Topology and Ambiguity in Omega Context Free Languages

We study the links between the topological complexity of an ω-context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel ω-context free languages which are recognized by Büchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation W → W , defined over finitary context free languages. We prove also that taking the adherence or the δ-limit of a finitary language preserves neither ambiguity nor inherent ambiguity. On the other side we show that methods used in the study of ω-context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Büchi 2-tape automata and we get first results in that direction.