Question 9558: Ranking Decrease, Stuck States, and Actual Termination

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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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R0018-09558 · q-20261013-advance-20261019-r05-algorithm-correctness-0058

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], [0, 3], [3, 4], [4, 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. Natural ranking vector=[13, 5, 4, 3, 2, 0]. Give (rank vector, reachable nonhalt Step edges failing strict decrease, ranking valid?, reachable stuck states, ranking valid and no stuck states?, actual all-path termination?).

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], [0, 3], [3, 4], [4, 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. Natural ranking vector=[13, 5, 4, 3, 2, 0]. Give (rank vector, reachable nonhalt Step edges failing strict decrease, ranking valid?, reachable stuck states, ranking valid and no stuck states?, actual all-path termination?).

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], [0, 3], [3, 4], [4, 5]]

After

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

After

expanded successors=[1, 3];newly reached=[1, 3]

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, 3];all reached=[0, 1, 3]

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

depth=1;state=3;halt=No;raw successors=[4]

After

expanded successors=[4];newly reached=[4]

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, S005

S007 · implies

Before

completed frontier=[1, 3];depth=1

After

next frontier=[2, 4];all reached=[0, 1, 2, 3, 4]

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, S006

S008 · implies

Before

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

After

expanded successors=[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, S007

S009 · implies

Before

depth=2;state=4;halt=No;raw successors=[5]

After

expanded successors=[5];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=[2, 4];depth=2

After

next frontier=[5];all reached=[0, 1, 2, 3, 4, 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

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, S010

S012 · implies

Before

completed frontier=[5];depth=3

After

next frontier=[];all reached=[0, 1, 2, 3, 4, 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, S011

S013 · implies

Before

reached state=0;declared halt=No;raw successors=[1, 3]

After

status=active-nonhalt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S012

S014 · implies

Before

reached state=1;declared halt=No;raw successors=[2]

After

status=active-nonhalt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S013

S015 · implies

Before

reached state=2;declared halt=No;raw successors=[5]

After

status=active-nonhalt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S014

S016 · implies

Before

reached state=3;declared halt=No;raw successors=[4]

After

status=active-nonhalt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S015

S017 · implies

Before

reached state=4;declared halt=No;raw successors=[5]

After

status=active-nonhalt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S016

S018 · implies

Before

reached state=5;declared halt=Yes;raw successors=[]

After

status=halt

Conditions: none

The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination.

Step identifiers and dependencies

Rule: halt_stuck_active_are_distinct · Uses: G001, S017

S019 · implies

Before

natural rank vector=[13, 5, 4, 3, 2, 0];reachable=[0, 1, 2, 3, 4, 5];Halt=[5]

After

all active ranking edges=[[0, 1], [0, 3], [1, 2], [2, 5], [3, 4], [4, 5]]

Conditions: none

A natural ranking must strictly decrease on every reachable nonhalt outgoing edge,including edges into halt.Raw outgoing halt edges are not execution steps.

Step identifiers and dependencies

Rule: ranking_reachable_nonhalt_edge_scope · Uses: G001, S018

S020 · calculation

Before

rank decrease on edge0,1 := sub(13, 5)

After

rank decrease on edge0,1 = 8

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

edge=[0, 1];ranks=[13, 5];difference=8

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S020

S022 · calculation

Before

rank decrease on edge0,3 := sub(13, 3)

After

rank decrease on edge0,3 = 10

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

edge=[0, 3];ranks=[13, 3];difference=10

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S022

S024 · calculation

Before

rank decrease on edge1,2 := sub(5, 4)

After

rank decrease on edge1,2 = 1

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

edge=[1, 2];ranks=[5, 4];difference=1

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S024

S026 · calculation

Before

rank decrease on edge2,5 := sub(4, 0)

After

rank decrease on edge2,5 = 4

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

edge=[2, 5];ranks=[4, 0];difference=4

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S026

S028 · calculation

Before

rank decrease on edge3,4 := sub(3, 2)

After

rank decrease on edge3,4 = 1

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

edge=[3, 4];ranks=[3, 2];difference=1

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S028

S030 · calculation

Before

rank decrease on edge4,5 := sub(2, 0)

After

rank decrease on edge4,5 = 2

Conditions: none

Subtract successor rank from predecessor rank;strict decrease requires a value>0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

edge=[4, 5];ranks=[2, 0];difference=2

After

strictly decreasing=Yes

Conditions: none

Equality is not decrease.Retain every nondecreasing original edge,not just the first.

