  Compact representations of automata are important for efficiency. In this
paper, we study methods to compute reduced automata, in which no two states
accept the same language. We do this for finitary automata (FA), an abstract
definition that encompasses probabilistic and weighted automata. Our procedure
makes use of Milius' locally finite fixpoint. We present a reduction algorithm
that instantiates to probabilistic and S-linear weighted automata (WA) for a
large class of semirings. Moreover, we propose a potential connection between
properness of a semiring and our provided reduction algorithm for WAs, paving
the way for future work in connecting the reduction of automata to the
properness of their associated coalgebras.
