Formal Languages and TQFTs with Defects

TL;DR

This paper explores a functorial construction linking finite automata, context-free grammars, and topological quantum field theories (TQFTs) with defects, revealing new categorical and cohomological structures.

Contribution

It demonstrates the functorial nature of a Boolean 1D TQFT construction for automata and extends it to context-free grammars via a categorical Chomsky-Schützenberger theorem.

Findings

Subregular languages correspond to cohomological structures on TQFTs

The construction generalizes to context-free grammars

TQFTs are described as morphisms of colored operads

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-Sch\"utzenberger representation theorem, due to Melli\`es and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.