Weighted finite automata over hemirings

Quantitative automata computing the maximal average consumption of resources are currently intensively investigated. We introduce Conway hemirings and show that a few equational axioms suffice to imply a Kleene theorem for the possible behaviors of quantitative automata characterizing them as rational series, and we derive several further natural identities for rational operations on such series. We also obtain a more abstract Kleene theorem for Conway hemiring automata. This extends classical results of Conway and Schützenberger for recognizable languages resp. semiring-weighted automata.