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Question 22499: Each Explicit Stack Step Must Respect Input and Epsilon Consumption

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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R0043-22499 · q-20261014-advance-20261110-r07-pushdown-automaton-0049

Original question and givens

Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a23, a23, a23, b23, b23, b23], supplied trace=[[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).

  • G001 — Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a23, a23, a23, b23, b23, b23], supplied trace=[[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).

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 operation=pda-trace-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[input, trace, transition_indices, valid_run, accepted, certificate]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL transition index=0;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push23, a23, Z23, push23, [Z23, A23]];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S001

S003 · implies

Before

ORIGINAL transition index=1;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push23, a23, A23, push23, [A23, A23]];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S002

S004 · implies

Before

ORIGINAL transition index=2;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push23, b23, A23, pop23, []];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S003

S005 · implies

Before

ORIGINAL transition index=3;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop23, b23, A23, pop23, []];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S004

S006 · implies

Before

ORIGINAL transition index=4;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push23, null, Z23, accept23, [Z23]];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S005

S007 · implies

Before

ORIGINAL transition index=5;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop23, null, Z23, accept23, [Z23]];bottom=Z23

After

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

Conditions: none

Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.

Step identifiers and dependencies

Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S006

S008 · calculation

Before

PDA complete ORIGINAL transition count := sum(1, 1, 1, 1, 1, 1)

After

PDA complete ORIGINAL transition count = 6

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

ORIGINAL states=[push23, pop23, accept23],input alphabet=[a23, b23],stack alphabet=[Z23, A23];start=push23,bottom=Z23,final states=[accept23]

After

finite unique pairwise-disjoint alphabets qualified;transition count=6;acceptance convention FULL input in final state,NOT empty stack;qualification alone proves NO run,determinism or CFG equivalence

Conditions: none

A qualified complete definition gives no accepting run,determinism or CFG-equivalence proof.Acceptance may retain a nonempty stack,including its bottom,but requires full input consumption in a final state.

Step identifiers and dependencies

Rule: pda_complete_definition_not_determinism_or_accepting_run · Uses: G001, S008

S010 · implies

Before

ORIGINAL input=[a23, a23, a23, b23, b23, b23];ALL source configurations=[[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]];declared start=push23,bottom=Z23

After

first configuration EXACTLY=[push23, 0, [Z23]];all state/position/stack values legal;this is NOT yet a valid-run or acceptance certificate

Conditions: none

Require the exact initial state,position0 and one bottom.All configurations must be structurally legal before checking their transitions.A legal configuration list alone need not be a valid run.

Step identifiers and dependencies

Rule: pda_trace_ALL_configurations_qualified_actual_initial_required · Uses: G001, S009

S011 · implies

Before

ORIGINAL trace step=0;from=[push23, 0, [Z23]];to=[push23, 1, [Z23, A23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push23, 1, [Z23, A23]]], [1, null], [2, null], [3, null], [4, [accept23, 0, [Z23]]], [5, null]];FIRST matching index=0

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S010

S012 · implies

Before

ORIGINAL trace step=1;from=[push23, 1, [Z23, A23]];to=[push23, 2, [Z23, A23, A23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push23, 2, [Z23, A23, A23]]], [2, null], [3, null], [4, null], [5, null]];FIRST matching index=1

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S011

S013 · implies

Before

ORIGINAL trace step=2;from=[push23, 2, [Z23, A23, A23]];to=[push23, 3, [Z23, A23, A23, A23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push23, 3, [Z23, A23, A23, A23]]], [2, null], [3, null], [4, null], [5, null]];FIRST matching index=1

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S012

S014 · implies

Before

ORIGINAL trace step=3;from=[push23, 3, [Z23, A23, A23, A23]];to=[pop23, 4, [Z23, A23, A23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop23, 4, [Z23, A23, A23]]], [3, null], [4, null], [5, null]];FIRST matching index=2

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S013

S015 · implies

Before

ORIGINAL trace step=4;from=[pop23, 4, [Z23, A23, A23]];to=[pop23, 5, [Z23, A23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop23, 5, [Z23, A23]]], [4, null], [5, null]];FIRST matching index=3

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S014

S016 · implies

Before

ORIGINAL trace step=5;from=[pop23, 5, [Z23, A23]];to=[pop23, 6, [Z23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop23, 6, [Z23]]], [4, null], [5, null]];FIRST matching index=3

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S015

S017 · implies

Before

ORIGINAL trace step=6;from=[pop23, 6, [Z23]];to=[accept23, 6, [Z23]];actual input=[a23, a23, a23, b23, b23, b23]

After

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept23, 6, [Z23]]]];FIRST matching index=5

Conditions: none

Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.

Step identifiers and dependencies

Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S016

S018 · implies

Before

ORIGINAL last config=[accept23, 6, [Z23]];input length=6;matched prior indices=[0, 1, 1, 2, 3, 3, 5]

After

valid_run=Yes;FULL input consumed=Yes;last state final=Yes;accepted=Yes;residual stack may be NONEMPTY

Conditions: none

Acceptance conjoins a valid run,full input consumption and a final state.A residual bottom or other stack symbols may remain.Validity and acceptance differ;the tag cannot turn a false flag into a certificate.

Step identifiers and dependencies

Rule: pda_run_validity_full_consumption_final_state_SEPARATE · Uses: G001, S017

Reverse checks

  • 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 — Recompute PDA complete ORIGINAL transition count with exact rational arithmetic, not rounded decimals.
  • 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

([a23, a23, a23, b23, b23, b23], [[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)

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: fd5f2eeac12d142849dadac19750cd23145e0d0eca976ece72834568018a3ada
  • candidate record SHA-256: 585c6fcbcb1d9c53891e446a964939730c688d02b2e9660d0d0b650f2025326d
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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