J. H astad and M. Goldmann. On the power of small-depth threshold circuits. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 610-618, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the central motivating problem we address. See [Gr99] and [Gr00] for somewhat di erent perspectives. This problem shares some of the diculties of nding lower bounds for depth 2 and depth 3 threshold circuits. Among the strongest lower bounds of this type is the result of H astad and Goldmann [HG] that says that the generalized inner product function requires exponential size MAJ MAJ AND O( 1=2 ) log n) circuits. This implies, of course, a similar lower bound against MAJMOD 2 AND (1=2 ) log n circuits computing generalized inner product. We suspect that even the simpler MOD 3 function ....

.... parity and a MOD 3 AND circuit can be written as an exponential sum (also variously known as a character sum or a generalized Gaussian sum) Evaluations of such sums were also instrumental in the communication complexity lower bound of Babai, Nisan and Szegedy [BNS] on which the H astad Goldmann [HG] result is based. Character sums, which originated with Gauss in the study of cyclotomic elds and quadratic reciprocity, have been intensively studied in the number theoretic literature (see, e.g. LN] and [Sch] Here we develop a new technique for evaluating the type of sums that arise in ....

J. H astad and M. Goldmann, On the power of small-depth threshold circuits, in Computational Complexity, 1 (1991) 113-129.

....the global G of C circuit. 2 Results The proof of the main theorem is based on the ffl discriminator lemma of Hajnal, Maass, Pudl ak, Szegedy and Tur an [HMPST] It is the basis for most of the lower bound proofs on depth three (or more) circuits with a threshold on top (e.g. HMPST] Gr] HG] Lemma 2.1. Let T be a threshold circuit consisting of a threshold gate over subcircuits c 1 ; c s , each taking up to n inputs. Thus, T outputs 1 on input x 2 f0; 1g n if and only if P s i=1 c i (x) t, for some integer t which is fixed for the circuit T . Let T compute the Boolean ....

J. H astad and M. Goldmann, On the power of small-depth threshold circuits, in Computational Complexity, 1 (1991) 113-129.

....that if the same result could be proved in the non monotone case, the Hstad switching lemma could be used to show the separation for all k. In the monotone setting, the separation between depth k and depth k Gamma 1 perceptrons for all k follows from a stronger result by Hstad and Goldmann [10] that separates boolean circuits of depth k from threshold circuits of depth k Gamma 1. In this paper we show that there are functions computable by linear size boolean circuits of depth k that require super polynomial size perceptrons of depth k Gamma 1, for k logn= 6log logn) and ....

Johan Hstad and Mikael Goldmann. On the power of small-depth threshold circuits. Computational Complexity, 1(2):113--129, 1991. 16

....This class includes the Generalized Inner Product function [BNS] and Majority of Majorities. We get polylog(n) upper bounds for this class of functions when the number of players is at least 2 log n. This work is motivated by an approach suggested by the results of H astad Goldmann [HG](1991) Yao [Y] 1990) and Beigel Tarui [BT] 1991) to give explicit functions outside the circuit complexity class ACC. We combine ideas from [HG] with our lower bounds on SM complexity to derive lower bounds for certain depth 2 circuits computing the GAF function. This is a signi cantly ....

....functions when the number of players is at least 2 log n. This work is motivated by an approach suggested by the results of H astad Goldmann [HG] 1991) Yao [Y] 1990) and Beigel Tarui [BT] 1991) to give explicit functions outside the circuit complexity class ACC. We combine ideas from [HG] with our lower bounds on SM complexity to derive lower bounds for certain depth 2 circuits computing the GAF function. This is a signi cantly expanded version of [BKL] y Email: laci cs.uchicago.edu. Partially supported by NSF Grant CCR 9014562. z Email: panni cs.utexas.edu. Work done ....

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J. Hastad, M. Goldmann. On the Power of Small-Depth Threshold Circuits. Computational Complexity, 1, 1991, pp. 113-129.

