Regular D-Length: a Tool for Improved Prefix-Stable Forward Ramsey Factorisations

Recently, Jecker has introduced and studied the regular D-length of a monoid, as the length of its longest chain of regular D-classes. We use this parameter in order to improve the construction, originally proposed by Colcombet, of a deterministic automaton that allows to map a word to one of its forward Ramsey splits: these are a relaxation of factorisation forests that enjoy prefix stability, thus allowing a compositional construction. For certain monoids that have a small regular D-length, our construction produces an exponentially more succinct deterministic automaton. Finally, we apply it to obtain better complexity result for the problem of fast infix evaluation.