Generating the Functions with Regular Graphs under Composition

TL;DR

This paper explores the generation of regular functions through compositions of functions derived from regular graphs, providing new insights into automata theory and decompositions using the Krohn-Rhodes theorem.

Contribution

It introduces a small, infinite set of functions that generate all regular functions under composition and offers new proofs of the Krohn-Rhodes theorem in modern notation.

Findings

A collection of functions generating regular functions under composition

Interpretation of powerset determinization via input-to-run maps

Detailed decomposition of generating sets using Krohn-Rhodes theorem

Abstract

While automata theory often concerns itself with regular predicates, relations corresponding to acceptance by a finite state automaton, in this article we study the regular functions, such relations which are also functions in the set-theoretic sense. Here we present a small (but necessarily infinite) collection of (multi-ary) functions which generate the regular functions under composition. To this end, this paper presents an interpretation of the powerset determinization construction in terms of compositions of input-to-run maps. Furthermore, known results using the Krohn-Rhodes theorem to further decompose our generating set are spelled out in detail, alongside some coding tricks for dealing with variable length words. This will include two clear proofs of the Krohn-Rhodes Theorem in modern notation.