Polynomial Threshold Functions, AC^0 Functions and Spectral Norms

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_∞^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived

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

kbstr act We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MOD, where p is a prime require Ezp(O(n’)) gates to calculate MOD, functions for any r # pm. This statement contains as special cases Yao’s PARITY result [ Ya 85] and Razborov’s new MAJORITY resul

Lower Bounds for Polynomial Evaluation and Interpolation Problems

We show that there is a set of points p 1 ; p 2 ; : : : ; p n such that any arithmetic circuit of depth d for polynomial evaluation (or interpolation) at these points has size \Omega ` n log n log(2 + d= log n) ' : Moreover, for circuits of sub-logarithmic depth, we obtain a lower bound of \Omega\Gamma dn 1+1=d ) on its size. 1 Introduction To prove a superlinear lower bound for a natural problem is one of the greatest challenges of theoretical computer science. Algebraic complexity theory is the study of a restricted class of algorithms that can perform arithmetic operations on data (i.e., add, subtract, multiply and divide), but that do not care how the data is represented. This is a reasonable class of algorithms to consider when solving algebraic problems. We shall use arithmetic circuits as our model of computation. Two very natural measures of the complexity of such a circuit are its size (number of gates and wires) and depth (length of longest path from input to output..

Randomization and the Computational Power of Analytic and Algebraic Decision Trees

We introduce a new powerful method for proving lower bounds on randomized and deterministic analytic decision trees, and give direct applications of our results towards some concrete geometric problems. We design also randomized algebraic decision trees for recognizing the positive octant in R n or computing MAX in R n+1 in depth log 0(1) n. Both problems are known to have linear lower lower bounds for the depth of any deterministic analytic decision tree recognizing them. The main new (and unifying) proof idea of the paper is in the reduction technique of the signs of testing functions in a decision tree to the signs of their leading terms at the specially chosen points. This allows us to reduce the complexity of a decision tree to the complexity of a certain boolean circuit. Dept. of Computer Science and Mathematics, Penn State University, University Park, PA 16802. Supported in part by NSF grant CCR-9424358. E-mail: [email protected] y Dept. of Computer Science, Universi..

A Lower Bound for Randomized Algebraic Decision Trees

We prove the first nontrivial (and superlinear) lower bounds on the depth of randomized algebraic decision trees (with two-sided error) for problems being finite unions of hyperplanes and intersections of halfspaces, solving a long standing open problem. As an application, among other things, we derive, for the first time, an \Omega\Gamma n 2 ) randomized lower bound for the Knapsack Problem, and an \Omega\Gamma n log n) randomized lower bound for the Element Distinctness Problem which were previously known only for deterministic algebraic decision trees. It is worth noting that for the languages being finite unions of hyperplanes our proof method yields also a new elementary lower bound technique for deterministic algebraic decision trees without making use of Milnor's bound on Betti number of algebraic varieties

The Bit Extraction Problem or t-Resilient Functions

\Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with uniform probability distribution). The adversary tries to prevent the outcome of f from being uniformly distributed in f0; 1g m . The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f : f0; 1g n 7! f0; 1g m , when being restricted to set t of the input bits of f . We present various lower and upper bounds on m's allowing an affirmative answer. These bounds are relatively close for t n=3 and for t 2n=3. Our results have applications in the fields of fault-tolerance and cryptography. 1. INTRODUCTION The bit extraction problem formulated above The bit extraction problem was suggested by Brassard and Robert [BRref] and by V..