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. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).
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. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).
Step-by-step derivation
S001 · implies
Before
source operation=pda-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention]
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