Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{\epsilon n}$.
We ... more >>>

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq poly(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive ... more >>>

We study a new model of space-bounded computation, the {\it random-query} model. The model is based on a branching-program over input variables $x_1,\ldots,x_n$. In each time step, the branching program gets as an input a random index $i \in \{1,\ldots,n\}$, together with the input variable $x_i$ (rather than querying an ... more >>>

We study a new type of separation between quantum and classical communication complexity which is obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits with oracle access to their inputs. More precisely, we give an explicit partial Boolean ... more >>>

A line of recent works showed that for a large class of learning problems, any learning algorithm requires either super-linear memory size or a super-polynomial number of samples [Raz16,KRT17,Raz17,MM18,BOGY18,GRT18]. For example, any algorithm for learning parities of size $n$ requires either a memory of size $\Omega(n^{2})$ or an exponential number ... more >>>

In 1985, Ben-Or and Linial (Advances in Computing Research '89) introduced the collective coin-flipping problem, where $n$ parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party ... more >>>

We present a distribution $D$ over inputs in $\{-1,1\}^{2N}$, such that:
(1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time $O(\log N)$, that distinguishes between $D$ and the uniform distribution with advantage $\Omega(1/\log N)$.
(2) No Boolean circuit of $\mathrm{quasipoly}(N)$ ... more >>>

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen ... more >>>

We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples.

Our result is stated in terms of the norm ... more >>>

We define a concept class ${\cal F}$ to be time-space hard (or memory-samples hard) if any learning algorithm for ${\cal F}$ requires either a memory of size super-linear in $n$ or a number of samples super-polynomial in $n$, where $n$ is the length of one sample.

A recent work shows ... more >>>

We prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples. This proves a recent conjecture of Steinhardt, Valiant and Wager and shows that for some learning problems a large storage space is crucial.

More formally, in the problem of ... more >>>

We show an exponential gap between communication complexity and external information complexity, by analyzing a communication task suggested as a candidate by Braverman [Bra13]. Previously, only a separation of communication complexity and internal information complexity was known [GKR14,GKR15].

More precisely, we obtain an explicit example of a search problem with ... more >>>

We continue the study of welfare maximization in unit-demand (matching) markets, in a distributed information model
where agent's valuations are unknown to the central planner, and therefore communication is required to determine an
efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed that if the market size is $n$, ... more >>>

Linear Programs are abundant in practice, and tremendous effort has been put into designing efficient algorithms for such problems, resulting with very efficient (polynomial time) algorithms. A fundamental question is: what is the space complexity of Linear Programming?

It is widely believed that (even approximating) Linear Programming requires a large ... more >>>

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity $\leq O(k)$, and distributional communication complexity $\geq 2^k$. This shows that a communication protocol for a partial boolean function cannot always be compressed to ... more >>>

We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity $\leq O(k)$, and distributional communication complexity $\geq 2^k$. This shows that a communication protocol cannot always be compressed to its internal information. By a result of ... more >>>

In the Coin Problem, one is given n independent flips of a coin that has bias $\beta > 0$ towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply ... more >>>

We construct a 1-round delegation scheme (i.e., argument-system) for every language computable in time t=t(n), where the running time of the prover is poly(t) and the running time of the verifier is n*polylog(t). In particular, for every language in P we obtain a delegation scheme with almost linear time verification. ... more >>>

We put forward a new approach for the design of efficient multiparty protocols:

1. Design a protocol for a small number of parties (say, 3 or 4) which achieves
security against a single corrupted party. Such protocols are typically easy
to construct as they may employ techniques that do not ... more >>>

In 1989, Babai, Nisan and Szegedy [BNS92] gave a construction of a pseudorandom generator for logspace, based on lower bounds for multiparty communication complexity. The seed length of their pseudorandom generator was $2^{\Theta(\sqrt n)}\,\,\,$, because the best lower bounds for multiparty communication complexity are relatively weak. Subsequently, pseudorandom generators for ... more >>>

We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).

Our technical contributions include a theorem that shows that the ``expected ... more >>>

We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical $\mathcal{AM}$ streaming algorithm for a wide class of data stream problems. The algorithm offers a tradeoff between the length of the proof and the space complexity that is needed to verify it.

