The following question is a modified version of this question https://gateoverflow.in/3785/gate2005-it-38, in this GATE question they have NPDA, but I'm asking about DPDA
Let D be a deterministic push-down automaton (DPDA) with exactly one state, q, and exactly one symbol, Z, in its stack alphabet. State q is both the starting as well as the accepting state of the DPDA. The stack is initialized with one Z before the start of the operation of the DPDA. Let the input alphabet of the DPDA be Σ. Let L(P) be the language accepted by the DPDA by reading a string and reaching its accepting state. Let N(P) be the language accepted by the DPDA by reading a string and emptying its stack.
Which of the following statements is TRUE?
- L(P) is necessarily Σ* but N(P) is not necessarily Σ*.
- N(P) is necessarily Σ* but L(P) is not necessarily Σ*.
- Both L(P) and N(P) are necessarily Σ*.
- Neither L(P) nor N(P) are necessarily Σ*.