Abstract. We construct a function in AC 0 that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n 1/5)). This solves an open problem due to Krause and Pudlák (1994) and matches Allender’s classic result (1989) that AC 0 can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, our work yields the first known function in AC 0 with exponentially small discrepancy, exp(−Ω(n 1/5)). Key words. Majority circuits, constant-depth AND/OR/NOT circuits, communication complexity, discrepancy, threshold degree of Boolean functions. AMS subject classifications. 03D15, 68Q15, 68Q17

The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.

We give the first representation-independent hardness results for PAC learning intersections of halfspaces, a central concept class in computational learning theory. Our hardness results are derived from two public-key cryptosystems due to Regev, which are based on the worstcase hardness of well-studied lattice problems. Specifically, we prove that a polynomialtime algorithm for PAC learning intersections of n ε halfspaces (for a constant ε> 0) in n dimensions would yield a polynomial-time solution to Õ(n 1.5)-uSVP (unique shortest vector problem). We also prove that PAC learning intersections of n ε low-weight halfspaces would yield a polynomial-time quantum solution to Õ(n 1.5)-SVP and Õ(n 1.5)-SIVP (shortest vector problem and shortest independent vector problem, respectively). Our approach also yields the first representation-independent hardness results for learning polynomialsize depth-2 neural networks and polynomial-size depth-3 arithmetic circuits. Key words: Cryptographic hardness results, intersections of halfspaces, computational learning theory, lattice-based cryptography 1

. We survey some of the central results in the complexity theory of discrete neural networks, with pointers to the literature. Our main emphasis is on the computational power of various acyclic and cyclic network models, but we also discuss briefly the complexity aspects of synthesizing networks from examples of their behavior. CR Classification: F.1.1 [Computation by Abstract Devices]: Models of Computation---neural networks, circuits; F.1.3 [Computation by Abstract Devices ]: Complexity Classes---complexity hierarchies Key words: Neural networks, computational complexity, threshold circuits, associative memory 1. Introduction The currently again very active field of computation by "neural" networks has opened up a wealth of fascinating research topics in the computational complexity analysis of the models considered. While much of the general appeal of the field stems not so much from new computational possibilities, but from the possibility of "learning", or synthesizing networks...

Given any linear threshold function f on n Boolean variables, we construct a linear threshold function g which disagrees with f on at most an ɛ fraction of inputs and has integer weights each of magnitude at most √ n · 2 Õ(1/ɛ2). We show that the construction is optimal in terms of its dependence on n by proving a lower bound of Ω ( √ n) on the weights required to approximate a particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately counting the fraction of satisfying assignments to an instance of the zero-one knapsack problem to within an additive ±ɛ. The algorithm runs in time polynomial in n (but exponential in 1/ɛ 2). In our second application, we show that any linear threshold function f is specified to within error ɛ by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive ±1/(n · 2 Õ(1/ɛ2)). This is the first such accuracy bound which is inverse polynomial in n (previous work of Goldberg [12] gave a 1/quasipoly(n) bound), and gives the first polynomial bound (in terms of n) on the number of examples required for learning linear threshold functions in the “restricted focus of attention ” framework.

. A set F of Boolean functions is called a pseudorandom function generator (PRFG) if communicating with a randomly chosen secret function from F cannot be efficiently distinguished from communicating with a truly random function. We ask for the minimal hardware complexity of a PRFG. This question is motivated by design aspects of secure secret key cryptosystems. Such cryptosystems should be efficient in hardware, but often are required to behave like PRFGs. By constructing efficient distinguishing schemes we show for a wide range of basic nonuniform complexity classes, induced by depth restricted branching programs and several types of constant depth circuits (including TC 0 2 ), that they do not contain PRFGs. On the other hand we show that the PRFG proposed by Naor and Reingold in [24] consists of TC 0 4 -functions. The question if TC 0 3 -functions can form PRFGs remains as an interesting open problem. We further discuss relations of our results to previous work on cryptographic ...

Constant-depth polynomial-size threshold circuits are usually classified according to their total depth. For example, the best known threshold circuits for iterated multiplication and division have depth four and three, respectively. In this paper, the complexity of threshold circuits is investigated from a different point of view: explicit AND, OR gates are allowed in the circuits, and a threshold circuit is said to have majority-depth d if no path traverses more than d threshold gates. It is then shown that iterated multiplication can be computed by polynomial-size threshold circuits of total depth five but of majority-depth three. Circuits of depth four and majority-depth two are obtained for division and powering. These results rely on a careful implementation of iterated addition and Chinese remaindering. In addition, a simple symbolic calculus for composing circuit classes is developed: this notation allows for a concise and elegant presentation of the results. 3 List of symbol...

We show that any distribution on {−1, +1} n that is k-wise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2 (1/ɛ)). Our result is tight up to log(1/ɛ) factors. Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G: {−1, +1} s → {−1, +1} n that fool halfspaces. Specifically, we fool halfspaces with error ɛ and seed length s = k · log n = O(log n · ɛ−2 log 2 (1/ɛ)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).

We introduce a new Boolean computing element related to the Boolean version of a neural element. Instead of the sign function in the Boolean neural element (also known as an LT element), it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. We call the new computing element an LTM element, which stands for Linear Threshold with Multiple transitions. The paper consists of the following main contributions related to our study of LTM circuits: (i) the characterization of the computing power of LTM relative to LT circuits, (ii) a proof that the area of the VLSI layout is reduced from O(n 2 ) in LT circuits to O(n) in LTM circuits, for n inputs symmetric Boolean functions, and (iii) the creation of efficient designs of LTM circuits for the addition of a multiple number of integers and the product of two integers. In particular, we show how to compute the addition of m integers with a single layer of LTM elements. Category : The...

We study the circuit complexity of the powering function, defined as POWm(Z) = Z m for an n-bit integer input Z and an integer exponent m � poly(n). Let � LTd denote the class of functions computable by a depth-d polynomial-size circuit of majority gates. We give a simple proof that POWm � ∈ � LT2 for any m � 2. Specifically, we prove a 2 Ω(n/logn) lower bound on the size of any depth-2 majority circuit that computes POWm. This work generalizes Wegener’s earlier result that the squaring function (i.e., POWm for the special case m = 2) is not in � LT2. Our depth lower bound is optimal due to Siu and Roychowdhury’s matching upper bound: POWm ∈ � LT3. The second part of this research note presents several counterintuitive findings about the membership of arithmetic functions in the circuit classes � LT1 and � LT2. For example, we construct a function f (Z) such that f � ∈ � LT1 but 5 f ∈ � LT1. We obtain similar findings for � LT2. This apparent brittleness of � LT1 and � LT2 highlights a difficulty in proving lower bounds for arithmetic functions.