Comparing the power of advice strings: a notion of complexity for infinite words

We investigate in this paper a certain notion of comparison between infinite words. In a general way, if M is a model of computation (e.g. Turing machines) and C a class of objects (e.g. languages), the complexity of an infinite word α can be measured with respect to the objects from C that are presentable with machines from M using α as an oracle. In our case, the model M is finite automata and the objects are either recognized languages or presentable structures, known respectively as advice regular languages and advice automatic structures. This leads to several different classifications of infinite words that are studied in detail; logical and computational characterizations are derived. Our main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers. They suggest a closer study of the resulting hierarchy over infinite words.