. In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following: 1. A single threshold gate with weights cannot in general be replaced by a polynomial fan-in unweighted threshold gate of parity gates. 2. On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depth d polynomial size threshold circuit with weights can be computed by a depth d + 1 polynomial size unweighted threshold circuit, where d is an arbitrary fixed integer. 3. A polynomial fan-in threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size...

. Define the MODm -degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 0-1 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm -degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm -degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm -degree of the MODm function is 1; we show that the MODm -degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...

We investigate the computational power of depth two circuits consisting of MOD r --gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold--MOD r circuits) and circuits with two levels of MOD gates (MOD p -MOD q circuits). In particular, we will show the following results. (i) For all prime numbers p and integers q; r, it holds that if p divides r but not q then all threshold--MOD q circuits for MOD r have exponentially many nodes. (ii) For all integers r, all problems computable by depth two fAND;OR;NOTg-- circuits of polynomial size have threshold--MOD r circuits with polynomially many edges. (iii) There is a problem computable by depth 3 fAND;OR;NOTg--circuits of linear size and constant bottom fan--in which for all r needs threshold--MOD r circuits with exponentially many nodes. (iv) For p; r different primes, and q 2; k positive integers, where r does not divide q; every MOD p k -MOD q circuit for MOD r has e...

The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. Two applications of such restricted variants are given. First, we show that our lower bound on a variant of rigidity implies lower bounds on size-depth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain restricted forms of rigidity to conclude several separation results such as separating the analogs of PH and PSPACE in communication complexity theory. Our res...

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.

Abstract We use a Ramsey-theoretic argument to obtain the firstlower bounds for approximations over Zm by nonlinearpolynomials: ffl A degree-2 polynomial over Zm (m odd) mustdiffer from the parity function on at least a

In this talk we will consider various classes defined by small depth polynomial size circuits which contain threshold gates and parity gates. We will describe various inclusions between many classes defined in this way and also classes whose definitions rely upon spectral properties of Boolean functions.

We show that at least =r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank below r. This improves upon the previously known bound ) [6]. If we additionally know that these changes are bounded from above in their absolute values by some n=r, we prove another bound on their number. This improves upon the bound ) from [13].

. 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 ...

: We prove that any depth--3 circuit with MOD m gates of unbounded fan-in on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that log(wM ) = \Omega\Gamma n), where w is the sum of the absolute values of the weights, and M is the maximum fan--in of the AND gates on level 2. Setting all weights to 1, we have got a trade--off between the numbers of the MOD m gates and the AND gates. By our knowledge, this is the first trade--off result involving hard--to--handle MOD m gates. In contrast, with n AND gates at the bottom and a single MOD 2 gate at the top one can compute the inner product function. The lower--bound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fan--in. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates. ...