Topology and Ambiguity in Omega Context Free Languages

TL;DR

This paper explores the relationship between topological complexity and ambiguity in omega context free languages, revealing that certain complexity levels imply maximum ambiguity and that ambiguity is not preserved under common language operations.

Contribution

It establishes that non-Borel omega context free languages recognized by Büchi pushdown automata have maximum ambiguity and extends methods to study ambiguity in infinitary rational relations.

Findings

Non-Borel omega context free languages have maximum ambiguity.

Ambiguity is not preserved under omega power, adherence, or delta-limit operations.

Methods can be applied to study ambiguity in infinitary rational relations.

Abstract

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