# Question 9665: Complete Reachability with Halt Outgoing Edges Truncated

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0018-09665 · q-20261013-advance-20261019-r04-computational-model-0015



## 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], [3, 4], [4, 5], [2, 1]]. 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. Give (n, normalized Init, normalized Halt, normalized raw Step, all reachable states). Include zero-step reachability.


- 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], [3, 4], [4, 5], [2, 1]]. 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. Give (n, normalized Init, normalized Halt, normalized raw Step, all reachable states). Include zero-step reachability.





## 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-computational-model](https://ocme.evemisslab.com/en/mko/mko-computational-model/)





## Step-by-step derivation

S001 · impliesBefore

n=6;raw Init=[0];raw Halt=[5];raw Step=[[0, 1], [1, 2], [2, 5], [3, 4], [4, 5], [2, 1]]

After

Init=[0];Halt=[5];canonical raw Step=[[0, 1], [1, 2], [2, 1], [2, 5], [3, 4], [4, 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 dependenciesRule: finite_transition_normalization · Uses: G001





S002 · impliesBefore

Init=[0]

After

zero-step reachable=[0]

Conditions: none

Every initial state is reachable in zero steps,including an initial halt.An empty Init has no reachable states.

Step identifiers and dependenciesRule: zero_step_reachability · Uses: G001, S001





S003 · impliesBefore

depth=0;state=0;halt=No;raw successors=[1]

After

expanded successors=[1];newly reached=[1]

Conditions: none

Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist.

Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S002





S004 · impliesBefore

completed frontier=[0];depth=0

After

next frontier=[1];all reached=[0, 1]

Conditions: none

The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff.

Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S003





S005 · impliesBefore

depth=1;state=1;halt=No;raw successors=[2]

After

expanded successors=[2];newly reached=[2]

Conditions: none

Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist.

Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S004





S006 · impliesBefore

completed frontier=[1];depth=1

After

next frontier=[2];all reached=[0, 1, 2]

Conditions: none

The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff.

Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S005





S007 · impliesBefore

depth=2;state=2;halt=No;raw successors=[1, 5]

After

expanded successors=[1, 5];newly reached=[5]

Conditions: none

Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist.

Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S006





S008 · impliesBefore

completed frontier=[2];depth=2

After

next frontier=[5];all reached=[0, 1, 2, 5]

Conditions: none

The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff.

Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S007





S009 · impliesBefore

depth=3;state=5;halt=Yes;raw successors=[]

After

expanded successors=[];newly reached=[]

Conditions: none

Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist.

Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S008





S010 · impliesBefore

completed frontier=[5];depth=3

After

next frontier=[];all reached=[0, 1, 2, 5]

Conditions: none

The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff.

Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S009









## Reverse checks


- 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

(6, [0], [5], [[0, 1], [1, 2], [2, 1], [2, 5], [3, 4], [4, 5]], [0, 1, 2, 5])

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: 06c7bfad77ecbe5712e6b2e0ee00648990c2ea26f79e166dfdb81093625477c7
- candidate record SHA-256: a588f3549cab759522f861bc4175cc9b1cc4c8c1649fd1c152711c8341f12773
- semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
- question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r04-computational-model-0015.json)