Step identifiers and dependencies

Rule: ranking_each_strict_decrease · Uses: G001, S030

S032 · implies

Before

reachable=[0, 1, 2, 3, 4, 5];Halt=[5]

After

reachable nonhalt graph vertices=[0, 1, 2, 3, 4]

Conditions: none

Inspect cycles only in the reachable nonhalt subgraph.Unreachable cycles and outgoing halt edges do not affect this execution question.

Step identifiers and dependencies

Rule: reachable_nonhalt_cycle_scope · Uses: G001, S031

S033 · implies

Before

peeling layer=0;remaining=[0, 1, 2, 3, 4]

After

vertices with no remaining nonhalt successor=[2, 4]

Conditions: none

Repeatedly remove vertices with no remaining nonhalt successor.A directed cycle cannot be removed this way.Stuck vertices may be removed too,so keep the separate stuck check.

Step identifiers and dependencies

Rule: finite_cycle_sink_peeling · Uses: G001, S032

S034 · implies

Before

peeling layer=1;remaining=[0, 1, 3]

After

vertices with no remaining nonhalt successor=[1, 3]

Conditions: none

Repeatedly remove vertices with no remaining nonhalt successor.A directed cycle cannot be removed this way.Stuck vertices may be removed too,so keep the separate stuck check.

Step identifiers and dependencies

Rule: finite_cycle_sink_peeling · Uses: G001, S033

S035 · implies

Before

peeling layer=2;remaining=[0]

After

vertices with no remaining nonhalt successor=[0]

Conditions: none

Repeatedly remove vertices with no remaining nonhalt successor.A directed cycle cannot be removed this way.Stuck vertices may be removed too,so keep the separate stuck check.

Step identifiers and dependencies

Rule: finite_cycle_sink_peeling · Uses: G001, S034

S036 · implies

Before

all reachable nonhalt vertices peeled

After

reachable nonhalt graph is acyclic

Conditions: none

Removing all vertices certifies absence of a directed cycle,not merely failure to encounter one on a single run.

Step identifiers and dependencies

Rule: complete_acyclic_nonhalt_certificate · Uses: G001, S035

S037 · implies

Before

reachable stuck=[];reachable nonhalt cycle=No

After

all paths reach Halt=Yes

Conditions: none

In a finite model an infinite nonhalt path repeats a state and forms a cycle;a finite maximal nonhalting path ends stuck.Exclude both to ensure every path reaches Halt.Empty Init is vacuous;this does not establish termination of arbitrary infinite-state programs.

Step identifiers and dependencies

Rule: finite_all_paths_termination_joint_gate · Uses: G001, S036

S038 · implies

Before

ranking valid=Yes;reachable stuck=[]

After

ranking plus no stuck=Yes;actual all-path termination=Yes

Conditions: none

Strict natural descent excludes infinite execution,but stuck states must also be excluded to reach successful halt.A failed ranking candidate does not mean the model fails to terminate;actual termination is independently checked from the full finite graph.

Step identifiers and dependencies

Rule: ranking_candidate_not_actual_termination_identity · Uses: G001, S037

Reverse checks

S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S030 · premise_and_result_audit — Recompute rank decrease on edge4,5 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute rank decrease on edge3,4 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S026 · premise_and_result_audit — Recompute rank decrease on edge2,5 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S024 · premise_and_result_audit — Recompute rank decrease on edge1,2 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Recompute rank decrease on edge0,3 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Recompute rank decrease on edge0,1 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

([13, 5, 4, 3, 2, 0], [], 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: 16940c0c0eabbd17275a964c7c4d21891a2f3a7800c4f48967d2a8b82378bd96
candidate record SHA-256: e1010f0b2ff8ebd0849289fdffa2f35ecdaf5fd4cd617a1e466bf2f0f705a032
semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
Lean: NOT_RUN · human review: NOT_PERFORMED

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