On Threshold Circuits and Polynomial Computation

A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field ZP(n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P(n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) &#x2265 n and depth D(n) can also be computed by ZP(n) Circuits of size O(S(n) log S(n) + nP(n) log P(n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by ZP(n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O((1/&#x22082)(S(n) log P(n))1+&#x2208) and depth O((1/&#x22085)D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, an integer reciprocal can be computed in size nO(1)M and depth O(1). More generally, any analytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2-nc , for any constant c, by Threshold Circuits of polynomial size and constant depth. In addition, integer and polynomial division, FFf, polynomial interpolation, Chinese Remaindering, all the elementary symmetric functions, banded matrix inverse, and triangular Toeplitz matrix inverse can be exactly computed by Threshold Circuits of polynomial size and constant depth. All these results and simulations hold for polytime uniform circuits. This paper also gives a corresponding simulation oflogspace uniform ZP(n) Circuits by logspace uniform Threshold Circuits requiring an additional multiplying factor of O(log log log P(n) depth. Finally, purely algebraic methods forlowerbounds for ZP(n) Circuits are developed. Using degree arguments, a Depth Hierarchy Theorem for ZP(n) Circuits is proved: for any S(n) &#x2265 n, D(n) = O(S(n)c') for some constant c' P(n) where 6(S(n)/D(n))D(n) P(n) c' 2n, there exist explicitly constructible functions computable by ZP(n) Circuits of size S(n) and depth D(n), but provably not computable by ZP(n) Circuits of size S(n)c and depth oD(n)) for any constant c &#x2265 1

Algorithms and Lower Bounds in Circuit Complexity

Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits

Neural computation of arithmetic functions

A neuron is modeled as a linear threshold gate, and the network architecture considered is the layered feedforward network. It is shown how common arithmetic functions such as multiplication and sorting can be efficiently computed in a shallow neural network. Some known results are improved by showing that the product of two n-bit numbers and sorting of n n-bit numbers can be computed by a polynomial-size neural network using only four and five unit delays, respectively. Moreover, the weights of each threshold element in the neural networks require O(log n)-bit (instead of n -bit) accuracy. These results can be extended to more complicated functions such as multiple products, division, rational functions, and approximation of analytic functions

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Challenges in computational lower bounds

We draw two incomplete, biased maps of challenges in computational complexity lower bounds

Power of Quantum Computation with Few Clean Qubits

This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0> state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor intermediate measurements. The main results of this paper are unexpectedly strong error-reducible properties of such quantum computations. It is proved that any problem solvable by a polynomial-time quantum computation with one-sided bounded error that uses logarithmically many clean qubits can also be solvable with exponentially small one-sided error using just two clean qubits, and with polynomially small one-sided error using just one clean qubit. It is further proved in the case of two-sided bounded error that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits can also be solvable with exponentially small two-sided error using just two clean qubits. If only one clean qubit is available, the problem is again still solvable with exponentially small error in one of the completeness and soundness and polynomially small error in the other. As an immediate consequence of the above result for the two-sided-error case, it follows that the TRACE ESTIMATION problem defined with fixed constant threshold parameters is complete for the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using logarithmically many clean qubits and just one clean qubit. The techniques used for proving the error-reduction results may be of independent interest in themselves, and one of the technical tools can also be used to show the hardness of weak classical simulations of one-clean-qubit computations (i.e., DQC1 computations).Comment: 44 pages + cover page; the results in Section 8 are overlapping with the main results in arXiv:1409.677

Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur