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On Transition Constructions for Automata -- A Categorical Perspective

Mike Cruchten

TL;DR

The transition monoid construction for deterministic automata in a categorical setting is investigated and established as an adjunction to obtain two endofunctors on deterministic automata, a comonad and a monad, which are closely related to the largest set of equations and the smallest set of coequations satisfied by an automaton.

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 Ω-automata, and show that they form adjunctions. We present some initial results on sets of equations and coequations for lasso automata.

On Transition Constructions for Automata -- A Categorical Perspective

TL;DR

The transition monoid construction for deterministic automata in a categorical setting is investigated and established as an adjunction to obtain two endofunctors on deterministic automata, a comonad and a monad, which are closely related to the largest set of equations and the smallest set of coequations satisfied by an automaton.

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 Ω-automata, and show that they form adjunctions. We present some initial results on sets of equations and coequations for lasso automata.
Paper Structure (13 sections, 25 theorems, 43 equations, 1 table)

This paper contains 13 sections, 25 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Preliminaries
  4. The Transition Monoid and Machine Constructions
  5. The Functors $T$ and $M$
  6. $T\vdash M$
  7. $\nu\text{C}$ and $\mu\text{PL}$
  8. An Example
  9. Instantiating to Lasso Automata
  10. Transition and Machine Constructions Anew
  11. $\Omega$-Automata and Wilke Algebras
  12. Conclusion
  13. Acknowledgements.

Key Result

Lemma 3.1

The categories $\text{Alg}_r(G_1)$ and $\Sigma^{\ast}\mathrel{\hbox{o}rigin=c]{-90}{$↠$}} \text{Mon}$ are both thin (or posetal).

Theorems & Definitions (51)

  • Definition 2.1: calbrix:1994:ultimatelyPeriodicWordscruchten:2022:omegaAutomata
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Proposition 4.1
  • proof
  • ...and 41 more