We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.

@article{paperbot2236,
  title = {A Finite Axiomatisation of Finite-State Automata Using String Diagrams},
  author = {Robin Piedeleu and Fabio Zanasi},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 19, Issue 1},
  year = {2023},
  doi = {10.46298/lmcs-19(1:13)2023}
}