Finite automata are the simplest model of computing machines. At the
cross-road of so many theorems, they are part of the base of theoretical
computer science as well as they appear as building bricks in many piece
of software for pattern recognition, formal verification, computational
linguistics, etc.

It is then easy, and natural, to enrich the model so that a finite
automaton is not a machine that accept or reject a sequence of symbols
anymore but rather a machine that 'computes' a multiplicity for every
sequence of symbols. According to where this value is taken, whether it
is an integer, a set of tuple of integers, another sequence of symbols,
etc. these 'weighted' automata have the capacity of modelizing as many
different kind of processes, from discrete event systems to timed
automata to probabilistic speech recognition systems. As always with
modelization, the decidability of the equivalence of two machines is a
problem that immediately arises.

In this talk, I will first sketch briefly the framework of the study of
weighted automata. A key result due to Schützenberger (1961) is the
decidability of equivalence of automata with multiplicity in a (skew)
field, via the computation of reduced representations. It is noteworthy
to quote that the long standing equivalence problem of deterministic
k-tape automata was eventually solved (Harju & Karhumäki 1988) by
reduction to that result.

I shall then present a recent result from a joint work with Marie-Pierre
Béal and Sylvain Lombardy, where we introduce the notion of conjugacy of
automata, a notion borrowed to symbolic dynamic. Two conjugated automata
are equivalent and we shall see that conversely, in the cases where
equivalence is known to be decidable, two equivalent automata are
conjugated to a same third automaton. This will give us the opportunity
to revisit classical constructions on automata such as determinization
or reduction of representations over a field. If time allows, we shall
also see how we have been able to deduce from this result on automata
with natural integer weights an answer to an open question in the theory
of automatic structures.