An Algebra for Kripke Polynomial Coalgebras

Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleenepsilas regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, labelled transition systems with explicit termination, and automata on guarded strings.