On Transition Constructions for Automata -- A Categorical Perspective

TL;DR

This paper explores the categorical structure of transition monoids in automata theory, establishing adjunctions and algebraic frameworks that deepen understanding of automata behaviors and their algebraic properties.

Contribution

It introduces a categorical perspective on transition monoids, forming adjunctions and algebraic structures for various automata types, extending classical automata theory.

Findings

Established an adjunction for transition monoid construction.

Defined related monad and comonad structures for automata.

Presented initial results on equations for lasso automata.

Abstract

We investigate the transition monoid construction for deterministic automata in a categorical setting and establish it as an adjunction. We pair this adjunction with two other adjunctions to obtain two endofunctors on deterministic automata, a comonad and a monad, which are closely related, respectively, to the largest set of equations and the smallest set of coequations satisfied by an automaton. Furthermore, we give similar transition algebra constructions for lasso and {\Omega}-automata, and show that they form adjunctions. We present some initial results on sets of equations and coequations for lasso automata.