On Müller context-free grammars

We define context-free grammars with Müller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Müller context-free languages including closure properties and others. We show that every Müller context-free grammar can be transformed into a normal form grammar in polynomial space, and then we show that many decision problems can be decided in polynomial time for Müller context-free grammars in normal form. These decision problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result, we establish a limitedness property of Müller context-free grammars: if the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.