autoAx: An Automatic Design Space Exploration and Circuit Building Methodology utilizing Libraries of Approximate Components

Approximate computing is an emerging paradigm for developing highly energy-efficient computing systems such as various accelerators. In the literature, many libraries of elementary approximate circuits have already been proposed to simplify the design process of approximate accelerators. Because these libraries contain from tens to thousands of approximate implementations for a single arithmetic operation it is intractable to find an optimal combination of approximate circuits in the library even for an application consisting of a few operations. An open problem is "how to effectively combine circuits from these libraries to construct complex approximate accelerators". This paper proposes a novel methodology for searching, selecting and combining the most suitable approximate circuits from a set of available libraries to generate an approximate accelerator for a given application. To enable fast design space generation and exploration, the methodology utilizes machine learning techniques to create computational models estimating the overall quality of processing and hardware cost without performing full synthesis at the accelerator level. Using the methodology, we construct hundreds of approximate accelerators (for a Sobel edge detector) showing different but relevant tradeoffs between the quality of processing and hardware cost and identify a corresponding Pareto-frontier. Furthermore, when searching for approximate implementations of a generic Gaussian filter consisting of 17 arithmetic operations, the proposed approach allows us to identify approximately $10^3$ highly important implementations from $10^{23}$ possible solutions in a few hours, while the exhaustive search would take four months on a high-end processor.Comment: Accepted for publication at the Design Automation Conference 2019 (DAC'19), Las Vegas, Nevada, US

Approximate Computing in Coarse Grained Reconfigurable Architecture

Approximate computing has emerged as a new computing paradigm capable of reducing the power requirements for or accelerating some workloads. Due to cascading error and the nature of binary arithmetic, it is difficult to predict the exact effects that approximation may have on an error tolerant workload. In this work, we implemented configurable levels of approximation into a Coarse Grained Reconfigurable Architecture (CGRA) to study the effects of error tolerant algorithms on an approximate CGRA. We will use the CGRA Compilation Framework which simulates a CGRA using gem5, and we will implement the approximate hardware using multiple different approximate arithmetic modules included in Low Power Approximate Computing Library. Finally, we will perform a hardware level simulation on approximate modules to estimate the reduction in power from using approximate hardware

Approximate Euclidean Ramsey theorems

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large grid, if $N$ is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval $[0,L]$ on the line contains an arbitrary long approximate arithmetic progression, if $L$ is large enough. (ii) every dense separated set of points in the $d$-dimensional cube $[0,L]^d$ in \RR^d contains an arbitrary large approximate grid, if $L$ is large enough. A further generalization for any finite pattern in \RR^d is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in \RR^d contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.Comment: 11 pages, 1 figure

Computation Over Gaussian Networks With Orthogonal Components

Function computation of arbitrarily correlated discrete sources over Gaussian networks with orthogonal components is studied. Two classes of functions are considered: the arithmetic sum function and the type function. The arithmetic sum function in this paper is defined as a set of multiple weighted arithmetic sums, which includes averaging of the sources and estimating each of the sources as special cases. The type or frequency histogram function counts the number of occurrences of each argument, which yields many important statistics such as mean, variance, maximum, minimum, median, and so on. The proposed computation coding first abstracts Gaussian networks into the corresponding modulo sum multiple-access channels via nested lattice codes and linear network coding and then computes the desired function by using linear Slepian-Wolf source coding. For orthogonal Gaussian networks (with no broadcast and multiple-access components), the computation capacity is characterized for a class of networks. For Gaussian networks with multiple-access components (but no broadcast), an approximate computation capacity is characterized for a class of networks.Comment: 30 pages, 12 figures, submitted to IEEE Transactions on Information Theor

Symbolic arithmetic knowledge without instruction

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations and their performance suffers if this nonsymbolic system is impaired. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is required. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics