A coalgebraic take on regular and $ω$-regular behaviours

Abstract

We present a general coalgebraic setting in which we define finite and infinite behaviour with Büchi acceptance condition for systems with internal moves. Since systems with internal moves are defined here as coalgebras for a monad, in the first part of the paper we present a construction of a monad suitable for modelling (in)finite behaviour. The second part of the paper focuses on presenting the concepts of a (coalgebraic) automaton and its (ω-) behaviour. We end the paper with coalgebraic Kleene-type theorems for (ω-) regular input. The framework is instantiated on non-deterministic (Büchi) automata, tree (Büchi) automata and probabilistic (Büchi) automata.