Data ω-words are ω-words where each position is additionally labelled by a data value from an infinite alphabet. They can be seen as graphs equipped with two sorts of edges: 'next position' and 'next position with the same data value'. Based on this view, an extension of Data Automata called Generalized Data Automata (GDA) is introduced. While the decidability of emptiness of GDA is open, the decidability for a subclass class called Büchi GDA is shown using Multicounter Automata. Next a natural

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... fixpoint logic is defined on the graphs of data ω-words and it is shown that the µ-fragment as well as the alternation-free fragment is undecidable. But the fragment which is defined by limiting the number of alternations between future and past formulas is shown to be decidable, by first converting the formulas to equivalent alternating Büchi automata and then to Büchi GDA.