7,735 research outputs found

    Size, Depth and Energy of Threshold Circuits Computing Parity Function

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    We investigate relations among the size, depth and energy of threshold circuits computing the n-variable parity function PAR_n, where the energy is a complexity measure for sparsity on computation of threshold circuits, and is defined to be the maximum number of gates outputting "1" over all the input assignments. We show that PAR_n is hard for threshold circuits of small size, depth and energy: - If a depth-2 threshold circuit C of size s and energy e computes PAR_n, it holds that 2^{n/(elog ^e n)} ? s; and - if a threshold circuit C of size s, depth d and energy e computes PAR_n, it holds that 2^{n/(e2^{e+d}log ^e n)} ? s. We then provide several upper bounds: - PAR_n is computable by a depth-2 threshold circuit of size O(2^{n-2e}) and energy e; - PAR_n is computable by a depth-3 threshold circuit of size O(2^{n/(e-1)} + 2^{e-2}) and energy e; and - PAR_n is computable by a threshold circuit of size O((e+d)2^{n-m}), depth d + O(1) and energy e + O(1), where m = max (((e-1)/(d-1))^{d-1}, ((d-1)/(e-1))^{e-1}). Our lower and upper bounds imply that threshold circuits need exponential size if both depth and energy are constant, which contrasts with the fact that PAR_n is computable by a threshold circuit of size O(n) and depth 2 if there is no restriction on the energy. Our results also suggest that any threshold circuit computing the parity function needs depth to be sparse if its size is bounded

    Three Puzzles on Mathematics, Computation, and Games

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    In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

    Exponential Lower Bounds for Threshold Circuits of Sub-Linear Depth and Energy

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    Circuit Complexity of Visual Search

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    We study computational hardness of feature and conjunction search through the lens of circuit complexity. Let x=(x1,...,xn)x = (x_1, ... , x_n) (resp., y=(y1,...,yn)y = (y_1, ... , y_n)) be Boolean variables each of which takes the value one if and only if a neuron at place ii detects a feature (resp., another feature). We then simply formulate the feature and conjunction search as Boolean functions FTRn(x)=i=1nxi{\rm FTR}_n(x) = \bigvee_{i=1}^n x_i and CONJn(x,y)=i=1nxiyi{\rm CONJ}_n(x, y) = \bigvee_{i=1}^n x_i \wedge y_i, respectively. We employ a threshold circuit or a discretized circuit (such as a sigmoid circuit or a ReLU circuit with discretization) as our models of neural networks, and consider the following four computational resources: [i] the number of neurons (size), [ii] the number of levels (depth), [iii] the number of active neurons outputting non-zero values (energy), and [iv] synaptic weight resolution (weight). We first prove that any threshold circuit CC of size ss, depth dd, energy ee and weight ww satisfies logrk(MC)ed(logs+logw+logn)\log rk(M_C) \le ed (\log s + \log w + \log n), where rk(MC)rk(M_C) is the rank of the communication matrix MCM_C of a 2n2n-variable Boolean function that CC computes. Since CONJn{\rm CONJ}_n has rank 2n2^n, we have ned(logs+logw+logn)n \le ed (\log s + \log w + \log n). Thus, an exponential lower bound on the size of even sublinear-depth threshold circuits exists if the energy and weight are sufficiently small. Since FTRn{\rm FTR}_n is computable independently of nn, our result suggests that computational capacity for the feature and conjunction search are different. We also show that the inequality is tight up to a constant factor if ed=o(n/logn)ed = o(n/ \log n). We next show that a similar inequality holds for any discretized circuit. Thus, if we regard the number of gates outputting non-zero values as a measure for sparse activity, our results suggest that larger depth helps neural networks to acquire sparse activity

    On the Computational Power of Radio Channels

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    Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

    08381 Abstracts Collection -- Computational Complexity of Discrete Problems

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    From the 14th of September to the 19th of September, the Dagstuhl Seminar 08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work as well as open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
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