  This paper is the extended version of On the Complexity of Infinite Advice
Strings (ICALP 2018). We investigate a notion of comparison between infinite
strings. In a general way, if M is a computation model (e.g. Turing machines)
and C a class of objects (e.g. languages), the complexity of an infinite word
alpha can be measured with respect to the amount of objects from C that are
presentable with machines from M using alpha as an oracle. In our case, the
model M is finite automata and the objects C 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; we also derive logical and
computational equivalent measures. 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.
