  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.
