Question

Whenever each transition of a PDA M pops a symbol, what can be said about the language of M?

1) The language of M must be empty.
2) The language of M must be context-free but need not be regular.
3) The language of M must be finite but need not be empty.
4) The language of M may not be context-free.
5) The language of M must be regular and could be infinite.

Answer 1

The correct answer is that the language of a PDA which pops a symbol with each transition must be context-free but need not be regular. PDAs can recognize an infinite variety of context-free languages, which can include complex nested structures.

Step-by-step explanation:

When considering the behavior of a pushdown automaton (PDA) in relation to the language it accepts, we can analyze the implications of popping a symbol with each transition. In the context of formal language theory, if a PDA pops a symbol from the stack during each transition, this behavior aligns with the acceptance of a certain class of languages.

Option 2 is the correct answer: The language of M must be context-free but need not be regular. A PDA is a computational model that can recognize context-free languages. Popping a symbol does not necessarily force the language to be finite or regular, nor does it imply that the language is empty or not context-free.

Context-free languages include many constructs that cannot be recognized by finite automata, like nested structures and balanced parentheses, which are characteristic of many programming languages' syntax. Therefore, the language recognized by a PDA which pops a symbol with each transition can indeed be infinite, demonstrating that it need not be regular but is within the class of context-free languages.