We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Fur-ther we show that there are functions computable in polynomial size and depth k but requires ex-ponential size when the depth is restricted to k-1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.

This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over GF (2)). This paper shows how to learn in polynomial time any function that can be approximated (in norm L 2 ) by a polynomially sparse function (i.e., a function with only polynomially many non-zero Fourier coefficients). The authors demonstrate that any function f whose L 1 -norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial L 1 -norm can be learned deterministically. The algorithm can a...

Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and linear size ^ � _-circuits but require exponential size to compute by a depth k; 1 monotone weighted threshold circuit. Key words. Circuit complexity, monotone circuits, threshold circuits, lower bounds Subject classi cations. 68Q15, 68Q99 1.

this paper, we consider circuits of bounded depth in the basis f; \Phig.

To appear in Izvestiya of the RAN Abstract We show that if strong pseudorandom generators exist then the statement "ff encodes a circuit of size n(log * n) for SATISFIABILITY " is not refutable in S22 (ff). For refutation in S12 (ff), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constant-depth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolation-like theorems for certain "split versions " of classical systems of Bounded Arithmetic introduced in this paper.

Traditionally, computational complexity has considered only static problems. Classical Complexity Classes such as NC, P, and NP are defined in terms of the complexity of checking -- upon presentation of an entire input -- whether the input satisfies a certain property. For many applications of computers it is more appropriate to model the process as a dynamic one. There is a fairly large object being worked on over a period of time. The object is repeatedly modified by users and computations are performed. We develop a theory of Dynamic Complexity. We study the new complexity class, Dynamic First-Order Logic (Dyn-FO). This is the set of properties that can be maintained and queried in first-order logic, i.e. relational calculus, on a relational database. We show that many interesting properties are in Dyn-FO including multiplication, graph connectivity, bipartiteness, and the computation of minimum spanning trees. Note that none of these problems is in static FO, and this f...

We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.

We show that a DNF with terms of size at most d can be approximated by a function with at most d O(d log 1=") non zero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ". This property is used to derive a learning algorithm for DNF, under the uniform distribution. The learning algorithm uses queries and learns, with respect to the uniform distribution, a DNF with terms of size at most d in time polynomial in n and d O(d log 1=") . The interesting implications are for the case when " is constant. In this case our algorithm learns a DNF with a polynomial number of terms in time n O(log log n) , and a DNF with terms of size at most O(log n= log log n) in polynomial time.

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this paper we shall assume that S is a commutative ring with identity. Then each instruction f i can be identified with the polynomial that is computed at f i , if\Omega and \Phi are interpreted as the ring operations in the polynomial ring S[x 1 ; \Delta \Delta \Delta ; x n ]. Among natural multivariate polynomials whose complexity in this model is of interest are Hamiltonian circuits (HC), the permanent (PERM) and the determinant (DET). These are defined over a matrix X of indeterminates fx 11 ; \Delta \Delta \Delta ; x nn g where x ij