....t 2 ;k and GAFZn;k , respectively. In [BGKL] upper bounds are also proved on a class of functions de ned by certain depth 2 circuits. This class included the Generalized Inner Product (GIP) function, which was a prime example in the study and applications of multiparty communication complexity [BNS,G,HG,RW], and the Majority of Majorities function. Babai, Kimmel, and Lokam [BKL] show an O(n 0:92 ) upper bound for GAF Z t 2 ;3 , i.e. on the 3 party SM complexity of GAF G;k , when G = Z t 2 . More generally, they show an O(n H(1=k) upper bound for GAF Z t 2 ;k . Pudl ak, R odl, and ....

J. Hastad, M. Goldmann. On the Power of Small-Depth Threshold Circuits. Computational Complexity, 1, 1991, pp. 113-129.

....and a decision tree lower bound. Department of Computer Science, Eotvos University, Budapest, Address: M uzeum krt.6 8, H 1088 Budapest, HUNGARY; E mail: grolmusz cs.elte.hu 1 1 INTRODUCTION Methods in communication complexity have become standard tools in circuit complexity theory ([19], 23] 22] 26] 29] 12] 15] 10] These methods are also used with success for giving lower bounds for the depth of decision trees with linear or low degree test functions [11] 26] 37] Another important tool in examining Boolean function complexity is representing the Boolean ....

....product function. Nisan [26] called a Boolean function f : f Gamma1; 1g n f Gamma1; 1g a threshold gate of degree d (or d threshold gate) if f can be expressed as a sign of a real polynomial of degree at most d. Then he has built a random (d 1) party protocol using the results of [19] which evaluates the d threshold gates with a small number of communicated bits, and then, using the BNS lower bound [2] the size lower bound of Omega Gamma c d n= log 2 n) follows for d = O(log n) We instead of symmetricity or degree conditions require the L 1 norms of the ....

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J. H astad and M. Goldmann, On the power of the small-depth threshold circuits, Computational Complexity, 1 (1991), pp. 113--129.

....contained in it. The computational model used in this paper assumes unit weight threshold circuits, where the weights w i 2 f1; Gamma1g, for all i = 1; 2; k. Unit weight threshold circuits are considerably less expensive to implement than threshold circuits of large weight. Recent papers [1, 2, 5, 3, 7, 8, 11] have begun to study the class of functions that are computable by threshold circuits that have constant depth and use a polynomial number of edges. The class of these functions is commonly called class TC 0 . Several important operations, such as iterated addition, multiplication, and the ....

H astad, J., Goldmann, M.: On the power of small-depth threshold circuits. Computational Complexity (1991) 113--129

....interest to characterize Boolean functions for which most of the weight of the Fourier coe cients is concentrated on a set of polynomial size in n. 7. 2 TC0 and noise sensitivity Noise sensitivity seems related to another class of boolean functions threshold circuits of bounded depths see [30, 17]. In a threshold circuit each gate is a weighted majority function. Conjecture 7.3. Let f be a boolean function given by a monotone threshold circuit of depth c and size M . Then J(f) O(1) log M) c 1 : Thus, for 1= O(1) log M) c 1 we expect that VAR(f; is bounded away from zero. ....

....where all the threshold gates are balanced are uniformly stable. And in particular, J(f) O(1) Possibly, functions in this class of functions approximate arbitrary well arbitrary uniform stable monotone Boolean functions. Conjecture 7. 3 implies theorems of Yau [30] and H astad and Goldmann [17]. They proved that the and or tree (or equivalently the example of ternary tree of Section 6) does not belong to monotone TC0; i.e. it cannot be expressed as a monotone bounded depth circuit of polynomial size. The results of Yau and H astad are still open for the non monotone case. This would ....

