Roman Smolensky

The bit extraction problem or t-resilient functions

We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : {0, 1}n → {0, 1}m be a function. An adversary, knowing the function f, sets t of the n input bits, while the rest (n-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 {0, 1}m. The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f : {0,1}n → {0, 1}m, when being restricted to set t of the input bits of f. We present various lower and upper bounds on ms 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 faulttolerance and cryptography.

The electrical resistance of a graph captures its commute and cover times

View ann-vertex,m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with thelengths of random walks on the graph. For example, thecommute time between two verticess andt (the expected length of a random walk froms tot and back) is precisely characterized by the effective resistanceRst betweens andt: commute time=2mRst. As a corollary, thecover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistanceR in the graph to within a factor of logn:mR<-cover time<-O(mRlogn). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multi-dimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

On lower bounds for read- k -times branching programs

AbstractA syntactic read-k-times branching program has the restriction that no variable occurs more thank times on any path (whether or not consistent) of the branching program. We first extend the result in [31], to show that the “n/2 clique only function”, which is easily seen to be computable by deterministic polynomial size read-twice programs, cannot be computed by nondeterministic polynomial size read-once programs, although its complement can be so computed. We then exhibit an explicit Boolean functionf such that every nondeterministic syntactic read-k-times branching program for computingf has size exp

On interpolation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates

\left( {\Omega \left( {\frac{n}{{4^k k^3 }}} \right)} \right).

Well-known bound for the VC-dimension made easy

This paper examines the class of polynomial threshold functions using harmonic analysis and applies the results to derive lower bounds related to

Lower bounds for polynomial evaluation and interpolation problems

AC^0

Representations of sets of Boolean functions by commutative rings

functions. A Boolean function is polynomial threshold if it can be represented as the sign of a sparse polynomial (one that consists of a polynomial number of terms). The main result of this paper is that the class of polynomial threshold functions can be characterized using their spectral representation. In particular, it is proved that an n-variable Boolean function whose