# A coalgebraic take on regular and $\omega$-regular behaviours

## TL;DR

This paper develops a coalgebraic framework for modeling finite and infinite behaviors with B"uchi acceptance, unifying various automata types and establishing Kleene-type theorems for regular and -regular behaviors.

## Contribution

It introduces a general coalgebraic approach to -regular behaviors, including constructions for suitable monads and coalgebraic automata, extending classical automata theory.

## Key findings

- Framework instantiated on non-deterministic Buchi automata
- Framework applied to tree automata and probabilistic automata
- Established coalgebraic Kleene-type theorems for -regular behaviors

## Abstract

We present a general coalgebraic setting in which we define finite and infinite behaviour with B\"uchi acceptance condition for systems whose type is a monad. The first part of the paper is devoted to presenting 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 ($\omega$-) behaviour. We end the paper with coalgebraic Kleene-type theorems for ($\omega$-) regular input. The framework is instantiated on non-deterministic (B\"uchi) automata, tree automata and probabilistic automata.