Question 24731: A Canonical Scan Schema Has an All-Input Ranking and Correct Answers

Original question and givens

Use deterministic single-tape states=[scan, accept, reject], input_alphabet=[a5, b5], tape_alphabet=[a5, b5, _], blank=_, start=scan, accept=accept, reject=reject, transitions=[[scan, a5, scan, a5, R], [scan, b5, accept, b5, S], [scan, _, reject, _, S]]. Each transition is [state,read,next,write,L/R/S]; keys [state,read] are unique and halting states have no outgoing transitions. Input symbols form a blank-free tape subset. The integer-indexed tape is blank at every unlisted cell, with configuration [state,head,sparse nonblank tape]; L may move left of0. Canonical answer transitions/tape cells retain the source sort convention. Target input symbol=b5. Give (states, input_alphabet, tape_alphabet, blank, start, accept, reject, transitions, target_symbol, ranking, decrease_rule, step_bound, schema_decider_certificate, certificate).

• G001 — Use deterministic single-tape states=[scan, accept, reject], input_alphabet=[a5, b5], tape_alphabet=[a5, b5, _], blank=_, start=scan, accept=accept, reject=reject, transitions=[[scan, a5, scan, a5, R], [scan, b5, accept, b5, S], [scan, _, reject, _, S]]. Each transition is [state,read,next,write,L/R/S]; keys [state,read] are unique and halting states have no outgoing transitions. Input symbols form a blank-free tape subset. The integer-indexed tape is blank at every unlisted cell, with configuration [state,head,sparse nonblank tape]; L may move left of0. Canonical answer transitions/tape cells retain the source sort convention. Target input symbol=b5. Give (states, input_alphabet, tape_alphabet, blank, start, accept, reject, transitions, target_symbol, ranking, decrease_rule, step_bound, schema_decider_certificate, certificate).

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-turing-machine-decision-model

Step-by-step derivation

source roles = [tm-global-decider-ranking]

derived = [[states, input_alphabet, tape_alphabet, blank, start, accept, reject, transitions, target_symbol, ranking, decrease_rule, step_bound, schema_decider_certificate, certificate]]

source roles = [[scan, accept, reject], [a5, b5], [a5, b5, _], _, scan, accept, reject, [[scan, a5, scan, a5, R], [scan, b5, accept, b5, S], [scan, _, reject, _, S]]]

derived = [[accept, reject, scan], [a5, b5], [_, a5, b5], [[scan, _, reject, _, S], [scan, a5, scan, a5, R], [scan, b5, accept, b5, S]], Yes]

source roles = [b5, No, [[accept, reject, scan], [a5, b5], [_, a5, b5], _, scan, accept, reject, [[scan, _, reject, _, S], [scan, a5, scan, a5, R], [scan, b5, accept, b5, S]]]]

derived = [[[accept, reject, scan], [a5, b5], [_, a5, b5], _, scan, accept, reject, [[scan, _, reject, _, S], [scan, a5, scan, a5, R], [scan, b5, accept, b5, S]]], Yes]

source roles = [b5, [a5, b5], [[scan, _, reject, _, S], [scan, a5, scan, a5, R], [scan, b5, accept, b5, S]]]

derived = [ANY legal w:scan head k<=|w|,first k symbols target-free,tape unchanged, scan rank=|w|-k+1;halting rank=0, nontarget R step lowers rank by1;target accept or blank reject lowers to0, EVERY word halts within |w|+1;accept iff target occurs]

Reverse checks

• 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

([accept, reject, scan], [a5, b5], [_, a5, b5], _, scan, accept, reject, [[scan, _, reject, _, S], [scan, a5, scan, a5, R], [scan, b5, accept, b5, S]], b5, ranking=unscanned_input_cells_plus_one, strictly_decreases_on_every_scan_step, |w|+1, Yes, global_termination_and_answer_correctness_certificate_for_all_words)

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.