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Question 17271: A State Invariant Covers Every Word, Not Just Test Cases

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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R0033-17271 · q-20261014-advance-20261102-r07-regular-language-0021

Original question and givens

Use alphabet=Fin(3)={0,…,2} and the language whose number of occurrences of target_symbol=1 is congruent to residue=1 modulo modulus=3. 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. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order.

  • G001 — Use alphabet=Fin(3)={0,…,2} and the language whose number of occurrences of target_symbol=1 is congruent to residue=1 modulo modulus=3. 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. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order.

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

mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=3

After

next state=(0+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001

S002 · implies

Before

mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=3

After

next state=(0+1) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S001

S003 · implies

Before

mod-count state=0;original symbol=2;target symbol=1;indicator=0;modulus=3

After

next state=(0+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S002

S004 · implies

Before

mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=3

After

next state=(1+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S003

S005 · implies

Before

mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=3

After

next state=(1+1) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S004

S006 · implies

Before

mod-count state=1;original symbol=2;target symbol=1;indicator=0;modulus=3

After

next state=(1+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S005

S007 · implies

Before

mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=3

After

next state=(2+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S006

S008 · implies

Before

mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=3

After

next state=(2+1) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S007

S009 · implies

Before

mod-count state=2;original symbol=2;target symbol=1;indicator=0;modulus=3

After

next state=(2+0) mod3=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.

Step identifiers and dependencies

Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S008

S010 · implies

Before

states Fin(3);complete typed table=[[0, 1, 0], [1, 2, 1], [2, 0, 2]];start0;accepting residue=1

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.

Step identifiers and dependencies

Rule: regular_mod_count_all_prefix_induction_base_step · Uses: G001, S009

S011 · implies

Before

for EVERY finite word,state equals target-symbol count mod3;accepting set=[1]

After

accepted iff count congruent to1;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.

Step identifiers and dependencies

Rule: regular_mod_count_language_characterized_not_test_suite · Uses: G001, S010

Reverse checks

  • 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

([0, 1, 2], [[0, 1, 0], [1, 2, 1], [2, 0, 2]], 0, [1], Yes, state_equals_target_symbol_count_mod_modulus_for_all_prefixes)

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: c251d76395e4d2bc8c30486f8b0c1c580b009b711ada7550e73471abd5c9d471
  • candidate record SHA-256: a86a0436d58dbea321e162785d7cf7bd5c02c4f667663b7a98c031a3a046c0e7
  • semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3
  • question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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