Threshold circuits of bounded depth

@article{Hajnal1987ThresholdCO,
  title={Threshold circuits of bounded depth},
  author={A. Hajnal and W. Maass and P. Pudl{\'a}k and M. Szegedy and Gy{\"o}rgy Tur{\'a}n},
  journal={28th Annual Symposium on Foundations of Computer Science (sfcs 1987)},
  year={1987},
  pages={99-110}
}
We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3, and from probabilistic polynomial size threshold circuits of depth 2. We also consider circuits of unreliable threshold gates, circuits of imprecise threshold gates and threshold quantifiers. 
On the power of small-depth threshold circuits
TLDR
It is proved that there are monotone functionsfk that can be computed in depthk and linear size ⋎, ⋏-circuits but require exponential size to compute by a depthk−1 monot one weighted threshold circuit. Expand
On Lower Bounds for the Depth of Threshold Circuits with Weights from {-1, 0, +1}
  • A. Albrecht
  • Mathematics, Computer Science
  • GOSLER Final Report
  • 1995
TLDR
The notation of sharp bounded density is introduced and it is proved that boolean functions f n (x) satisfying this property cannot be realized by threshold circuits of depth two with weights from {−1,0,+1}. Expand
On the power of small-depth threshold circuits
  • J. Håstad, M. Goldmann
  • Mathematics, Computer Science
  • Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
  • 1990
TLDR
It is proved that there are monotone functions f/sub k/ that can be computed on depth k and linear size AND, OR circuits but require exponential-size to be computed by a depth-(k-1) monot one weighted threshold circuit. Expand
Optimal Lower Bounds on the Depth of Polynomial-Size Threshold Circuits for Some Arithmetic Functions
  • I. Wegener
  • Mathematics, Computer Science
  • Inf. Process. Lett.
  • 1993
Abstract Depth 3 is necessary for threshold circuits of polynomial size to compute squaring, powering, inversion or division. These lower bounds are optimal since they match existing upper bounds.
Computing Boolean functions by polynomials and threshold circuits
TLDR
A function with small size unbounded weight threshold—AND circuits for which all threshold—XOR circuits have exponentially many nodes is presented, which answers the basic question of separating subsets of the hypercube by hypersurfaces induced by sparse real polynomials. Expand
Exact Threshold Circuits
TLDR
Many of the results can be seen as evidence that this class is a strict subclass of depth two threshold circuits --- thus it is argued that efforts in proving lower bounds should be directed towards this class. Expand
On the computational power of sigmoid versus Boolean threshold circuits
TLDR
It turns out that, for any constant depth d, polynomial size sigmoid threshold circuits with polynomially bounded weights compute exactly the same Boolean functions as the corresponding circuits with Boolean threshold gates. Expand
Simulating Threshold Circuits by Majority Circuits
TLDR
It is proved that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size, depth-2 majority circuit and it is shown that such a simulation is possible even if the depth d grows with the number of variables n. Expand
On Realizing Iterated Multiplication by Small Depth Threshold Circuits
TLDR
It is obtained that for iterated multiplication of n-bit numbers, in contrast to multiplication, powering, and division, decomposition via Chinese Remaindering does not yield efficient depth 3 threshold circuits. Expand
Lower bounds on threshold and related circuits via communication complexity
TLDR
Using communication complexity concepts and techniques, linear and almost-linear lower bounds on the size of circuits implementing certain functions and some applications to threshold circuit complexity are derived. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 74 REFERENCES
Log Depth Circuits for Division and Related Problems
TLDR
This work presents optimal depth Boolean circuits for integer division, powering, and multiple products and describes an algorithm for testing divisibility that is optimal for both depth and space. Expand
Parity, circuits, and the polynomial-time hierarchy
  • M. Furst, J. Saxe, M. Sipser
  • Mathematics, Computer Science
  • 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
  • 1981
TLDR
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy. Expand
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
TLDR
It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm. Expand
Almost optimal lower bounds for small depth circuits
  • J. Håstad
  • Computer Science, Mathematics
  • STOC '86
  • 1986
TLDR
Improved lower bounds for the size of small depth circuits computing several functions are given and it is shown that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k 1. Expand
Separating the polynomial-time hierarchy by oracles
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all theExpand
Constant Depth Reducibility
The purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing AND’s, OR’s and NOT’s, with no bound placedExpand
Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version)
  • A. Yao
  • Mathematics, Computer Science
  • FOCS
  • 1985
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all theExpand
The Depth of All Boolean Functions
Every Boolean function of n arguments has a circuit of depth $n + 1$ over the basis $\{ f|f:\{ 0,1\} ^2 \to \{ 0,1\} \} $.
A Logic for Constant-Depth Circuits
TLDR
This note presents an extended first-order logic designed to be exactly equivalent in expressiveness to polynomialsize, constant-depth, unbounded-fan-in circuits constructed by Turing machines of bounded computational complexity. Expand
A theorem on probabilistic constant depth Computations
TLDR
Stockmeyer [St] showed that probabilistic bounded depth circuits can approximate the exact number of ones in the input with very low probability of error. Expand
...
1
2
3
4
5
...