Question 23955: Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0046-23955 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0080

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. Supplied full input=[b4]. 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=[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. Supplied full input=[b4]. 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.

mko-subset-construction-nfa-to-dfa

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=[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 whole-word 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 whole-word 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

S005 · implies

Before

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d4, m4, s4];symbol=b4;ALL matching ORIGINAL edge rows=[[3, m4, f4]]

After

direct symbol targets=[f4];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=[f4];previous set=[f4]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s4, m4, No], [1, s4, d4, No], [4, f4, e14, Yes], [5, e14, e24, No]];next set=[e14, f4];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, S005

S007 · implies

Before

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=1;seed set=[f4];previous set=[e14, f4]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s4, m4, No], [1, s4, d4, No], [4, f4, e14, Yes], [5, e14, e24, Yes]];next set=[e14, e24, f4];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, S006

S008 · implies

Before

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=2;seed set=[f4];previous set=[e14, e24, f4]

After

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s4, m4, No], [1, s4, d4, No], [4, f4, e14, Yes], [5, e14, e24, Yes]];next set=[e14, e24, f4];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, S007

S009 · implies

Before

ORIGINAL word=[b4];direct NFA configuration stage=0;previous configs=[[s4, 0]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s4, 0], 0, [m4, 0]], [[s4, 0], 1, [d4, 0]]];next configs=[[s4, 0], [m4, 0], [d4, 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, S008

S010 · implies

Before

ORIGINAL word=[b4];direct NFA configuration stage=1;previous configs=[[s4, 0], [m4, 0], [d4, 0]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s4, 0], 0, [m4, 0]], [[s4, 0], 1, [d4, 0]], [[m4, 0], 3, [f4, 1]]];next configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 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, S009

S011 · implies

Before

ORIGINAL word=[b4];direct NFA configuration stage=2;previous configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s4, 0], 0, [m4, 0]], [[s4, 0], 1, [d4, 0]], [[m4, 0], 3, [f4, 1]], [[f4, 1], 4, [e14, 1]]];next configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1], [e14, 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, S010

S012 · implies

Before

ORIGINAL word=[b4];direct NFA configuration stage=3;previous configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1], [e14, 1]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s4, 0], 0, [m4, 0]], [[s4, 0], 1, [d4, 0]], [[m4, 0], 3, [f4, 1]], [[f4, 1], 4, [e14, 1]], [[e14, 1], 5, [e24, 1]]];next configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1], [e14, 1], [e24, 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=[b4];direct NFA configuration stage=4;previous configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1], [e14, 1], [e24, 1]]

After

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s4, 0], 0, [m4, 0]], [[s4, 0], 1, [d4, 0]], [[m4, 0], 3, [f4, 1]], [[f4, 1], 4, [e14, 1]], [[e14, 1], 5, [e24, 1]]];next configs=[[s4, 0], [m4, 0], [d4, 0], [f4, 1], [e14, 1], [e24, 1]];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, S012

S014 · implies

Before

ORIGINAL word=[b4];ALL DFA prefix subsets=[[d4, m4, s4], [e14, e24, f4]];independent direct NFA prefix subsets=[[d4, m4, s4], [e14, e24, f4]]

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, S013

Reverse checks

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

([b4], [[d4, m4, s4], [e14, e24, f4]], [[d4, m4, s4], [e14, e24, f4]], 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: f52fd87d48644a79aa103cd1d0b898934298829d656aa637700fe3d8269f8c35
candidate record SHA-256: 0b32aa47090174e445b93effdee4c4e8779c26423006a3d04afcd740f60570d3
semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
Lean: NOT_RUN · human review: NOT_PERFORMED

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