The set of (non-generalized) regular expressions over an alphabet $\Sigma$ is the set of expressions generated by the following grammar, where $a\in \Sigma$ ranges over symbols in the alphabet: $$ \pi ::= \epsilon \mid \emptyset \mid a \mid \pi_1\cdot\pi_2 \mid \pi_1\cup\pi_2 \mid \pi^*. $$ Here, $\epsilon$ stands for the language containing only the empty word and $\emptyset$ stands for the empty language. The star height $h(\pi)$ of a regular expression $\pi$ is defined as the maximum nesting depth of Kleene stars in $\pi$, i.e., we have:
- $h(\epsilon) := 0$, $h(\emptyset) := 0$ and $h(a):= 0$ for all $a\in \Sigma$,
- $h(\pi_1\cdot\pi_2) := \max\{h(\pi_1), h(\pi_2)\}$ and $h(\pi_1\cup\pi_2) := \max\{ h(\pi_1), h(\pi_2)\}$,
- $h(\pi^*) := h(\pi) +1$.
Finally, the star height $H(L)$ of a regular language $L$ is the minimum star height among all regular expressions describing $L$, so we have: $$ H(L) := \min \{ h(\pi)\mid \text{$\pi$ is a regular expression and $L=L(\pi)$}\}. $$ It was shown by Eggan (1963) and Dejean & Schützenberger (1966) that there exist regular languages of arbitrary star height, even over a two-element alphabet. In particular, it is possible to prove that, for any $n\geq 0$, the regular expression $\pi_n$ inductively defined below describes a language of star height $n$ (that is, it holds $H(L(\pi_n)) = n$).
- $\pi_1 := (ab)^*$,
- $\pi_{n+1} := (\underbrace{a\cdots a}_{2^n} \cdot e_n \cdot \underbrace{b\cdots b}_{2^n} \cdot e_n)^*$.
However, as shown by Cohen & Brzozowski (1970, Proposition 5.4), taking the complement of a regular language may increase the star height of the language. More specifically, for every integer $m\geq 0$, there exists a regular language $L$ such that $H(\overline{L}) \geq H(L) + m$.
My question is now whether we can also find, for every $n\geq 1$, a regular language $L$ of star height $n$ such that the star height is not increased by taking the complement. More precisely: is it true that, for every $n\geq 1$, there exists a regular language $L$ such that $H(L) = n$ and $H(\overline{L}) \leq n$?
