What is the difference between a single-tape and a multi-tape Turing machine?

A single-tape Turing machine has only one tape, which is used for both input and storage, while a multi-tape Turing machine has multiple tapes, each of which can be used for input or storage. Multi-tape Turing machines are more powerful than single-tape Turing machines, as they can solve some problems more efficiently.

Can a Turing machine recognize a language that is not recursively enumerable?

No, a Turing machine can only recognize recursively enumerable languages, which are languages that can be listed by a computable algorithm. If a language is not recursively enumerable, then it cannot be recognized by a Turing machine.

What does it mean for a language to be decidable?

A language is decidable if there exists an algorithm that can determine whether a given string belongs to the language or not. In other words, a decidable language is a language for which we can construct a Turing machine that halts on every input and correctly decides whether the input belongs to the language or not.

What is the difference between a decidable and an undecidable language?

A decidable language is a language for which there exists an algorithm that can determine whether a given string belongs to the language or not. An undecidable language is a language for which there is no algorithm that can determine whether a given string belongs to the language or not.

What is the halting problem?

The halting problem is the problem of determining, given a Turing machine M and an input w, whether M halts on input w or not. It is a classic example of an undecidable problem, which means that there is no algorithm that can solve it for all possible inputs.

Can all context-free languages be decided?

No, not all context-free languages can be decided. The language universality problem, which asks whether a given context-free grammar generates all possible strings over a given alphabet, is an example of an undecidable problem. Therefore, there is no algorithm that can decide whether a given context-free language is universal or not.

Are all regular languages decidable?

Yes, all regular languages are decidable. This is because regular languages can be recognized by a finite automaton, which can always be constructed to halt on every input and correctly decide whether the input belongs to the language or not. Therefore, there exists an algorithm that can decide any regular language.

What is an example of using the pumping lemma to prove that a language is not regular?

An example of using the pumping lemma to prove that a language is not regular is to consider the language L = {0ⁿ1ⁿ | n >= 0}. We assume that L is regular and choose a string w = 0ⁿ1ⁿ in the language with length greater than the pumping constant n. We then show that for any way of splitting w into xyz, the string xyⁱz is not in the language when i = 2, contradicting the pumping lemma and showing that L is not regular.