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We develop a fully diagrammatic approach to the theory of finite-stateautomata, based on reinterpreting their usual state-transition graphicalrepresentation as a two-dimensional syntax of string diagrams. Moreover, weprovide an equational theory that completely axiomatises language equivalencein this new setting. This theory has two notable features. First, the Kleenestar is a derived concept, as it can be decomposed into more primitivealgebraic blocks. Second, the proposed axiomatisation is finitary -- a resultwhich is provably impossible to obtain for the one-dimensional syntax ofregular expressions.

A String Diagrammatic Axiomatisation of Finite-State Automata