Partial derivatives on graphs for Kleene allegories

Abstract

Brunet and Pous showed at LICS 2015 that the equational theory of identity-free relational Kleene lattices (a fragment of Kleene allegories) is decidable in EXPSPACE. In this paper, we show that the equational theory of Kleene allegories is decidable, and is EXPSPACE-complete, answering the first open question posed by their work. The proof proceeds by designing partial derivatives on graphs, which are generalizations of partial derivatives on strings for regular expressions, called Antimirov's partial derivatives. The partial derivatives on graphs give a finite automata construction algorithm as with the partial derivatives on strings.