A String Diagrammatic Axiomatisation of Finite-State Automata

Abstract

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— a result which is provably impossible for the one-dimensional syntax of regular expressions. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.

A Complete Diagrammatic Calculus for Boolean Satisfiability

“…To prove completeness for m → 0 diagrams, we adopt a normal form argument that is common for the completeness proof of many diagrammatic calculi [24,25]. In a nutshell, a normal form defines a certain syntactic representative for each semantic object in the image of the interpretation, and shows that if two diagrams have the same semantics, then they are both equal in SATA to this syntactic representative.…”

Axiomatization of Blockchain Theory

“…Both in the first and second case, mathematical methods are used for formalization and verification. Moreover, verification using finite automata can be transferred to the language of logic since there is an axiomatization [42,43] of the theory of finite automata.…”

A Compositional Approach to Parity Games

“…The functor R M is not faithful (a one-node example can be easily given). In recent works on props such as [5,30], the main interest is in the faithfulness of a semantics functor whose codomain is a well-known semantic category (that of linear relations [5], automata [30], etc.). In this case, faithfulness amounts to the completeness of equational axioms.…”

“…In particular, props have been used extensively as graphical languages. They define graphical languages as models for some mathematical structures (signal flow diagrams [5], networks [1], Petri nets [3], automata [30], and the ZX-calculus [6,7] respectively) and prove that the graphical language is equivalent to the category that they are studying. They can therefore transfer properties of the graphical language (for example, decidability of equivalence of diagrams) to the category they are studying.…”

“…Note that our work uses a 2-colored prop for modeling the two possible directions for edges in an open parity game, while [3,5,27] only have a single type of edges, which are undirected. In [30], the authors use a colored prop to model different kinds of edges, and in particular, they use two colors to model directed edges.…”