# 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.

#### 342 Citations

On the power of small-depth threshold circuits

- Mathematics, Computer Science
- computational complexity
- 2005

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}

- Mathematics, Computer Science
- GOSLER Final Report
- 1995

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

- Mathematics, Computer Science
- Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

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

- 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

- Mathematics, Computer Science
- computational complexity
- 1998

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

- Mathematics, Computer Science
- 2010 IEEE 25th Annual Conference on Computational Complexity
- 2010

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

- Mathematics, Computer Science
- [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
- 1991

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

- Mathematics, Computer Science
- SIAM J. Comput.
- 1998

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

- Computer Science, Mathematics
- STACS
- 1995

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

- Mathematics, Computer Science
- IEEE Trans. Inf. Theory
- 1994

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

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