Closure properties of the class of inherently ambiguous CFLs

Question

is set of inherently ambiguous context free languages close under operations such that union, intersection, kleene star, concatenation, reverse, complementation and etc. how many of theme are answered?

Answer 1

Reversal

The class of inherently ambiguous context-free languages is closed under reversal (exercise).

Intersection

The class of inherently ambiguous context-free languages is not closed under intersection. Indeed, take some inherently ambiguous context-free language $L$. Now take $L'$ to be a copy of $L$ which uses a different alphabet. Then $L \cap L' = \emptyset$.

Complementation

The class of inherently ambiguous context-free languages is not closed under complementation. Indeed, consider the Goldstine language $$ G = \{ a^{n_1} b \cdots a^{n_p} b : p \geq 1 \text{ and } n_i \neq i \text{ for some } i \}. $$ This language is context-free and inherently ambiguous (see slides of Cyril Nicaud). Also, $$ \overline{G} \cap \Sigma^* b = \{ aba^2\cdots a^p b : p \geq 1 \}, $$ which isn't even context-free.

Union

The class of inherently ambiguous context-free languages is not closed under union. Indeed, the following variant of the Goldstine language is probably also context-free and inherently ambiguous: $$ G' = \{ a^{n_1} b \cdots a^{n_p} b : p \geq 1 \text{ and } n_i \neq i+1 \text{ for some } i \}. $$ However, $G \cup G' = (a^*b)^+$.

Iteration (Kleene star)

The class of inherently ambiguous context-free languages is not closed under Kleene star. Indeed, consider the following language: $$ \Omega_3 = \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b \text{ or } |u|_a \neq |u|_c \}. $$ Here $|u|_a$ is the number of $a$s in $u$. This language is context-free and inherently ambiguous (see the slides of Cyril Nicaud). However, $\Omega_3^* = \{a,b,c\}^*$.

Concatenation

The class of inherently ambiguous context-free languages is not closed under concatenation. Indeed, the following variants of $\Omega_3$ are probably also context-free and inherently ambiguous: $$ \begin{align*} \Omega'_3 &= \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b \text{ or } |u|_a \neq |u|_c \text{ or } u = \epsilon \}, \\ \Omega''_3 &= \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b+1 \text{ or } |u|_a \neq |u|_c+1 \text{ or } u = \epsilon \}. \end{align*} $$ However, $\Omega'_3 \Omega''_3 = \{a,b,c\}^*$.