OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 24706: A Finite Configuration Trace Has Only Its Declared Input and Budget Scope

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

Return to catalogue page 124

← Question derivation index · Original question batch

R0048-24706 · q-20261015-advance-20261113-r05-turing-machine-decision-model-0031

Original question and givens

Use deterministic single-tape states=[q05, q15, qa5, qr5], input_alphabet=[05], tape_alphabet=[05, X5, _5], blank=_5, start=q05, accept=qa5, reject=qr5, transitions=[[q05, 05, q15, X5, R], [q05, _5, qr5, _5, S], [q15, _5, qa5, _5, L]]. 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. Input=[05], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary).

  • G001 — Use deterministic single-tape states=[q05, q15, qa5, qr5], input_alphabet=[05], tape_alphabet=[05, X5, _5], blank=_5, start=q05, accept=qa5, reject=qr5, transitions=[[q05, 05, q15, X5, R], [q05, _5, qr5, _5, S], [q15, _5, qa5, _5, L]]. 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. Input=[05], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, 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.

Step-by-step derivation

S001 · implies

Before

source roles = [tm-exact-step-trace]

After

derived = [[input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary]]

Conditions: none

List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[q05, q15, qa5, qr5], [05], [05, X5, _5], _5, q05, qa5, qr5, [[q05, 05, q15, X5, R], [q05, _5, qr5, _5, S], [q15, _5, qa5, _5, L]]]

After

derived = [[q05, q15, qa5, qr5], [05], [05, X5, _5], [[q05, _5, qr5, _5, S], [q05, 05, q15, X5, R], [q15, _5, qa5, _5, L]], Yes]

Conditions: none

All three lists are nonempty and unique;input is a blank-free tape subset,but state and alphabet labels may overlap.Source NUL-joined(from,read) keys are unique,and halting states have no outgoing rules.A partial table is legal,not filled with rejection;transition collation and UTF16 set order differ.

Step identifiers and dependencies

Rule: TM_complete_signature_partial_transition_table_allowed · Uses: G001, S001

S003 · implies

Before

source roles = [[05], q05, _5]

After

derived = [[q05, 0, [[0, 05]]]]

Conditions: none

Initially head0,input occupies integer cells from0,and every other cell is blank.The sparse record contains nonblank cells only;negative indices are legal.

Step identifiers and dependencies

Rule: TM_exact_initial_configuration_blank_elsewhere · Uses: G001, S002

S004 · calculation

Before

TM exact integer head move0 := sum(0, 1)

After

TM exact integer head move0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,checking nonzero denominators without floats.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [0, [q05, 0, [[0, 05]]], 05, [q05, 05, q15, X5, R]]

After

derived = [step, [q15, 1, [[0, X5]]]]

Conditions: none

Write at the old head first,delete a blank cell,then update state and move exactly L/R/S according to the source rule.

Step identifiers and dependencies

Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S004

S006 · calculation

Before

TM exact integer head move1 := sum(1, -1)

After

TM exact integer head move1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,checking nonzero denominators without floats.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [1, [q15, 1, [[0, X5]]], _5, [q15, _5, qa5, _5, L]]

After

derived = [step, [qa5, 0, [[0, X5]]]]

Conditions: none

Write at the old head first,delete a blank cell,then update state and move exactly L/R/S according to the source rule.

Step identifiers and dependencies

Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S006

S008 · implies

Before

source roles = [[05], 2, [[q05, 0, [[0, 05]]], [q15, 1, [[0, X5]]], [qa5, 0, [[0, X5]]]], [step, step]]

After

derived = [qa5, accept, No]

Conditions: none

Only actual halting states accept or reject.Exhaustion or stuck is neither ordinary rejection nor a proof of infinite running.One finite input trace supplies no all-input decider certificate.

Step identifiers and dependencies

Rule: TM_finite_trace_input_budget_scope_NOT_global_decider · Uses: G001, S007

Reverse checks

  • 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 — Recompute TM exact integer head move1 with exact rational arithmetic, not rounded decimals.
  • 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 — Recompute TM exact integer head move0 with exact rational arithmetic, not rounded decimals.
  • 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

([05], 2, [[q05, 0, [[0, 05]]], [q15, 1, [[0, X5]]], [qa5, 0, [[0, X5]]]], [step, step], qa5, accept, No, finite_trace_is_input_local_and_does_not_certify_a_global_decider)

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: 5b82846dd4e53a6606ddfc007d51ce7fe3a4dfdb8c0f1a03bb1cd0a0d88104fe
  • candidate record SHA-256: 11b24038b880dc90ae9f6e3379f91024a3b479e094aa1527e3c496c9a88b8dcf
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON