Noncommutative Valiant's Classes

Abstract

In this article, we explore the noncommutative analogues, VP nc and VNP nc , of Valiant’s algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the following: — We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for the class VP nc under ≤ abp reductions. To the best of our knowledge, these are the first natural polynomial families shown to be VP nc -complete. Likewise, it turns out that PAL (palindrome polynomials defined from palindromes) are complete for the class VSKEW nc (defined by polynomial-size skew circuits) under ≤ abp reductions. The proof of these results is by suitably adapting the classical Chomsky-Schützenberger theorem showing that Dyck languages are the hardest CFLs. — Assuming that VP nc ≠ VNP nc , we exhibit a strictly infinite hierarchy of p-families, with respect to the projection reducibility, between the complexity classes VP nc and VNP nc (analogous to Ladner’s theorem [Ladner 1975]). — Additionally, inside VP nc , we show that there is a strict hierarchy of p-families (based on the nesting depth of Dyck polynomials) with respect to the ≤ abp reducibility (defined explicitly in this article).