  Weighted automata over the max-plus semiring S are closely related to
finitely generated semigroups of matrices over S. In this paper, we use results
in automata theory to study two quantities associated with sets of matrices:
the joint spectral radius and the ultimate rank. We prove that these two
quantities are not computable over the tropical semiring, i.e. there is no
algorithm that takes as input a finite set of matrices M and provides as output
the joint spectral radius (resp. the ultimate rank) of M. On the other hand, we
prove that the joint spectral radius is nevertheless approximable and we
exhibit restricted cases in which the joint spectral radius and the ultimate
rank are computable. To reach this aim, we study the problem of comparing
functions computed by weighted automata over the tropical semiring. This
problem is known to be undecidable and we prove that it remains undecidable in
some specific subclasses of automata.
