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Formal Languages and TQFTs with Defects

Luisa Boateng, Matilde Marcolli

TL;DR

The paper develops a categorical extension of the Boolean 1D TQFT with defects associated to finite automata, proving functoriality under transducers and linking language-theoretic subregularities to cohomological data in the TQFT. It then generalizes to context-free grammars using a categorical Chomsky–Schützenberger framework based on operads of spliced arrows and cobordisms with defects, producing TQFTs as operad morphisms. Subregular classes such as ${\rm SL}_k$ and ${\rm LT}_k$ yield additional structures (e.g., defect operator cohomologies) on the TQFTs, highlighting a deep interaction between formal language theory and topological field theories. The approach further extends to categorical FSAs and CF grammars, with tree-contour grammars providing a universal Dyck-language basis, and it outlines future directions toward higher-dimensional automata and 2D TQFTs with defects. Overall, the work integrates category theory, automata, operads, and TQFTs to create a unified, functorial framework connecting formal languages to defect-laden cobordism theories.

Abstract

A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky-Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.

Formal Languages and TQFTs with Defects

TL;DR

The paper develops a categorical extension of the Boolean 1D TQFT with defects associated to finite automata, proving functoriality under transducers and linking language-theoretic subregularities to cohomological data in the TQFT. It then generalizes to context-free grammars using a categorical Chomsky–Schützenberger framework based on operads of spliced arrows and cobordisms with defects, producing TQFTs as operad morphisms. Subregular classes such as and yield additional structures (e.g., defect operator cohomologies) on the TQFTs, highlighting a deep interaction between formal language theory and topological field theories. The approach further extends to categorical FSAs and CF grammars, with tree-contour grammars providing a universal Dyck-language basis, and it outlines future directions toward higher-dimensional automata and 2D TQFTs with defects. Overall, the work integrates category theory, automata, operads, and TQFTs to create a unified, functorial framework connecting formal languages to defect-laden cobordism theories.

Abstract

A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky-Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.

Paper Structure

This paper contains 17 sections, 14 theorems, 64 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Categories of Formal Languages and Automata
  3. A category of formal languages
  4. Regular and context-free languages
  5. Category of formal languages
  6. Boolean 1D TQFTs with defects
  7. Finite State Automata and Boolean 1D TQFTs with Defects
  8. Transducers and functorial properties
  9. Subregular classes
  10. 1D TQFTs: from Regular to Context-free Languages
  11. Categorical finite state automata
  12. Operads of spliced arrows and species
  13. Operad structure and cobordisms with defects
  14. Categorical rational transductions
  15. Categorical context-free grammars and cobordisms with defects
  16. ...and 2 more sections

Key Result

Proposition 2.3

The action of rational transductions as morphisms of context-free languages induces an action of transducers on PSA and FSA automata, of the following form. Let ${\mathcal{M}}$ be an automaton that is either a PSA ${\mathcal{M}}=({\mathfrak Q}_{\mathcal{M}}, {\mathfrak A}_{\mathcal{M}}, \Gamma_{\mat where ${\mathfrak A}_{{\mathcal{T}}({\mathcal{M}})}={\mathfrak B}$, $\Gamma_{{\mathcal{T}}({\mathca

Figures (9)

  • Figure 1: The generating morphisms of the category of one-dimensional cobordisms. The cup and cap morphisms and permutations which compose with them to get the opposite cup and cap, respectively.
  • Figure 2: All four types of morphisms in the category of 1-dimensional cobordisms with defects, $Cob_{1,\sum}$, with $\sum=\{a,b,c\}$.
  • Figure 3: Two finite state automata which compute the same regular language $(AB)^n$.
  • Figure 4: Operad composition $f \circ_1 g$ of the sequence $g=u_0\square u_1 \square u_2$ into the sequence $f= w_0 \square w_1 \square w_2 \square w_3$, from MelZei2.
  • Figure 5: Operations in the operad ${\mathcal{O}}_{{\rm Cob},{\mathcal{C}}}$ of 1D cobordisms with defects.
  • ...and 4 more figures

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.2
  • Theorem 3.3
  • Definition 4.1
  • ...and 15 more