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Topoi of automata I: Four topoi of automata and regular languages

Ryuya Hora

TL;DR

It is shown that the four different notions of automata form four types of Grothendieck topoi, illustrating how the technical details of automata theory are described by topos theory.

Abstract

Both topos theory and automata theory are known for their multi-faceted nature and relationship with topology, algebra, logic, and category theory. This paper aims to clarify the topos-theoretic aspects of automata theory, particularly demonstrating through two main theorems how regular (and non-regular) languages arise in topos-theoretic calculation. First, it is shown that the four different notions of automata form four types of Grothendieck topoi, illustrating how the technical details of automata theory are described by topos theory. Second, we observe that the four characterizations of regular languages (DFA, Myhill-Nerode theorem, finite monoids, profinite words) provide Morita-equivalent definitions of a single Boolean-ringed topos, situating this within the context of Olivia Caramello's 'Toposes as Bridges.' This paper also serves as a preparation for follow-up papers, which deal with the relationship between hyperconnected geometric morphisms and algebraic/geometric aspects of formal language theory.

Topoi of automata I: Four topoi of automata and regular languages

TL;DR

It is shown that the four different notions of automata form four types of Grothendieck topoi, illustrating how the technical details of automata theory are described by topos theory.

Abstract

Both topos theory and automata theory are known for their multi-faceted nature and relationship with topology, algebra, logic, and category theory. This paper aims to clarify the topos-theoretic aspects of automata theory, particularly demonstrating through two main theorems how regular (and non-regular) languages arise in topos-theoretic calculation. First, it is shown that the four different notions of automata form four types of Grothendieck topoi, illustrating how the technical details of automata theory are described by topos theory. Second, we observe that the four characterizations of regular languages (DFA, Myhill-Nerode theorem, finite monoids, profinite words) provide Morita-equivalent definitions of a single Boolean-ringed topos, situating this within the context of Olivia Caramello's 'Toposes as Bridges.' This paper also serves as a preparation for follow-up papers, which deal with the relationship between hyperconnected geometric morphisms and algebraic/geometric aspects of formal language theory.

Paper Structure

This paper contains 22 sections, 23 theorems, 25 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. 1. Introducing four topoi of automata.
  3. 2. Proving that four characterizations of regular languages provide four descriptions of a single boolean-ringed topos $({\Sigma\text{-}\mathbf{Set}}_{\mathrm{o.f.}}, \mathcal{R})$.
  4. On the follow-up papers
  5. Acknowledgement
  6. Four topoi of automata
  7. Aset: The presheaf topos of word actions
  8. Word actions form a topos
  9. The canonical point and the internal Boolean algebra of languages
  10. Atmt: The presheaf topos of (coalgebraic) automata
  11. (Coalgebraic) automata form a topos
  12. Language recognition
  13. ofAset: The Grothendieck topos of orbit-finite word actions
  14. Orbit-finite word actions form a topos
  15. The canonical point and the internal Boolean algebra of regular languages
  16. ...and 7 more sections

Key Result

Proposition 2.2

The category of $\Sigma$-sets is equivalent to category of right ${{\Sigma}^{\ast}}$-actions In particular, it is a presheaf topos (and hence, a Grothendieck topos).

Figures (1)

  • Figure 1: Four characterizations of regular languages are Morita equivalent.

Theorems & Definitions (68)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3: Studies on the topos of word actions.
  • Definition 2.4
  • Lemma 2.5: Canonical boolean algebra in a pointed topos
  • proof
  • Definition 2.6
  • Lemma 2.7
  • Proposition 2.8: The canonical Boolean algebra consists of all languages
  • ...and 58 more