OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 23934: 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

Return to catalogue page 120

← Question derivation index · Original question batch

R0046-23934 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0009

Original question and givens

Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. 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=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. 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=[s8, m8, f8, e18, e28, d8],alphabet=[a8, b8],start=s8;ORIGINAL transition order=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]

After

final set normalized UTF16=[f8];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=[s8];previous set=[s8]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s8, m8, Yes], [1, s8, d8, Yes], [4, f8, e18, No], [5, e18, e28, No]];next set=[d8, m8, s8];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=[s8];previous set=[d8, m8, s8]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s8, m8, Yes], [1, s8, d8, Yes], [4, f8, e18, No], [5, e18, e28, No]];next set=[d8, m8, s8];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

(s8, [d8, m8, s8], 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: 0af19ece5ffed3de365c3085f43f3cbe84d0e23bb2a0425cce084904db2824ce
  • candidate record SHA-256: 130bb90e303e084dc201d737ab5417d21208f3a3530d92faa6f54119b075145c
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON