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · Uses: G001, S003

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

(s4, [d4, m4, s4], 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 1e86ed5559b3b90a31d5e42d8b8576e8369dd9ec96a5f8f2059d1c4b0f392c27
  • candidate record SHA-256: 8bdd335fb22c4c3d130b920caf83c80c72d615bd2ccdf6c0c94ba0ab8910ffdd
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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