Characterizing NC1 with Typed Monoids

TL;DR

This paper characterizes the complexity class NC1 using typed monoids and logical definability, extending algebraic automata theory methods to better understand computational complexity.

Contribution

It extends the algebraic characterization of complexity classes from TC0 to NC1 using typed monoids and logical frameworks, answering open questions about quantifier collapse.

Findings

NC1 characterized by unary quantifiers over regular languages

Finite monoid multiplication quantifiers can be replaced by unary quantifiers

General collapse result for interpretations over strings

Abstract

Krebs et al. (2007) gave a characterization of the complexity class TC0 as the class of languages recognized by a certain class of typed monoids. The notion of typed monoid was introduced to extend methods of algebraic automata theory to infinite monoids and hence characterize classes beyond the regular languages. We advance this line of work beyond TC0 by giving a characterization of NC1. This is obtained by first showing that NC1 can be defined as the languages expressible in an extension of first-order logic using only unary quantifiers over regular languages. The expressibility result is a consequence of a general result showing that finite monoid multiplication quantifiers of higher dimension can be replaced with unary quantifiers in the context of interpretations over strings, which also answers a question of Lautemann et al. (2001). We establish this collapse result for a much more general class of interpretations using results on interpretations due to Boja\'nczyk et al. (2019), which may be of independent interest.