J. Hastad and M. Goldmann, On the power of small-depth threshold circuits, Computational Complexity, 1 (1991), 113-129.

....proving lower bounds on the computational complexity of explicit Boolean functions. However, in spite of many results since then, no lower bounds for multiparty games with more than log n players have been proved. Attacking the barrier of log n is especially interesting, since by the results of Hastad Goldmann (1991) such bounds are connected to lower bounds on ACC circuits. In this paper we introduce a restricted model of conservative one way protocols, and prove lower bounds for pointer jumping with up to about n 1=3 players. Due to the restriction to conservative protocols, our lower bounds do not imply ....

....and non conservative protocols. Open problems are discussed in Section 7. The lower and upper bounds for small k were also reported in Damm Jukna (1995) The preliminary version of this paper appeared in Damm, Jukna Sgall (1996) 2. Related work Our main motivation is the result of Hastad Goldmann (1991), based on Yao (1990) and following also easily from an improvement of Yao (1990) by Beigel Tarui (1994) They show that any function in ACC (i.e. computed by polynomial size, bounded depth and unbounded fan in circuit with gates computing AND, OR, NOT, and MODm for a fixed m) can be computed ....

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J. H astad and M. Goldmann, On the power of small-depth threshold circuits. Comput complexity 1 (1991), 113--129.

....the rigidity of certain matrices. The combinatorial approach (which can be also characterized as information theoretic) leads to the multiparty communication complexity introduced by Chandra, Furst, and Lipton [16] which was used to prove lower bounds on circuit complexity in various situations [7, 15, 20], and to the concept of computation with common bits introduced by Valiant [32, 33] In this paper, we generalize the concept of the rigidity of a matrix to the rigidity of a tensor, where a tensor is essentially a set of matrices. This gives a tool for proving lower bounds on semilinear circuits ....

.... k ) for the generalized inner product [7] This means that we have no lower bounds at all for k =# (log n) If we could extend the lower bounds on the almost simultaneous communication complexity to k = polylog(n) it would yield a lower bound on ACC circuits, which is a major open problem; see [8, 15, 34]. One function often considered in this context is pointer jumping in a directed acyclic graph with one source and k additional levels with n vertices on each level. The out degree of each vertex is 1, except for the last level. The inputs are divided between the players so that every player ....

J. H astad and M. Goldmann, On the power of small-depth threshold circuits, Comput. Complexity, 1 (1991), pp. 113--129.

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J. H astad and M. Goldmann. On the power of small-depth threshold circuits. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 610-618, 1990.

....All lower bounds known so far are for very limited classes. In particular there are good lower bounds for depth 2 circuits with small weights by Hajnal et al. 6] and more recently by Krause [10] and Krause Waack [11] The techniques of Hajnal et al. were extended by Hastad and Goldmann [7] to deal with depth 3 circuits with small weights and small bottom fanin. These lower bounds agree very well with our intuition, as do the results about monotone threshold circuits by Yao [21] extended in [7] The first surprise was presented by Allender [1] who, inspired by the results of Toda ....

....Krause Waack [11] The techniques of Hajnal et al. were extended by Hastad and Goldmann [7] to deal with depth 3 circuits with small weights and small bottom fanin. These lower bounds agree very well with our intuition, as do the results about monotone threshold circuits by Yao [21] extended in [7]) The first surprise was presented by Allender [1] who, inspired by the results of Toda [18] proved that depth 3 threshold circuits of subexponential size could do all of AC 0 . Yao [22] extended this to ACC 0 which consists of all functions computable by polynomial size constant depth ....

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J. H astad and M. Goldmann, On the power of small-depth threshold circuits, in Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, 1990, 610--618.

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J. Hastad and M. Goldmann. On the power of small-depth threshold circuits. Computational Complexity, 1(2):113-129, 1991.

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J. H astad and M. Goldmann, On the power of the small-depth threshold circuits, Computational Complexity, 1 (1991), pp. 113--129.

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Johan H astad and Mikael Goldmann, On the power of the small-depth threshold circuits. Computational Complexity 1 (1991), 113--129.

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