Original question and givens
Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. 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=[a1, a1, b1, b1], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).
G001 — Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. 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=[a1, a1, b1, b1], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=pda-bounded-search-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary]
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]=[push1, a1, Z1, push1, [Z1, A1]];bottom=Z1
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]=[push1, a1, A1, push1, [A1, A1]];bottom=Z1
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]=[push1, b1, A1, pop1, []];bottom=Z1
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]=[pop1, b1, A1, pop1, []];bottom=Z1
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]=[push1, null, Z1, accept1, [Z1]];bottom=Z1
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]=[pop1, null, Z1, accept1, [Z1]];bottom=Z1
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=[push1, pop1, accept1],input alphabet=[a1, b1],stack alphabet=[Z1, A1];start=push1,bottom=Z1,final states=[accept1]
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=[a1, a1, b1, b1];initial configuration=[push1, 0, [Z1]];max_steps=1;stack cap=7
After
depth0 fully-consumed final-state acceptance=No;both STEP and STACK limits apply;not an unbounded rejection test
Conditions: none
The initial state may already accept empty input.Search bounds both steps and stack length;do not replace final-state acceptance with an empty-stack convention.
Step identifiers and dependencies
Rule: pda_bounded_initial_acceptance_and_TWO_search_limits · Uses: G001, S009
S011 · implies
Before
ORIGINAL search depth=0;ALL frontier configs=[[push1, 0, [Z1]]];stack cap=7
After
ALL [from,ORIGINAL transition index,next_or_null,within_stack_cap,accepts_IN_THIS_SEARCH] rows=[[[push1, 0, [Z1]], 0, [push1, 1, [Z1, A1]], Yes, No], [[push1, 0, [Z1]], 1, null, No, No], [[push1, 0, [Z1]], 2, null, No, No], [[push1, 0, [Z1]], 3, null, No, No], [[push1, 0, [Z1]], 4, [accept1, 0, [Z1]], Yes, No], [[push1, 0, [Z1]], 5, null, No, No]];next distinct frontier=[[accept1, 0, [Z1]], [push1, 1, [Z1, A1]]];hit=No
Conditions: none
Try every configuration and original transition.Filter candidates by the declared stack cap before testing acceptance in this search.Show the full layer rather than hidden pruning;within-layer deduplication preserves existence.
Step identifiers and dependencies
Rule: pda_ALL_nondeterministic_frontier_steps_stack_filter_BEFORE_acceptance · Uses: G001, S010
S012 · implies
Before
ORIGINAL input=[a1, a1, b1, b1];step bound=1;stack bound=7;found_accepting_run=No
After
rejection_or_nonmembership_claim fixed No;longer or larger-stack accepting runs have NOT been excluded
Conditions: none
A found run witnesses this bounded search;failure does not show that all runs reject.The final No does not negate a true found flag.No determinism or CFG equivalence is inferred.
Step identifiers and dependencies
Rule: pda_bounded_nohit_NOT_rejection_or_nonmembership · Uses: G001, S011