GATE CS 2021 Set 1 — Question 61
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Theory of Computation → Context-Free Languages → Pushdown Automata (PDA)
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Question
In a pushdown automaton , a transition of the form,
where , , and , representsConsider the following pushdown automaton over the input alphabet and stack alphabet .
The number of strings of length 100 accepted by the above pushdown automaton is __________.
Correct answer
50 to 50
Solution
Let's analyze the transitions of the PDA:
For a string of length 100, we have .
The only string of length 100 in is .
Therefore, the number of such strings is 1.
1.: Pushes the bottom-of-stack marker and moves to .
2.: For every 'a' read, an 'A' is pushed onto the stack. After 'a's, the stack is .
3.: Non-deterministically moves to without consuming input or changing the stack.
4.: For every 'b' read, an 'A' is popped from the stack. To eventually reach , all 'A's must be popped by reading exactly 'b's.
5.: Once the stack top is (meaning all 'A's are popped), it pops and moves to the final state . The stack is now empty.
6.: This loop requires an 'A' on the stack to proceed. However, the stack is empty upon entering . Thus, this transition is unreachable.
The language accepted is .For a string of length 100, we have .
The only string of length 100 in is .
Therefore, the number of such strings is 1.
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