GATE CS 2016 Set 1 — Question 53
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Theory of Computation → Context-Free Languages → DPDA vs NPDA
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Question
Consider the transition diagram of a PDA given below with input alphabet and stack alphabet . is the initial stack symbol. Let denote the language accepted by the PDA.
Which one of the following is TRUE?
Correct answer
(D) L = \aⁿ n ≥ 0\ ∪ \aⁿ bⁿ n ≥ 0\ and is deterministic context-free
Solution
The given PDA operates as follows:
1.Start State: The transition reads an '', pops the initial stack symbol , and pushes . This ensures the string starts with at least one ''.
2.Second State: The loop pushes an for every subsequent ''. If there are ''s total, there will be 's on the stack (since the first '' pushed an via ).
3.Transition to Third State: The transition reads a '' and pops an . This begins the matching process.
4.Third State: The loop continues to pop an for every '' read.
5.Final Transition: The transition to the final state is only possible if the stack top is . This happens only when all 's have been popped, meaning the number of ''s equals the number of ''s.
Thus, the language accepted is .Analyzing the options:- (A) This option describes the language as (with , which is the standard form, though this specific PDA requires ). Crucially, the language is a Context-Free Language (CFL) but not a Regular Language. Therefore, it cannot be accepted by any finite automata. This statement is TRUE.
- (B) & (D) The PDA does not accept strings of the form (where no ''s follow), so the language is not the union .
- (C) The language is Context-Free, which implies it is Recursive. Any Recursive language is accepted by a Turing Machine that halts on every input.
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