The Power of Programs over Monoids in DA

Nathan Grosshans (1, 2) , Pierre Mckenzie (2) , Luc Segoufin (3)

1 LSV - Laboratoire Spécification et Vérification [Cachan]
2 DIRO - Département d'Informatique et de Recherche Opérationnelle [Montreal]
3 VALDA - Value from Data

The program-over-monoid model of computation originates with Barrington's proof that it captures the complexity class NC^1. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that the class known as DA satisfies tameness and hence that the regular languages recognized by programs over monoids in DA are precisely those recognizable in the classical sense by morphisms from QDA. Third, we show by contrast that the well studied class of monoids called J is not tame and we exhibit a regular language, recognized by a program over a monoid from J, yet not recognizable classically by morphisms from the class QJ. Finally, we exhibit a program-length-based hierarchy within the class of languages recognized by programs over monoids from DA.

Keywords: Programs over monoids; DA; lower-bounds

Domains: Computational Complexity [cs.CC]; Formal Languages and Automata Theory [cs.FL]

Nathan Grosshans, Pierre Mckenzie, Luc Segoufin. The Power of Programs over Monoids in DA. 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), Aug 2017, Aalborg, Denmark. pp.2:1-2:20, ⟨10.4230/LIPIcs.MFCS.2017.2⟩. ⟨hal-02303526⟩