Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Valiant's Classes: Structure and Complete Problems   In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and
$\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some
striking connections to classical formal language theory. Our main results are
the following: (1) We show that Dyck polynomials (defined from the Dyck
languages of formal language theory) are complete for the class
$\mathrm{VP}_{nc}$ under $\le_{abp}$ reductions. Likewise, it turns out that
$\mathrm{PAL}$ (Palindrome polynomials defined from palindromes) are complete
for the class $\mathrm{VSKEW}_{nc}$ (defined by polynomial-size skew circuits)
under $\le_{abp}$ reductions. The proof of these results is by suitably
adapting the classical Chomsky-Sch\"{u}tzenberger theorem showing that Dyck
languages are the hardest CFLs. (2) Next, we consider the class
$\mathrm{VNP}_{nc}$. It is known~\cite{HWY10a} that, assuming the
sum-of-squares conjecture, the noncommutative polynomial
$\sum_{w\in\{x_0,x_1\}^n}ww$ requires exponential size circuits. We
unconditionally show that $\sum_{w\in\{x_0,x_1\}^n}ww$ is not
$\mathrm{VNP}_{nc}$-complete under the projection reducibility. As a
consequence, assuming the sum-of-squares conjecture, we exhibit a strictly
infinite hierarchy of p-families under projections inside $\mathrm{VNP}_{nc}$
(analogous to Ladner's theorem~\cite{Ladner75}). In the final section we
discuss some new $\mathrm{VNP}_{nc}$-complete problems under
$\le_{abp}$-reductions. (3) Inside $\mathrm{VP}_{nc}$ too we show there is a
strict hierarchy of p-families (based on the nesting depth of Dyck polynomials)
under the $\le_{abp}$ reducibility.

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The categories of this paper are: cmp-lg
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The categories of this paper are: math
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph