IN620

pushdown automaton

Input word = i0, I, K

• i0 total length of input word
• I = i1, J
  • i1, size of automaton input word
  • J automaton input, list of integers
• K, list of transitions

transition (q, a, A, w, q')

• q : state before
• q' : state after
  • 0 : start state
  • 1 : accepting state
• a : symbol in input
  • 2 : ε
• A : symbol on stack (pop operation)
  • 2 : ε (no pop)
  • 3 : stack start symbol
• w : word to push on stack
  • w = u, V
    u : length of V, 0 if nothing to push
    V : list of integers to push (Vn, Vn-1... V0)
    Leftmost element of the list will be at the top of the stack after the transition (Vn here).

output:

• 0 : accept
• 1 : reject

examples :

transition (q, a, A, w, q') = (0, 3, 4, 54, 1)

(0, 3, 4, 2, 5, 4, 1)

transition (q, a, A, w, q') = (0, ε, 1, ε, 1)

(0, 2, 1, 2, 0, 1)

0ⁿ1ⁿ

0ⁿ1ⁿ automaton:

(0, 0, 3, 2, 1, 3, 0)	// stack empty, read 0, push 1, stay in state 0
(0, 0, 1, 2, 1, 1, 0)	// stack with 1 on top, read 0, push 1, stay in state 0
(0, 1, 1, 0, 2)		// stack with 1 on top, read 1, pop 1, go to state 2
(2, 1, 1, 0, 2)		// stack with 1 on top, read 1, pop 1, stay in state 2
(2, 2, 3, 1, 3, 1)	// stack empty in state 2, go to accepting state

0ⁿ1ⁿ inputs

0ⁿ1ⁿ
input 000111 (output should be 0)

(37, 6, 0, 0, 0, 1, 1, 1, 0, 0, 3, 2, 1, 3, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 3, 1)

0ⁿ1^m n<m
input 001111 (output should be 1)

(37, 6, 0, 0, 1, 1, 1, 1, 0, 0, 3, 2, 1, 3, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 3, 1)

0ⁿ1^m n>m
input 000011 (output should be 1)

(37, 6, 0, 0, 0, 0, 1, 1, 0, 0, 3, 2, 1, 3, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 3, 1)

!=0ⁿ1^m
input 001011 (output should be 1)

(37, 6, 0, 0, 1, 0, 1, 1, 0, 0, 3, 2, 1, 3, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 3, 1)