Threshold circuits of bounded depth

Hajnal, A.; Maass, Wolfgang; Pudlák, Pavel; Turán, György; Szegedy, Márió

doi:10.1016/0022-0000(93)90001-d

Cited by 213 publications

(92 citation statements)

References 9 publications

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“…† Supported in part by an NSF Graduate Fellowship. a number of models of computation; it appears to be at the heart of complexity questions, especially in highly parallel models (shallow circuits) [13,14,25,6,20], space-bounded computation (including the related models of decision trees and branching programs) [8,5,11], and hardness results required for the construction of pseudorandom generators for shallow circuits [17] and for space-bounded computation [5]. Nisan applied communication complexity to threshold circuits with no depth restriction [19].…”

Section: Brief Summarymentioning

confidence: 99%

The Cost of the Missing Bit: Communication Complexity with Help

2001

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We generalize the multiparty communication model of Chandra, Furst, and Lipton (1983) to functions with b-bit output (b = 1 in the CFL model). We allow the players to receive up to b−1 bits of information from an all-powerful benevolent Helper who can see all the input. Extending results of Babai, Nisan, and Szegedy (1992) to this model, we construct families of explicit functions for which Ω(n/c k ) bits of communication are required to find the "missing bit," where n is the length of each player's input and k is the number of players. As a consequence we settle the problem of separating the one-way vs. multiround communication complexities (in the CFL sense) for k ≤ (1 − ) log n players, extending a result of Nisan and Wigderson (1991) who demonstrated this separation for k = 3 players. As a by-product we obtain Ω(n/c k ) lower bounds for the multiparty complexity (in the CFL sense) of new families of explicit boolean functions (not derivable from BNS). The proofs exploit the interplay between two concepts of multicolor discrepancy; discrete Fourier analysis is the basic tool. We also include an unpublished lower bound by A. Wigderson regarding the one-way complexity of the 3-party pointer jumping function.

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Section: Brief Summarymentioning

confidence: 99%

The Cost of the Missing Bit: Communication Complexity with Help

2001

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“…This leaves the depth unchanged and at most doubles the size which makes no difference for our asymptotic results.) We recall the "discriminator lemma" of Hajnal et al [15]: Our goal is to show that as long as D does not assign too much probability weight to any single point then there is some parity function χA (where A is small as described below) which has nonnegligible correlation with f under D. Toward this end we use the main lemma of Linial et al [23]: Since each Fourier coefficientĈD(A) has magnitude at most 1, we have that |A| k |ĈD(A)| is at most n k , and thus |ED[fχA]| 1/(2sn k ) for some |A| k. Thus we have proved the following lemma:…”

Section: The Algorithmmentioning

confidence: 99%

Learnability beyond AC ⁰

Jackson

Klivans

Servedio

2002

Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing

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We give an algorithm to learn constant-depth polynomialsize circuits augmented with majority gates under the uniform distribution using random examples only. For circuits which contain a polylogarithmic number of majority gates the algorithm runs in quasipolynomial time. This is the first algorithm for learning a more expressive circuit class than the class AC 0 of constant-depth polynomial-size circuits, a class which was shown to be learnable in quasipolynomial time by Linial, Mansour and Nisan in 1989. Our approach combines an extension of some of the Fourier analysis from Linial et al. with hypothesis boosting. We also show that under a standard cryptographic assumption our algorithm is essentially optimal with respect to both running time and expressiveness (number of majority gates) of the circuits being learned.

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“…For example, for constant depth threshold circuits only slightly super linear lower 1 Since gv is not necessarily a symmetric function, we need to order the inputs to v, so we know which input is the first variable of gv etc. bounds are known [12] (exponential lower bounds are known for depth 2 [10]). …”

Section: Circuits With Arbitrary Gatesmentioning

confidence: 99%

Lower bounds for matrix product, in bounded depth circuits with arbitrary gates

Raz

Shpilka

2001

Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing

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We prove super-linear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bounds apply for several explicit functions, and, most importantly, for matrix product. In particular, we obtain the following results:1. We show that the number of edges in any constant depth arithmetic circuit for matrix product (over any field) is super-linear in m 2 (where m × m is the size of each matrix). That is, the lower bound is superlinear in the number of input variables. Moreover, if the circuit is bilinear the result applies also for the case where the circuit gets for free any product of two linear functions.2. We show that the number of edges in any constant depth arithmetic circuit for the trace of the product of 3 matrices (over fields with characteristic 0) is superlinear in m 2 . (Note that the trace is a single-output function).3. We give explicit examples for n Boolean functions f1, . . . , fn, such that any constant depth Boolean circuit with arbitrary gates for f1, ..., fn has a superlinear number of edges. The lower bound is proved also for circuits with arbitrary gates over any finite field. The bound applies for matrix product over finite fields as well as for several other explicit functions.

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Threshold circuits of bounded depth

Cited by 213 publications

References 9 publications

The Cost of the Missing Bit: Communication Complexity with Help

The Cost of the Missing Bit: Communication Complexity with Help

Learnability beyond AC ⁰

Lower bounds for matrix product, in bounded depth circuits with arbitrary gates

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Threshold circuits of bounded depth

Cited by 213 publications

References 9 publications

The Cost of the Missing Bit: Communication Complexity with Help

The Cost of the Missing Bit: Communication Complexity with Help

Learnability beyond AC 0

Lower bounds for matrix product, in bounded depth circuits with arbitrary gates

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Product

Resources

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Learnability beyond AC ⁰