# Theory of Computation: Question Set – 21

#### How can you convert a context-free grammar to a pushdown automaton?

To convert a context-free grammar to a pushdown automaton, you can create a PDA that reads the input string from left to right, pushing symbols onto the stack as it expands non-terminal symbols according to the grammar’s productions. The PDA can use its finite state machine to keep track of its current state and transition to a new state based on the input symbol and top of stack symbol. If the PDA reaches the end of the input string and is in an accepting state with an empty stack, the input string is accepted.

#### What is the pumping lemma for context-free languages?

The pumping lemma for context-free languages is a tool used to prove that a language is not context-free. It states that any sufficiently long string in a context-free language can be divided into three parts: a prefix, a repeating part, and a suffix. The repeating part can be pumped (repeated any number of times) and still be in the language, and the length of the repeating part and prefix together cannot exceed a certain value based on the grammar.

#### Can a pushdown automaton recognize all context-free languages?

Yes, a pushdown automaton can recognize all context-free languages. This is because context-free languages are precisely the set of languages that can be recognized by a pushdown automaton.

#### What is the difference between a PDA and a Turing machine?

A pushdown automaton (PDA) is a type of automaton that can recognize context-free languages, while a Turing machine is a more powerful type of automaton that can recognize all recursively enumerable languages. The main difference is that a Turing machine has an infinite tape instead of a stack, allowing it to perform arbitrary computations and simulate any algorithm. A PDA, on the other hand, has a more limited computational power and can only recognize context-free languages.

#### What is the difference between a deterministic and non-deterministic pushdown automaton?

A deterministic pushdown automaton (DPDA) is a type of pushdown automaton in which for every state and input symbol, there is at most one transition that can be taken. A non-deterministic pushdown automaton (NPDA), on the other hand, allows for multiple transitions to be taken from a given state and input symbol. This means that an NPDA can potentially explore multiple paths simultaneously, whereas a DPDA can only explore a single path.

#### What is the significance of the empty stack symbol in pushdown automata?

The empty stack symbol is a special symbol that is used to represent an empty stack in pushdown automata. Its significance lies in the fact that it allows the PDA to recognize when it has consumed all the input symbols and emptied its stack, indicating that it has successfully recognized the input string.

#### Can a pushdown automaton recognize languages that are not context-free?

No, a pushdown automaton can only recognize context-free languages. This is because the pushdown automaton model is designed specifically to recognize context-free languages, and it does not have the computational power to recognize languages that are not context-free.

#### What is the pumping lemma for context-free languages used for?

The pumping lemma for context-free languages is a tool that is used to prove that certain languages are not context-free. It states that for any sufficiently long string in a context-free language, there exists a substring that can be “pumped” (repeated any number of times) and still remain in the language. If a language fails the pumping lemma, it is not context-free.