Design a language L such that neither L nor its complement has an infinite regular subset?

Question

I'm having a class on automata theory, and right now we are learning the pumping lemma. There is an exercise question asking us to "Design a language L such that neither L nor its complement has an infinite regular subset?" But I don't understand the question. What is an infinite regular subset? How should I find a language that can meet this requirement?

Can anyone shed some light on this question?

Thanks!

Answer 1

More precisely, you need to find a language L so that there is no subset of L that is a infinite regular language and there is no subset of the complement of L that is an infinite regular language.

Here's an incorrect example: L = the union of a^n and a^n b^n. Since a^n is a regular language and it is a subset of L, this wouldn't work for the answer.

For finding a L that meets the requirements, I've found that these kind of questions are more like puzzles. You try some things out, check if they work or not, and try to think about why they don't solve the problem. Eventually you get your mind around the situation and come up with a solution.

Answer 2

Well, an infinite regular subset of a language is a subset that is infinite and regular. Okay, that's probably not very helpful.

So "subset" is pretty clear.

A "regular subset" is one that is accepted by a deterministic finite state machine (there's actually an amazing theorem in the theory of languages that says this condition is equivalent to a handful of other conditions).

An "infinite set" is a set that is not finite, or equivalently, a set that has infinitely many elements.

So let's say that a language L is special if it has an infinite regular subset.

It's your job to find a language L such that both L and the complement of L are not special.

To get somewhere with this problem, you need to wrap your head around that definition first. Take some of the examples of languages from your notes and your text. Figure out if they are regular. If you find one that isn't, think about what makes it not regular, and see if you design a language that has the property that all of its infinite subsets are not regular. Then see what happens when you look at its complement.

You have to build your intuition, and the only way to do that is to get your hands dirty.