unambiguous context-free languages and complementation

Question

I was considering the following two natural questions about the relationship between unambiguity and complementation for the class of context-free languages:

1. Is the complement of an unambiguous context-free language also a context-free language?
2. If a language is context-free, and its complement is context-free as well (i.e., a so-called strongly context-free language), is it the case that the language is unambiguous context-free?

The first question above is motivated by the fact that it holds for deterministic context-free languages, which are a strict subclass of the unambiguous context-free languages.

Answer 1

Both questions turn out to have negative answers, as shown in [this][1] article. In particular, the authors construct

1. An unambiguous context-free language whose complement is not context-free.
2. An inherently ambiguous (i.e., non-unambiguous) context-free language whose complement is context-free.

Consequently, there is no relationship between the notion of unambiguity for context-free languages and the complementation operation.

[1] The Independence of Inherent Ambiguity From Complementedness Among Context-Free Languages" by Hibbard and Ullian, JACM, Volume 13 Issue 4, Oct. 1966, Pages 588-593 https://dl.acm.org/citation.cfm?doid=321356.321366