Bringing Algebraic Hierarchical Decompositions to Concatenative Functional Languages

Programming languages tend to evolve over time to use more and more concepts from theoretical computer science. Still, there is a gap between programming and pure mathematics. Not all theoretical results have realized their promising applications. The algebraic decomposition of finite state automata (Krohn-Rhodes Theory) constructs an emulating hierarchical structure from simpler components for any computing device. These decompositions provide ways to understand and control computational processes, but so far the applications were limited to theoretical investigations. Here, we study how to apply algebraic decompositions to programming languages. We use recent results on generalizing the algebraic theory to the categorical level (from semigroups to semigroupoids) and work with the special class of concatenative functional programming languages. As a first application of semigroupoid decompositions, we start to design a family of programming languages with an explicit semigroupoid representation.