On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits

International audienceIn this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d = o(n), must have size at least (n/r^1.1)^{\sqrt{d/r}} when r is o(d) and is strictly less than n^1/1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r. We here prove a lower bound which does not deteriorate with r , however for a specific instance of d = d (n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^{O(1)} variables and degree Θ(log^2(n)) such that any depth four circuit of bounded individual degree r < n^0.2 must have size at least exp(Ω (log^2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017)

Abrasive Blasting Process Optimization: Enhancing Productivity, and Reducing Consumption and Solid/Hazardous Wastes

Abrasive blasting process optimization is aimed at establishing relationships between applied feed rates and resulting productivity and consumption rates. It is clear that the high costs of disposal of the multimedia wastes generated by the dry abrasive blasting processes are of increasing concern in the future of shipbuilding industry. In such circumstances essential care has to be given to all components of the process to enhance productivity and decrease consumption rates. This study discusses most of the process components and their respective effects on blasting productivity and consumption rates briefly and concentrates on two important process parameters, nozzle pressure and abrasive feed rate. Feed rate is a vital process parameter that contributes to the productivity and consumption rates of the process. Subsequently feed rates also can significantly impact the costs bore by Shipbuilding Industry in the form of disposal and environmental costs. Most commonly used abrasives were identified through a rigorous survey and were opted to be used in this study. The approach adopted to develop the relationships consists of a mass balance equation between the expended abrasives and disposed wastes to clean a predetermined area of a plate. The obtained data was further analysed to develop productivity rates and consumption rates for each sample runs. The data was then evaluated to formulate relationships that would enable the derivation of optimum feed rates for desirable productivity and reduced waste generation

Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{\omega(\sqrt{n}\log n)} size depth-4 circuits. Soon after Tavenas's result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{\Omega(\sqrt{n}\log n)} have been proved Kayal et al. and Fournier et al. In particular, using combinatorial design Kayal et al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{\Omega(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix multiplication polynomial (which is in VP) also requires 2^{\Omega(\sqrt{n}\log n)} size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies the property would achieve similar circuit size lower bound for depth-4 circuits. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a very simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare between our current knowledge of depth-4 circuit size lower bounds and determinantal complexity lower bounds. We prove the that the determinantal complexity of iterated matrix multiplication polynomial is \Omega(dn) where d is the number of matrices and n is the dimension of the matrices. So for d=n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. To the best of our knowledge, a \Theta(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for constant d>1 by Jansen.Comment: Extension of the previous uploa

Abrasive Blasting Process Optimization: Enhancing Productivity, and Reducing Consumption and Solid/Hazardous Wastes

Abrasive blasting process optimization is aimed at establishing relationships between applied feed rates and resulting productivity and consumption rates. It is clear that the high costs of disposal of the multimedia wastes generated by the dry abrasive blasting processes are of increasing concern in the future of shipbuilding industry. In such circumstances essential care has to be given to all components of the process to enhance productivity and decrease consumption rates. This study discusses most of the process components and their respective effects on blasting productivity and consumption rates briefly and concentrates on two important process parameters, nozzle pressure and abrasive feed rate. Feed rate is a vital process parameter that contributes to the productivity and consumption rates of the process. Subsequently feed rates also can significantly impact the costs bore by Shipbuilding Industry in the form of disposal and environmental costs. Most commonly used abrasives were identified through a rigorous survey and were opted to be used in this study. The approach adopted to develop the relationships consists of a mass balance equation between the expended abrasives and disposed wastes to clean a predetermined area of a plate. The obtained data was further analysed to develop productivity rates and consumption rates for each sample runs. The data was then evaluated to formulate relationships that would enable the derivation of optimum feed rates for desirable productivity and reduced waste generation

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the constant implicit in the double exponent for all $\Delta = o(\log n/\log \log n).$ This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case $\Delta = 1.$ Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits

Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{\Omega(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in $ACC^0$ can also be computed by algebraic $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits (i.e., circuits of the form -- sums of powers of polynomials) of $2^{\log^{O(1)}n}$ size. Thus they argued that a $2^{\omega(\log^{O(1)}{n})}$ "functional" lower bound for an explicit polynomial $Q$ against $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits would imply a lower bound for the "corresponding Boolean function" of $Q$ against non-uniform $ACC^0$. In their work, they ask if their lower bound be extended to $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits. In this paper, for large integers $n$ and $d$ such that $\omega(\log^2n)\leq d\leq n^{0.01}$, we show that any $\Sigma\mathord{\wedge}\Sigma\Pi$ circuit of bounded individual degree at most $O\left(\frac{d}{k^2}\right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($\in VP$) over $\{0,1\}^{n^2d}$ must have size $n^{\Omega(k)}$. Since Iterated Matrix Multiplication $IMM_{n,d}$ over $\{0,1\}^{n^2d}$ is functionally in $GapL$, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of $ACC^0$ from $GapL$

Enculturational practices in the teaching of proof in mathematics

Mathematics education reform is informed by constructivist theories that forefront student learning of concepts, and by sociocultural theories whose focus is on students’ mastery of mathematical practices. As Cobb (1994) pointed out, these theorizations are inconsistent with one another, leading to conflict as some theorists seek to promote their approach as the correct one. Alternatively, Cobb, and many others in the social constructivism or the situated cognition camps, seek some sort of integration or balancing of these priorities in pedagogical theorizing. Kirshner (2002, 2004, 2008) argued that instead of either selecting one theory or balancing/coordinating the two theories, we should regard each theory as an independent basis for pedagogical practice, and articulate a separate genre of teaching for each. In that spirit, the current study sought to explore pedagogical methods directed exclusively to enculturating students into mathematical practices, particularly, practices of argumentation characteristic of mathematical proof. The researcher worked with a group of 11 average-ability students in the 11-12 age range, over 24, half-hour sessions. At first, students were called upon to discuss various basic geometric terms, and then to present arguments establishing the truth of 10 basic geometric theorems. Students worked together in groups to discuss the problems, and presented their proofs. All sessions were videotaped and transcribed, and each student’s arguments were coded for sophistication on a 4-level system based on the work of Lolli (2005) and Douek (2009). The results indicated that all students advanced in their level of sophistication, most moving from level 1 in which one understands that an explanation is required, but one does not understand the obligation for the explanation to be logically persuasive to level 3 in which one coordinates the elements of the argument in a way that is consistent with logically sound deductive reasoning. The qualitative analysis of interactional processes illustrates the influence of the group’s level of discourse on individual development

Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within P. In this paper, we study the algebraic formula complexity of multiplying d many 2x2 matrices, denoted IMM_d, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear. Formally, for each depth Delta <= log d, we show that any product-depth Delta multilinear formula for IMM_d must have size exp(Omega(Delta d^{1/Delta})). It also follows from this that any multilinear circuit of product-depth Delta for the same polynomial of the above form must have a size of exp(Omega(d^{1/Delta})). In particular, any polynomial-sized multilinear formula for IMM_d must have depth Omega(log d), and any polynomial-sized multilinear circuit for IMM_d must have depth Omega(log d/log log d). Both these bounds are tight up to constant factors. Our lower bound has the following consequences for multilinear formula complexity. Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size s can be converted to one of size s^{O(1)} and depth O(log s); further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to o(log s) without a superpolynomial blow-up in size. Separations from general formulas: Shpilka and Yehudayoff (FnTTCS 2010) asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a non-trivial upper bound for IMM_d implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size s and product-depth Delta = o(log s), general formulas of size s and product-depth Delta cannot be converted to multilinear formulas of size s^{O(1)} and product-depth Delta, when the underlying field has characteristic zero

Storage characterization through gravity-meter experiments and stream flow analysis: Consequences for dry weather flow prediction

Dry weather ow prediction is important as stream ow during dry or rain-less periods, i.e. water available for various usages, is generally very low. Generally hydrological models are employed to predict dry weather ow. However, they keep several parameters whose values need to be determined through calibration, which is a cumbersome process. Furthermore, models usually under perform during low ow periods. The main aim of this study is to predict low ow by utilizing as much less information as possible