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R0033-17233 · q-20261014-advance-20261102-r07-regular-language-0083
Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=9. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Ordered test_words=[[], [1, 1, 1, 0, 0]]. Give (tested_words,sufficient?,finite_test_suite_not_regular_language_or_equivalence_certificate), with each tested row [word,accepted?].
G001 — Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=9. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Ordered test_words=[[], [1, 1, 1, 0, 0]]. Give (tested_words,sufficient?,finite_test_suite_not_regular_language_or_equivalence_certificate), with each tested row [word,accepted?].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.
Before
mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(0+0) mod9=0
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001
Before
mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(0+1) mod9=1
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S001
Before
mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(1+0) mod9=1
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S002
Before
mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(1+1) mod9=2
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S003
Before
mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(2+0) mod9=2
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S004
Before
mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(2+1) mod9=3
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S005
Before
mod-count state=3;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(3+0) mod9=3
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S006
Before
mod-count state=3;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(3+1) mod9=4
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S007
Before
mod-count state=4;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(4+0) mod9=4
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S008
Before
mod-count state=4;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(4+1) mod9=5
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S009
Before
mod-count state=5;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(5+0) mod9=5
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S010
Before
mod-count state=5;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(5+1) mod9=6
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S011
Before
mod-count state=6;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(6+0) mod9=6
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S012
Before
mod-count state=6;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(6+1) mod9=7
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S013
Before
mod-count state=7;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(7+0) mod9=7
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S014
Before
mod-count state=7;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(7+1) mod9=8
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S015
Before
mod-count state=8;original symbol=0;target symbol=1;indicator=0;modulus=9
After
next state=(8+0) mod9=8
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S016
Before
mod-count state=8;original symbol=1;target symbol=1;indicator=1;modulus=9
After
next state=(8+1) mod9=0
Conditions: none
Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S017
Before
states Fin(9);complete typed table=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 0]];start0;accepting residue=0
After
empty prefix count=0 equals start;append any symbol adds exactly its target indicator to BOTH count and state modulo the modulus
Conditions: none
Both count and state start at0.Appending any symbol updates both by the same indicator,so induction preserves state equal to target-symbol count modulo m for every prefix of every word.
Rule: regular_mod_count_all_prefix_induction_base_step · Uses: G001, S018
Before
for EVERY finite word,state equals target-symbol count mod9;accepting set=[0]
After
accepted iff count congruent to0;whole declared language has this complete DFA certificate
Conditions: none
Final-state membership in the specified residue characterizes the entire declared language.This uses the complete DFA and universal invariant,not finite tests,and adds no regex,epsilon moves,minimization or pumping lemma.
Rule: regular_mod_count_language_characterized_not_test_suite · Uses: G001, S019
Before
original test0:original ordered word=[];start=0
After
trace before consuming symbols=[0]
Conditions: none
Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide.
Rule: regular_word_initial_state_empty_word_retained · Uses: G001, S020
Before
original test0 exact target-symbol indicator count := sum()
After
original test0 exact target-symbol indicator count = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
original test0:original word=[];exact target count=0;final state=0;accepting residue=0
After
accepted=Yes;final state equals count modulo9
Conditions: none
Independently count original-symbol indicators and cross-check the final trace state.An empty word has empty sum0 and no invented transition.
Rule: regular_selected_word_count_reverse_check · Uses: G001, S022
Before
original test1:original ordered word=[1, 1, 1, 0, 0];start=0
After
trace before consuming symbols=[0]
Conditions: none
Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide.
Rule: regular_word_initial_state_empty_word_retained · Uses: G001, S023
Before
original test1:word position=0;symbol=1;current state=0;complete original row=[0, 1]
After
next state=1;ordered prefix trace=[0, 1]
Conditions: none
Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.
Rule: regular_every_symbol_source_bound_transition · Uses: G001, S024
Before
original test1:word position=1;symbol=1;current state=1;complete original row=[1, 2]
After
next state=2;ordered prefix trace=[0, 1, 2]
Conditions: none
Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.
Rule: regular_every_symbol_source_bound_transition · Uses: G001, S025
Before
original test1:word position=2;symbol=1;current state=2;complete original row=[2, 3]
After
next state=3;ordered prefix trace=[0, 1, 2, 3]
Conditions: none
Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.
Rule: regular_every_symbol_source_bound_transition · Uses: G001, S026
Before
original test1:word position=3;symbol=0;current state=3;complete original row=[3, 4]
After
next state=3;ordered prefix trace=[0, 1, 2, 3, 3]
Conditions: none
Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.
Rule: regular_every_symbol_source_bound_transition · Uses: G001, S027
Before
original test1:word position=4;symbol=0;current state=3;complete original row=[3, 4]
After
next state=3;ordered prefix trace=[0, 1, 2, 3, 3, 3]
Conditions: none
Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state.
Rule: regular_every_symbol_source_bound_transition · Uses: G001, S028
Before
original test1 exact target-symbol indicator count := sum(1, 1, 1, 0, 0)
After
original test1 exact target-symbol indicator count = 3
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
original test1:original word=[1, 1, 1, 0, 0];exact target count=3;final state=3;accepting residue=0
After
accepted=No;final state equals count modulo9
Conditions: none
Independently count original-symbol indicators and cross-check the final trace state.An empty word has empty sum0 and no invented transition.
Rule: regular_selected_word_count_reverse_check · Uses: G001, S030
Before
original finite test rows in supplied order=[[[], Yes], [[1, 1, 1, 0, 0], No]]
After
test-suite sufficiency=No;finite_test_suite_not_regular_language_or_equivalence_certificate
Conditions: none
The test list may repeat words or be empty;retain supplied order.The No rejects inference from tests alone to full-language regularity/equivalence.It does not deny the separate explicit DFA/invariant certificate for this model.
Rule: regular_finite_tests_not_universal_language_certificate · Uses: G001, S031
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Recompute original test1 exact target-symbol indicator count with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute original test0 exact target-symbol indicator count with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([[[], Yes], [[1, 1, 1, 0, 0], No]], No, finite_test_suite_not_regular_language_or_equivalence_certificate)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a71cc44e248f65e0ce387323f2364e2960976f39218d38f2efb735d3d12d7cd6candidate record SHA-256: 28c17b8b802432c227294db36c06683fd3a45f2182e2a0a95c82301ac5311420semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93Lean: NOT_RUN · human review: NOT_PERFORMED