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Question 23972: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

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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R0046-23972 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0097

Original question and givens

Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. 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. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

  • G001 — Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. 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. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).

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-full-word-trace-invariant;original snapshot and mathematics retained

After

explicit ordered source output fields=[input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary]

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=[s21, m21, f21, e121, e221, d21],alphabet=[a21, b21],start=s21;ORIGINAL transition order=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]

After

final set normalized UTF16=[f21];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 whole-word start:stage=0;seed set=[s21];previous set=[s21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, Yes], [1, s21, d21, Yes], [4, f21, e121, No], [5, e121, e221, No]];next set=[d21, m21, s21];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 whole-word start:stage=1;seed set=[s21];previous set=[d21, m21, s21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, Yes], [1, s21, d21, Yes], [4, f21, e121, No], [5, e121, e221, No]];next set=[d21, m21, s21];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

S005 · implies

Before

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d21, m21, s21];symbol=a21;ALL matching ORIGINAL edge rows=[[2, m21, m21]]

After

direct symbol targets=[m21];NO EPS preclosure inserted into S;now close ONLY these targets

Conditions: none

An arbitrary supplied S need not be EPS-closed;preclosing it would change this operation.Apply move(S,a) and then close its targets.Empty targets yield the legal empty dead state.

Step identifiers and dependencies

Rule: subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · Uses: G001, S004

S006 · implies

Before

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m21];previous set=[m21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, No], [5, e121, e221, No]];next set=[m21];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, S005

S007 · implies

Before

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m21];symbol=b21;ALL matching ORIGINAL edge rows=[[3, m21, f21]]

After

direct symbol targets=[f21];NO EPS preclosure inserted into S;now close ONLY these targets

Conditions: none

An arbitrary supplied S need not be EPS-closed;preclosing it would change this operation.Apply move(S,a) and then close its targets.Empty targets yield the legal empty dead state.

Step identifiers and dependencies

Rule: subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · Uses: G001, S006

S008 · implies

Before

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[f21];previous set=[f21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, No]];next set=[e121, f21];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, S007

S009 · implies

Before

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=1;seed set=[f21];previous set=[e121, f21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];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, S008

S010 · implies

Before

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=2;seed set=[f21];previous set=[e121, e221, f21]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s21, m21, No], [1, s21, d21, No], [4, f21, e121, Yes], [5, e121, e221, Yes]];next set=[e121, e221, f21];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, S009

S011 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=0;previous configs=[[s21, 0]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]]];next configs=[[s21, 0], [m21, 0], [d21, 0]];stable=No

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S010

S012 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=1;previous configs=[[s21, 0], [m21, 0], [d21, 0]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1]];stable=No

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S011

S013 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=2;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2]];stable=No

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S012

S014 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=3;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2]];stable=No

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S013

S015 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=4;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]], [[e121, 2], 5, [e221, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]];stable=No

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S014

S016 · implies

Before

ORIGINAL word=[a21, b21];direct NFA configuration stage=5;previous configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s21, 0], 0, [m21, 0]], [[s21, 0], 1, [d21, 0]], [[m21, 0], 2, [m21, 1]], [[m21, 1], 3, [f21, 2]], [[f21, 2], 4, [e121, 2]], [[e121, 2], 5, [e221, 2]]];next configs=[[s21, 0], [m21, 0], [d21, 0], [m21, 1], [f21, 2], [e121, 2], [e221, 2]];stable=Yes

Conditions: none

Compute all reachable configurations in the finite state×input-position graph.Do not replace all states by one chosen run or simply repeat the same DFA routine as a different observation.

Step identifiers and dependencies

Rule: subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · Uses: G001, S015

S017 · implies

Before

ORIGINAL word=[a21, b21];ALL DFA prefix subsets=[[d21, m21, s21], [m21], [e121, e221, f21]];independent direct NFA prefix subsets=[[d21, m21, s21], [m21], [e121, e221, f21]]

After

prefix_invariant=Yes;accepted=Yes;base is EPS closure of start;for ANY prefix,append one symbol move then finite EPS suffix to obtain EXACTLY all next reachable states

Conditions: none

General equivalence uses arbitrary-prefix induction with both inclusions:every DFA move has an NFA path witness,and any NFA path for the next prefix decomposes into prior prefix,symbol edge and EPS suffix.Intersection with finals then corresponds to a full accepting run.The fixed No only withholds general equivalence from finite word tests alone;it negates neither this invariant/acceptance nor the theorem.

Step identifiers and dependencies

Rule: subsetdfa_GENERAL_arbitrary_prefix_two_inclusions_acceptance_rule · Uses: G001, S016

Reverse checks

  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

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: 2dce1f6ae419945d3fb54e870c36e059efd2034ed2f45132184df6a4ca9e0f4c
  • candidate record SHA-256: fae6727e0cf6fdd9e84fa5f6794859731d203835f8ad8a22f954f58384ed1260
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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