# New algorithms and lower bounds for circuits with linear threshold gates

@article{Williams2014NewAA, title={New algorithms and lower bounds for circuits with linear threshold gates}, author={Ryan Williams}, journal={Proceedings of the forty-sixth annual ACM symposium on Theory of computing}, year={2014} }

Let ACC o THR be the class of constant-depth circuits comprised of AND, OR, and MODm gates (for some constant m > 1), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a "midpoint" between ACC (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of 2no(1) ACC o THR circuits of size 2no(1), on…

## 80 Citations

A Satisfiability Algorithm for Depth Two Circuits with a Sub-Quadratic Number of Symmetric and Threshold Gates

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

A deterministic algorithm that, given a circuit with n variables andm gates, counts the number of satisfying assignments in time 2 n−Ω, which runs in time super-polynomially faster than 2 n if m= O(n2/ logbn) for some constant b> 0.

Fooling Constant-Depth Threshold Circuits

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2021

This work presents new constructions of pseudorandom generators (PRGs) for two of the most widely studied non-uniform circuit classes in complexity theory: linear threshold (LTF) circuits of arbitrary constant depth and super-linear size and De Morgan formulas of size s.

Super-linear gate and super-quadratic wire lower bounds for depth-two and depth-three threshold circuits

- Computer ScienceSTOC
- 2016

It is proved that for all ε ≪ √log(n)/n, the linear-time computable Andreev’s function cannot be computed on a (1/2+ε)-fraction of n-bit inputs by depth-two circuits of o(ε3 n3/2/log3 n) gates, nor can it be computed with o(�3 n5/ 2/log7/2 n) wires.

Smaller ACC0 Circuits for Symmetric Functions

- Computer ScienceITCS
- 2022

This paper shows how to construct MODm circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC0[m] circuits when m is a prime power, and shows that depth-3 CC0 circuits can compute any symmetric function in subexponential size.

Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

: We prove that for all constants a, NQP = NTIME[n^polylog(n)] cannot be (1/2 + 2^(-log^a n) )-approximated by 2^(log^a n)-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of threshold…

Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2015

The techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds, which give satisfiability algorithms beating brute force search for depth-d threshold circuits with a superlinear number of wires.

Circuit Complexity: New Techniques and Their Limitations

- Computer Science, Mathematics
- 2017

The weighted gate elimination method is introduced, which runs a more sophisticated induction than gate elimination, and it is shown that this method gives a much simpler proof of a stronger lower bound of 3.11n for quadratic dispersers.

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2018

It is shown that, for any large enough constant c, given a depth-d circuit with (n2−1/c)-sparse PTF gates that has at most n1+εd wires, the number of satisfying assignments can be computed in randomized time 2n−nε with zero error.

A #SAT Algorithm for Small Constant-Depth Circuits with PTF gates

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2018

There is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs, counts the number of satisfying assignments to C in significantly better than brute-force time.

N ov 2 01 7 Quantified Derandomization of Linear Threshold Circuits

- Computer Science, Mathematics
- 2018

This work constructs an algorithm that gets as input a T C circuit C over n input bits with depth d and n1+exp(−d) wires, runs in almostpolynomial-time, and distinguishes between the case that C rejects at most 2n 1−1/5d inputs and the cases that C accepts at most2n 1-2-3d inputs, which implies that NEXP 6⊆ T C.

## References

SHOWING 1-10 OF 104 REFERENCES

Complex polynomials and circuit lower bounds for modular counting

- Mathematics, Computer Sciencecomputational complexity
- 2005

It is shown that a constant-depth circuit of size 2 cannot determine if the sum of the input bits is divisible byk; moreover, such a circuit must give the wrong answer on a constant fraction of the inputs, and this result was previously known only fork=2.

Correlation bounds for poly-size AC 0 circuits with n 1-o(1) symmetric gates

- Computer Science, Mathematics
- 2011

Average-case circuit lower bounds are one of the hardest problems tackled by computational complexity, and are essentially only known for bounded-depth circits with AND,OR,NOT gates (i.e. AC0). Faced…

Threshold Circuits of Small Majority-Depth

- Computer Science, MathematicsInf. Comput.
- 1998

This thesis presents the explicit use of AND-OR gates as a general tool for computing functions with integer values and uses it to obtain depth-four threshold circuits of majority-depth two for other arithmetic problems such as the logarithm and power series approximation.

On Threshold Circuits and Polynomial Computation

- Mathematics, Computer ScienceSIAM J. Comput.
- 1992

It is proved that all functions computed by Threshold Circuits of size S ( n ) ≥ n and depth D (n) can also be computed by Z P(n) Circuits, and a Depth Hierarchy Theorem for Z P (n ) Circuits is proved.

Multiplicity Automata , Polynomials and the Complexity of Small-Depth Boolean Circuits

- Computer Science, Mathematics
- 2012

A simple method based on multiplicity automata is developed which is used to prove lower bounds on the size of some classes of circuits and remark that p, k and the weights are not bounded above by any function and can in fact be exponential.

SeparatingAC0 from Depth-2 Majority Circuits

- Computer ScienceSIAM J. Comput.
- 2009

The first known function in ${AC}^0$ with exponentially small discrepancy is exhibited, thereby establishing the separations $\Sigma_2^{cc}\not\subseteq{PP}^{cc}$ and $\Pi_2_{cc}\ not\sub seteq{ PP}^{CC}$ in communication complexity.

Algebraic methods in the theory of lower bounds for Boolean circuit complexity

- Computer Science, MathematicsSTOC
- 1987

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.

Non-uniform ACC Circuit Lower Bounds

- Computer Science2011 IEEE 26th Annual Conference on Computational Complexity
- 2011

The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms can be applied to obtain the above lower bounds.

Pseudorandom bits for constant depth circuits with few arbitrary symmetric gates

- Computer Science20th Annual IEEE Conference on Computational Complexity (CCC'05)
- 2005

It is concluded that every function computable by uniform poly(n)-size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME.

Exact Threshold Circuits

- Computer Science2010 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.