Abstract
In this article, we explore the noncommutative analogues, VPnc and VNPnc, 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 VPnc under ≤ abp reductions. To the best of our knowledge, these are the first natural polynomial families shown to be VPnc-complete. Likewise, it turns out that PAL (palindrome polynomials defined from palindromes) are complete for the class VSKEWnc (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 VPnc ≠ VNPnc, we exhibit a strictly infinite hierarchy of p-families, with respect to the projection reducibility, between the complexity classes VPnc and VNPnc (analogous to Ladner’s theorem [Ladner 1975]).
— Additionally, inside VPnc, 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).
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Index Terms
Noncommutative Valiant's Classes: Structure and Complete Problems
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