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A generalized regular expression is like a regular expression but with one more operation allowed: complementation. The (generalized) star-height of a generalized regular expression is the maximal number of nested Kleene stars. The star-height of a language is the minimal star-height of a regular expression describing it. It is not known if there is even a language of star-height 2.

So I imagine the langage $((aa)^*bb(aa)^*bb(aa)^*)^*(aa)^*$ of words consisting of concatenations of $aa$ and $bb$, with an even number of $bb$, is of generalized star-height $1$. But I couldn't prove it. In the paper Some results on the generalized star-height problem (Pin, Straubing, Thérien), lemma 6.1 (the transfer lemma) cannot be applied because both $(bb)^*$ and $(aa)^*$ are of star-height 1.

I saw elsewhere some languages which were conjectured to be of star-height 2 and they were more complex than the one I give, so it is very probably of star-height 1. Is it the case?

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    $\begingroup$ Did you look at the complement? Either strings not of the form "(aa+bb)*", or strings with an odd number of bb with in between "a (not bb) a". The "not bb" are the complement of (a+b)*bb(a+b)*. As you know (a+b)* is starheight zero. $\endgroup$ Aug 27 '20 at 15:17
  • $\begingroup$ @HendrikJan Hi and thank you but I couldn't understand what you said. So yes first we can restrict our attention to strings of the form $(aa+bb)^*$ (we don't care about the other ones). Strings that do not contain $bb$ are also of star-height 1, ok. But what do you mean by ``strings with an odd number of bb with in between "a (not bb) a"''? The problem is I don't know how to count parity. $\endgroup$
    – Idéophage
    Aug 28 '20 at 0:18
  • $\begingroup$ @HendrikJan I don't see how it helps to shift attention to the complement, because when restricting the attention to strings of the form $(aa+bb)^*$, we will need to count strings with an odd number of $bb$, which seems to be of the same difficulty. $\endgroup$
    – Idéophage
    Aug 28 '20 at 0:19
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(This is an attempt to an answer, I hope the details are right.)

Your language consists of all strings in $(aa+bb)^*$ with an even number of $bb$.

We are allowed to use complementation, so we start by looking at the complement of the language. I think we can split the complement into two (overlapping) parts

  • strings not of the form $(aa+bb)^*$, that language has starheight one.
  • strings of the form $w_0\cdot bb\cdot w_1\cdot bb\cdot w_2 \dots bb\cdot w_n$, where (1) the number of $bb$ is odd, and (2) the $w_i$ do not contain any $bb$ (but I will have to be more precise on that)

Now we will try to find an expression for the language $W$ of the $w_i$ without using star. That means we can use the star to count the odd number of $bb$, as in $(W\cdot bb\cdot W\cdot bb)^*\cdot W\cdot bb\cdot W$.

This is the tricky part. Strings from $W$ are either

  • the empty string $\varepsilon$
  • just one single $a$
  • of the form $axa$, where $a$ does not contain $bb$. Not "containing $bb$" is the complement of $(a+b)^*bb(a+b)^*$, and as you know $(a+b)^*$ is again star-free.

The strings we now have specified can have longer stretches of $b$'s, but whenever that happens the $b$'s come in pairs.

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  • $\begingroup$ Aaah yes, thank you very much. Your idea is right but actually it is useless to take the complementary. The language I gave is explicitely $(aa+bb)^* ∩ (a^*bba^*bba^*)^*a^*$. They are simply the words of $(aa+bb)^*$ that contain a number of $b$s divisible by 4. I'm stupid... $\endgroup$
    – Idéophage
    Aug 28 '20 at 10:54
  • $\begingroup$ @Idéophage Sorry, but now you have to help me: why does it help? How do you get rid of the intersection? $\endgroup$ Aug 28 '20 at 23:13
  • $\begingroup$ Well, since we can use complementation, we can actually use all boolean operations : $X ∩ Y = \overline{\overline{X} ∪ \overline{Y}}$. :) Maybe you were thinking too much about restricted star-height. $\endgroup$
    – Idéophage
    Aug 28 '20 at 23:37
  • $\begingroup$ Thanks, I simply did not realize this... We all have our stupid days it seems ... $\endgroup$ Aug 29 '20 at 1:41

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