A String Diagrammatic Axiomatisation of Finite-State Automata

We develop a fully diagrammatic approach to the theory of finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. Moreover, we provide an equational theory that completely axiomatises language equivalence in this new setting. This theory has two notable features. First, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks. Second, the proposed axiomatisation is finitary -- a result which is provably impossible to obtain for the one-dimensional syntax of regular expressions.Comment: Minor corrections, in particular in the proof of completeness (including the ordering of the steps of Brzozowski's algorithm

Compactly accessible categories and quantum key distribution

Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finite-dimensional, they cannot accomodate (co)limit-based constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choice-of-duals functor on the compact part extends canonically to the whole compactly accessible category. As an example, we model a quantum key distribution protocol and prove its correctness categorically.Comment: 26 pages in Logical Methods in Computer Science, Volume 4, Issue 4 (November 17, 2008) lmcs:112