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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.

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  • $\begingroup$ Please ask only one question per post. The two here don't seem to be related at all are better asked separately. $\endgroup$ – Raphael Nov 19 '18 at 18:14
  • $\begingroup$ The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! $\endgroup$ – Raphael Nov 19 '18 at 18:14
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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

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