We introduce and study Certificate Game complexity, a measure of complexity based on the probability of winning a game where two players are given inputs with different function values and are asked to output some index $i$ such that $x_i\neq y_i$, in a zero-communication setting.

We give upper and lower ... more >>>

In this paper, we introduce a measure of Boolean functions we call diameter, that captures the relationship between certificate complexity and several other measures of Boolean functions. Our measure can be viewed as a variation on alternating number, but while alternating number can be exponentially larger than certificate complexity, we ... more >>>

Nisan and Szegedy conjectured that block sensitivity is at most
polynomial in sensitivity for any Boolean function.
Until a recent breakthrough of Huang, the conjecture had been
wide open in the general case,
and was proved only for a few special classes
of Boolean functions.
Huang's result implies that block ... more >>>

Comparator circuits are a natural circuit model for studying the concept of bounded fan-out computations, which intuitively corresponds to whether or not a computational model can make "copies" of intermediate computational steps. Comparator circuits are believed to be weaker than general Boolean circuits, but they can simulate Branching Programs and ... more >>>

We show that any Boolean function with approximate rank $r$ can be computed by bounded error quantum protocols without prior entanglement of complexity $O( \sqrt{r} \log r)$. In addition, we show that any Boolean function with approximate rank $r$ and discrepancy $\delta$ can be computed by deterministic protocols of complexity ... more >>>

Nisan and Szegedy conjectured that block sensitivity is at most polynomial in sensitivity for any Boolean function. There is a huge gap between the best known upper bound on block sensitivity in terms of sensitivity - which is exponential, and the best known separating examples - which give only a ... more >>>

We define new functions based on the Andreev function and prove that they require $n^{3}/polylog(n)$ formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the ... more >>>

SUBSET SUM is a well known NP-complete problem:
given $t \in Z^{+}$ and a set $S$ of $m$ positive integers, output YES if and only if there is a subset $S^\prime \subseteq S$ such that the sum of all numbers in $S^\prime$ equals $t$. The problem and its search ... more >>>

Consider a large database of $n$ data items that need to be stored using $m$ servers.
We study how to encode information so that a large number $k$ of read requests can be performed \textit{in parallel} while the rate remains constant (and ideally approaches one).
This problem is equivalent ... more >>>

We consider arithmetic complexity classes that are in some sense dual to the classes VP(Fp) that were introduced by Valiant. This provides new characterizations of the complexity classes ACC^1 and TC^1, and also provides a compelling example of
a class of high-degree polynomials that can be simulated via arithmetic circuits ... more >>>

Spira showed that any Boolean formula of size $s$ can be simulated in depth $O(\log s)$. We generalize Spira's theorem and show that any Boolean circuit of size $s$ with segregators of size $f(s)$ can be simulated in depth $O(f(s)\log s)$. If the segregator size is at least $s^{\varepsilon}$ for ... more >>>

Locally decodable codes (LDCs) are error correcting codes with the extra property that it is sufficient to read just a small number of positions of a possibly corrupted codeword in order to recover any one position of the input. To achieve this, it is necessary to use randomness in the ... more >>>

We bound the minimum number $w$ of wires needed to compute any (asymptotically good) error-correcting code
$C:\{0,1\}^{\Omega(n)} \to \{0,1\}^n$ with minimum distance $\Omega(n)$,
using unbounded fan-in circuits of depth $d$ with arbitrary gates. Our main results are:

(1) If $d=2$ then $w = \Theta(n ({\log n/ \log \log n})^2)$.

(2) ... more >>>

Locally decodable codes
are error correcting codes with the extra property that, in order
to retrieve the correct value of just one position of the input with
high probability, it is
sufficient to read a small number of
positions of the corresponding,
possibly corrupted ... more >>>

In this paper we propose the study of a new model of restricted
branching programs which we call incremental branching programs.
This is in line with the program proposed by Cook in 1974 as an
approach for separating the class of problems solvable in logarithmic
space from problems solvable ... more >>>

This paper provides logspace and small circuit depth analogs
of the result of Valiant-Vazirani, which is a randomized (or
nonuniform) reduction from NP to its arithmetic analog ParityP.
We show a similar randomized reduction between the
Boolean classes NL and semi-unbounded fan-in Boolean circuits and
their arithmetic counterparts. These ... more >>>

The model of span programs is a linear algebraic model of
computation. Lower bounds for span programs imply lower bounds for
contact schemes, symmetric branching programs and for formula size.
Monotone span programs correspond also to linear secret-sharing schemes.
We present a new technique for proving lower bounds for ... more >>>