# Two new criteria to prove the inherent ambiguity of bounded context-free languages

Florent Koechlin (LORIA, Nancy)

A context-free language is inherently ambiguous if any grammar that recognizes it is ambiguous, i.e. there exists a word that is generated in two different ways. Deciding the inherent ambiguity of a context-free language is a difficult problem, undecidable in general. The first examples of inherently ambiguous languages were discovered in the 1960s, using iteration techniques on derivation trees. They belonged to a particular subfamily of context-free languages, the bounded context-free languages.

Although they made it possible to prove the inherent ambiguity of several languages, as for example the language $$L = a^n b^m c^p$$ with $$n=m$$ or $$m=p$$, iteration techniques are still very laborious to implement, and are very specific to the language studied, even sometimes unadapted to the studied language. For instance, the proof of inherent ambiguity of the language $$L$$ completely collapses by replacing the constraint “$$n=m$$ or $$m=p$$” by “$$n\not=m$$ or $$m\not=p$$”.

In this talk, I will present two new criteria based on generating series that allow us to prove the inherent ambiguity of bounded languages. These languages, which have a rational generating series, resisted both the classical iteration techniques developed in the 1960’s and the analytic methods introduced by Philippe Flajolet in 1987.