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Question 9676: Legal Finite Trace Prefix and Post-Halt Continuation Guard

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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← Question derivation index · Original question batch

R0018-09676 · q-20261013-advance-20261019-r04-computational-model-0026

Original question and givens

Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. trace=[0, 1, 2, 5]. Give (trace, per-state type mask, nonempty?, first state in Init?, per-adjacent-pair raw Step flags, continued after a halt?, legal prefix?). A legal prefix need not end at Halt.

  • G001 — Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. trace=[0, 1, 2, 5]. Give (trace, per-state type mask, nonempty?, first state in Init?, per-adjacent-pair raw Step flags, continued after a halt?, legal prefix?). A legal prefix need not end at Halt.

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

n=6;raw Init=[0];raw Halt=[5];raw Step=[[0, 1], [1, 2], [2, 5]]

After

Init=[0];Halt=[5];canonical raw Step=[[0, 1], [1, 2], [2, 5]]

Conditions: none

Type the complete finite data and deduplicate directed edges.Retain raw outgoing halt edges as data,but execution stops at halt.Stuck is not successful halt,and one prefix proves no general program termination.

Step identifiers and dependencies

Rule: finite_transition_normalization · Uses: G001

S002 · implies

Before

trace position=0;raw state=0;domain Fin(6)

After

typed=Yes

Conditions: none

Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index.

Step identifiers and dependencies

Rule: trace_each_state_type_guard · Uses: G001, S001

S003 · implies

Before

trace position=1;raw state=1;domain Fin(6)

After

typed=Yes

Conditions: none

Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index.

Step identifiers and dependencies

Rule: trace_each_state_type_guard · Uses: G001, S002

S004 · implies

Before

trace position=2;raw state=2;domain Fin(6)

After

typed=Yes

Conditions: none

Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index.

Step identifiers and dependencies

Rule: trace_each_state_type_guard · Uses: G001, S003

S005 · implies

Before

trace position=3;raw state=5;domain Fin(6)

After

typed=Yes

Conditions: none

Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index.

Step identifiers and dependencies

Rule: trace_each_state_type_guard · Uses: G001, S004

S006 · implies

Before

trace pair index=0;pair=[0, 1];both typed=Yes

After

raw Step edge=Yes;continues from Halt=No

Conditions: none

Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution.

Step identifiers and dependencies

Rule: trace_edge_and_halt_continuation_separate · Uses: G001, S005

S007 · implies

Before

trace pair index=1;pair=[1, 2];both typed=Yes

After

raw Step edge=Yes;continues from Halt=No

Conditions: none

Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution.

Step identifiers and dependencies

Rule: trace_edge_and_halt_continuation_separate · Uses: G001, S006

S008 · implies

Before

trace pair index=2;pair=[2, 5];both typed=Yes

After

raw Step edge=Yes;continues from Halt=No

Conditions: none

Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution.

Step identifiers and dependencies

Rule: trace_edge_and_halt_continuation_separate · Uses: G001, S007

S009 · implies

Before

nonempty=Yes;initial in Init=Yes;state masks=[Yes, Yes, Yes, Yes];edge flags=[Yes, Yes, Yes];continued after halt=No

After

legal finite prefix=Yes

Conditions: none

A legal prefix is nonempty,typed,starts in Init,follows each edge and never continues after halt.It need not yet reach Halt and does not prove all branches terminate.

Step identifiers and dependencies

Rule: finite_trace_prefix_not_termination · Uses: G001, S008

Reverse checks

  • 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, 5], [Yes, Yes, Yes, Yes], Yes, Yes, [Yes, Yes, Yes], No, Yes)

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: 8ca0d93e5357dbddf95e9079003e7782216d034aa8ab559685b2aead8a9276ad
  • candidate record SHA-256: 555e16dd3934189edde86b2feb7f58626042ff7a3a1163752520a62ca45a0c91
  • semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
  • question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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