In contrast to deterministic or nondeterministic computation, it is
a fundamental open problem in randomized computation how to separate
different randomized time classes (at this point we do not even know
how to separate linear randomized time from ${\mathcal O}(n^{\log n})$
randomized time) or how to ... more >>>

We examine the power of Boolean functions with low L_1 norms in several
settings. In large part of the recent literature, the degree of a polynomial
which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function.
However, some functions ... more >>>

We propose definitions of $\QAC^0$, the quantum analog of the
classical class $\AC^0$ of constant-depth circuits with AND and OR
gates of arbitrary fan-in, and $\QACC^0[q]$, the analog of the class
$\ACC^0[q]$ where $\Mod_q$ gates are also allowed. We show that it is
possible to make a `cat' state on ... more >>>

Let A(x) be the characteristic function of A. Consider the function
F_k^A(x_1,...,x_k) = A(x_1)...A(x_k). We show that if F_k^A can be
computed with fewer than k queries to some set X, then A can be
computed by polynomial size circuits. A generalization of this result
has applications to bounded query ... more >>>

In this paper we consider the problem of determining whether an
unknown arithmetic circuit, for which we have oracle access,
computes the identically zero polynomial. Our focus is on depth-3
circuits with a bounded top fan-in. We obtain the following
results.

1. A quasi-polynomial time deterministic black-box identity testing algorithm ... more >>>

We show that the rank of a depth-3 circuit (over any field) that is simple,
minimal and zero is at most O(k^3\log d). The previous best rank bound known was
2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).
This almost resolves the rank question first posed by ... more >>>

Polynomial identity testing (PIT) is the problem of checking whether a given
arithmetic circuit is the zero circuit. PIT ranks as one of the most important
open problems in the intersection of algebra and computational complexity. In the last
few years, there has been an impressive progress on this ... more >>>

Assume that Alice is running a program $P$ on a RAM, and an adversary
Bob would like to get some information about the input or output of the
program. At each time, during the execution of $P$, Bob is able to see
the addresses of the memory cells involved in ... more >>>

We give new characterizations and lower bounds relating classes in the communication complexity polynomial hierarchy and circuit complexity to limited memory communication models.

We introduce the notion of rectangle overlay complexity of a function $f: \{0,1\}^n\times \{0,1\}^n\to\{0,1\}$. This is a natural combinatorial complexity measure in terms of combinatorial rectangles in ... more >>>

We show an efficient method for converting a logic circuit of gates with fan-out 1 into an equivalent circuit that works even if some fraction of its gates are short-circuited, i.e., their output is short-circuited to one of their inputs. Our conversion can be applied to any circuit with fan-in ... more >>>

We study computation by formulas over $(min, +)$. We consider the computation of $\max\{x_1,\ldots,x_n\}$
over $\mathbb{N}$ as a difference of $(\min, +)$ formulas, and show that size $n + n \log n$ is sufficient and necessary. Our proof also shows that any $(\min, +)$ formula computing the minimum of all ... more >>>

We study bounded depth $(\min, +)$ formulas computing the shortest path polynomial. For depth $2d$ with $d \geq 2$, we obtain lower bounds parametrized by certain fan-in restrictions on all $+$ gates except those at the bottom level. For depth $4$, in two regimes of the parameter, the bounds are ... more >>>

Deep neural networks are the dominant machine learning model. We show that this model is missing a crucial complexity parameter. Today, the standard neural network (NN) model is a circuit whose gates (neurons) are ReLU units. The complexity of a NN is quantified by the depth (number of layers) and ... more >>>

We introduce tri-state circuits (TSCs). TSCs form a natural model of computation that, to our knowledge, has not been considered by theorists. The model captures a surprising combination of simplicity and power. TSCs are simple in that they allow only three wire values ($0$,$1$, and undefined – $Z$) and three ... more >>>

We initiate an in-depth proof-complexity analysis of polynomial calculus (Q-PC) for Quantified Boolean Formulas (QBF). In the course of this we establish a tight proof-size characterisation of Q-PC in terms of a suitable circuit model (polynomial decision lists). Using this correspondence we show a size-degree relation for Q-PC, similar in ... more >>>