Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions: - An explicit hitting set of quasipolynomial size for UPT circuits, - An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), - An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016]

Near-optimal Bootstrapping of Hitting Sets for Algebraic Models

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel [Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is an explicit hitting set for all $n$-variate degree $s$, size $s$ algebraic circuits of size $(s+1)^n$. In this paper, we prove the following results: - Let $\epsilon > 0$ be a constant. For a sufficiently large constant $n$ and all $s > n$, if we have an explicit hitting set of size $(s+1)^{n-\epsilon}$ for the class of $n$-variate degree $s$ polynomials that are computable by algebraic circuits of size $s$, then for all $s$, we have an explicit hitting set of size $s^{\exp \circ \exp (O(\log^\ast s))}$ for $s$-variate circuits of degree $s$ and size $s$. That is, if we can obtain a barely non-trivial exponent compared to the trivial $(s+1)^{n}$ sized hitting set even for constant variate circuits, we can get an almost complete derandomization of PIT. - The above result holds when "circuits" are replaced by "formulas" or "algebraic branching programs". This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18] who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most $(s^{n^{0.5 - \delta}})$ (where $\delta>0$ is any constant). Hence, our work significantly weakens the hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial saving over the trivial hitting set, and also presents the first such result for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the older version of the paper. Additionally, the stronger theorem now holds even for subclasses of algebraic circuits, such as algebraic formulas and algebraic branching program

On Annihilators of Explicit Polynomial Maps

We study the algebraic complexity of annihilators of polynomials maps. In particular, when a polynomial map is `encoded by' a small algebraic circuit, we show that the coefficients of an annihilator of the map can be computed in PSPACE. Even when the underlying field is that of reals or complex numbers, an analogous statement is true. We achieve this by using the class VPSPACE that coincides with computability of coefficients in PSPACE, over integers. As a consequence, we derive the following two conditional results. First, we show that a VP-explicit hitting set generator for all of VP would separate either VP from VNP, or non-uniform P from PSPACE. Second, in relation to algebraic natural proofs, we show that proving an algebraic natural proofs barrier would imply either VP $\neq$ VNP or DSPACE($\log^{\log^{\ast}n} n$) $\not\subset$ P

On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, M\"oller and Mora (1993), and M\"oller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.Comment: Accepted to STACS 202

If VNP Is Hard, Then so Are Equations for It

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee, Kumar, Ramya, Saptharishi and Tengse (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations

Monotone Classes Beyond VNP

We study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat '08, Koiran-Perifel '09, Malod '11, Mahajan-Rao '13) that is believed to be larger than VNP. We show an exponential separation between the monotone version of Poizat's definition, and monotone VNP. We also show that unlike their non-monotone counterparts, these monotone analogues are not equivalent, with exponential separations in some cases. The primary motivation behind our work is to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrube\v{s} and Yehudayoff (2021). In that context, we are able to show that transparent polynomials of large sparsity are hard for the monotone analogues of all known definitions of VPSPACE, except for the one due to Poizat.Comment: 26 pages; the draft has been shortened and simplified to now focus solely on monotone classes beyond VN