Question 9511: Inductiveness versus Merely Covering Reachable States

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0018-09511 · q-20261013-advance-20261019-r05-algorithm-correctness-0011

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], [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. Proposed invariant=[0, 1, 2, 5]. Give (invariant, contains Init?, every Step edge from an invariant nonhalt to outside the invariant, step-preserved?, inductive?, covers all reachable states?). Inspect all invariant states for preservation, including unreachable ones.

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], [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. Proposed invariant=[0, 1, 2, 5]. Give (invariant, contains Init?, every Step edge from an invariant nonhalt to outside the invariant, step-preserved?, inductive?, covers all reachable states?). Inspect all invariant states for preservation, including unreachable ones.

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-algorithm-correctness

Step-by-step derivation

S001 · implies

Before

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

After

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

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

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 dependencies

Rule: zero_step_reachability · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: reachability_halt_truncates_expansion · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: complete_finite_reachability_layer · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: reachability_halt_truncates_expansion · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: complete_finite_reachability_layer · Uses: G001, S005

S007 · implies

Before

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 dependencies

Rule: reachability_halt_truncates_expansion · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: complete_finite_reachability_layer · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: reachability_halt_truncates_expansion · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: complete_finite_reachability_layer · Uses: G001, S009

S011 · implies

Before

candidate invariant=[0, 1, 2, 5];Init=[0]

After

contains every initial state=Yes

Conditions: none

An inductive invariant must first contain every initial state,not just one selected initial point.

Step identifiers and dependencies

Rule: invariant_initial_obligation · Uses: G001, S010

S012 · implies

Before

raw edge=[0, 1];source in invariant=Yes;source halt=No

After

preservation obligation active=Yes;violates invariant=No

Conditions: none

Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation.

Step identifiers and dependencies

Rule: invariant_all_states_not_only_reachable · Uses: G001, S011

S013 · implies

Before

raw edge=[1, 2];source in invariant=Yes;source halt=No

After

preservation obligation active=Yes;violates invariant=No

Conditions: none

Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation.

Step identifiers and dependencies

Rule: invariant_all_states_not_only_reachable · Uses: G001, S012

S014 · implies

Before

raw edge=[2, 1];source in invariant=Yes;source halt=No

After

preservation obligation active=Yes;violates invariant=No

Conditions: none

Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation.

Step identifiers and dependencies

Rule: invariant_all_states_not_only_reachable · Uses: G001, S013

S015 · implies

Before

raw edge=[2, 5];source in invariant=Yes;source halt=No

After

preservation obligation active=Yes;violates invariant=No

Conditions: none

Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation.

Step identifiers and dependencies

Rule: invariant_all_states_not_only_reachable · Uses: G001, S014

S016 · implies

Before

raw edge=[3, 4];source in invariant=No;source halt=No

After

preservation obligation active=No;violates invariant=No

Conditions: none

Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation.

Step identifiers and dependencies

Rule: invariant_all_states_not_only_reachable · Uses: G001, S015

S017 · implies

Before

bad preservation edges=[];initial obligation=Yes;reachable=[0, 1, 2, 5]

After

step preserved=Yes;inductive=Yes;covers reachable=Yes

Conditions: none

Initial validity plus one-step preservation gives inductiveness.Compute reachable-state coverage separately;true coverage does not replace preservation.

Step identifiers and dependencies

Rule: inductiveness_and_reachable_coverage_separate · Uses: G001, S016

Reverse checks

S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

([0, 1, 2, 5], Yes, [], Yes, Yes, 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: 5f67f49f360367ab13a5b03bd6232c96f04e4723621108246d0c9969b81f3830
candidate record SHA-256: 79ea1ff302cd4c20ee5a633680d0a4f9358ae7ef5b1f03f4cbbac5991ce366df
semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
Lean: NOT_RUN · human review: NOT_PERFORMED

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