We present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions are assigned a value in a monoid that represents cost, duration, probability, etc. Such systems are represented as coalgebras and (1) and (2) above are derived in a modular fashion from the underlying (functor) type of these coalgebras. In previous work, we applied a similar approach to a class of systems (without weights) that generalizes both the results of Kleene (on rational languages and DFA{\textquoteright}s) and Milner (on regular behaviours and finite LTS{\textquoteright}s), and includes many other systems such as Mealy and Moore machines.
In the present paper, we extend this framework to deal with quantitative systems. As a consequence, our results now include languages and axiomatizations, both existing and new ones, for many different kinds of probabilistic systems.
}, attachments = {https://haslab.uminho.pt/sites/default/files/xana/files/ic-concur.pdf}, author = {Alexandra Silva and Filippo Bonchi and Marcello Bonsangue and Jan Rutten} }