Borel Hierarchy and Omega Context Free Languages

TL;DR

This paper explores the topological complexity of omega context free languages, demonstrating some are non-Borel and undecidable to classify, and extends these properties to recursive analogues.

Contribution

It provides new topological results on omega-CFLs, including examples of non-Borel sets and undecidability results, addressing open questions in the field.

Findings

Existence of non-Borel omega-CFLs

Undecidability of Borel classification for omega-CFLs

Non-Borel omega powers of finitary languages

Abstract

We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621], proving new topological properties of omega context free languages : there exist some omega-CFL which are non Borel sets. And one cannot decide whether an omega-CFL is a Borel set. We give also an answer to questions of Niwinski and Simonnet about omega powers of finitary languages, giving an example of a finitary context free language L such that L^omega is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an omega-CFL is an arithmetical set.