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

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Inferring Energy Bounds via Static Program Analysis and Evolutionary Modeling of Basic Blocks

The ever increasing number and complexity of energy-bound devices (such as the ones used in Internet of Things applications, smart phones, and mission critical systems) pose an important challenge on techniques to optimize their energy consumption and to verify that they will perform their function within the available energy budget. In this work we address this challenge from the software point of view and propose a novel parametric approach to estimating tight bounds on the energy consumed by program executions that are practical for their application to energy verification and optimization. Our approach divides a program into basic (branchless) blocks and estimates the maximal and minimal energy consumption for each block using an evolutionary algorithm. Then it combines the obtained values according to the program control flow, using static analysis, to infer functions that give both upper and lower bounds on the energy consumption of the whole program and its procedures as functions on input data sizes. We have tested our approach on (C-like) embedded programs running on the XMOS hardware platform. However, our method is general enough to be applied to other microprocessor architectures and programming languages. The bounds obtained by our prototype implementation can be tight while remaining on the safe side of budgets in practice, as shown by our experimental evaluation.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854). Improved version of the one presented at the HIP3ES 2016 workshop (v1): more experimental results (added benchmark to Table 1, added figure for new benchmark, added Table 3), improved Fig. 1, added Fig.