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).