Formal Languages and TQFTs with Defects
This work provides a formal categorical framework for connecting automata theory and topological quantum field theories, which is incremental in extending prior results to broader language classes.
The paper demonstrates that a known construction linking finite state automata to Boolean 1D TQFTs with defects is functorial with respect to automata and transducers, and extends this to context-free grammars via categorical representation, describing the TQFTs as morphisms of colored operads.
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.