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Question 23933: 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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R0046-23933 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0008

Original question and givens

Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. 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=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. 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=[s7, m7, f7, e17, e27, d7],alphabet=[a7, b7],start=s7;ORIGINAL transition order=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]

After

final set normalized UTF16=[f7];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=[s7];previous set=[s7]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, Yes], [1, s7, d7, Yes], [4, f7, e17, No], [5, e17, e27, No]];next set=[d7, m7, s7];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=[s7];previous set=[d7, m7, s7]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s7, m7, Yes], [1, s7, d7, Yes], [4, f7, e17, No], [5, e17, e27, No]];next set=[d7, m7, s7];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

(s7, [d7, m7, s7], 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: 61f04f618f2a9cb4dd104a2188a9417a80e1061c5d32d3e37115e748505d9f85
  • candidate record SHA-256: 0c3036bf7c9b37f088ac113a492e50d37ed22d78c9d11c063b9953764c6fdd44
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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