Question 27330: One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity

Original question and givens

Use mode=single_split, complete model={"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).

• G001 — Use mode=single_split, complete model={"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_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-pumping-lemma-regular-languages

Step-by-step derivation

source roles = [pump-boundary]

derived = [[mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary]]

source roles = [5, [a, a, a, a, a, b], 4, 1]

derived = [1, Yes, No]

Reverse checks

• S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
• S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(single_split, 5, [a, a, a, a, a, b], 4, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)

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.