We study the interactive channel capacity of an $\epsilon$-noisy channel. The interactive channel capacity $C(\epsilon)$ is defined as the minimal ratio between the communication complexity of a problem (over a non-noisy channel), and the communication complexity of the same problem over the binary symmetric channel with noise rate $\epsilon$, where ... more >>>

We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size $s$ is concentrated on coefficients of degree up to ... more >>>

We give an explicit function $h:\{0,1\}^n\to\{0,1\}$ such that any deMorgan formula of size $O(n^{2.499})$ agrees with $h$ on at most $\frac{1}{2} + \epsilon$ fraction of the inputs, where $\epsilon$ is exponentially small (i.e. $\epsilon = 2^{-n^{\Omega(1)}}$). Previous lower bounds for formula size were obtained for exact computation.

The same ... more >>>

Let $C$ be a (fan-in $2$) Boolean circuit of size $s$ and depth $d$, and let $x$ be an input for $C$. Assume that a verifier that knows $C$ but doesn't know $x$ can access the low degree extension of $x$ at one random point. Two competing provers try to ... more >>>

Motivated by the classical problem of privacy amplification, Dodis and Wichs (STOC '09) introduced the notion of a non-malleable extractor, significantly strengthening the notion of a strong extractor. A non-malleable extractor is a function $nmExt : \{0,1\}^n \times \{0,1\}^d \rightarrow \{0,1\}^m$ that takes two inputs: a weak source $W$ and ... more >>>

The parallel repetition theorem states that for any Two
Prover Game with value at most $1-\epsilon$ (for $\epsilon<1/2$),
the value of the game repeated $n$ times in parallel is at most
$(1-\epsilon^3)^{\Omega(n/s)}$, where $s$ is the length of the
answers of the two provers. For Projection
Games, the bound on ... more >>>

We give new pseudorandom generators for \emph{regular} read-once branching programs of small width.
A branching program is regular if the in-degree of every vertex in it is (0 or) $2$.
For every width $d$ and length $n$,
our pseudorandom generator uses a seed of length $O((\log d + \log\log n ... more >>>

We show that any explicit example for a tensor $A:[n]^r \rightarrow \mathbb{F}$ with tensor-rank
$\geq n^{r \cdot (1- o(1))}$,
(where $r = r(n) \leq \log n / \log \log n$), implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over $\mathbb{F}$. This shows that strong enough ... more >>>

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>>

We show that the NP-Complete language 3Sat has a PCP
verifier that makes two queries to a proof of almost-linear size
and achieves sub-constant probability of error $o(1)$. The
verifier performs only projection tests, meaning that the answer
to the first query determines at most one accepting answer to the
more >>>

The parallel repetition theorem states that for any two-prover game,
with value $1- \epsilon$ (for, say, $\epsilon \leq 1/2$), the value of
the game repeated in parallel $n$ times is at most
$(1- \epsilon^c)^{\Omega(n/s)}$, where $s$ is the answers' length
(of the original game) and $c$ is a universal ... more >>>

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth $d$ ... more >>>

A basic fact in linear algebra is that the image of the curve
$f(x)=(x^1,x^2,x^3,...,x^m)$, say over $C$, is not contained in any
$m-1$ dimensional affine subspace of $C^m$. In other words, the image
of $f$ is not contained in the image of any polynomial-mapping
$G:C^{m-1} ---> C^m$ ... more >>>

We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

1. Noise-resistant. A syntactically multilinear ... more >>>

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. ... more >>>

An interactive-PCP (say, for the membership $x \in L$) is a
proof that can be verified by reading only one of its bits, with the
help of a very short interactive-proof.
We show that for membership in some languages $L$, there are
interactive-PCPs that are significantly shorter than the known
more >>>

We show a construction of a PCP with both sub-constant error and
almost-linear size. Specifically, for some constant alpha in (0,1),
we construct a PCP verifier for checking satisfiability of
Boolean formulas that on input of size n uses log n + O((log
n)^{1-alpha}) random bits to query a constant ... more >>>

We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for ... more >>>

We construct an explicit polynomial $f(x_1,...,x_n)$, with
coefficients in ${0,1}$, such that the size of any syntactically
multilinear arithmetic circuit computing $f$ is at least
$\Omega( n^{4/3} / log^2(n) )$. The lower bound holds over any field.

We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following:

1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial ... more >>>

An $(n,k)$-bit-fixing source is a distribution $X$ over $\B^n$ such that
there is a subset of $k$ variables in $X_1,\ldots,X_n$ which are uniformly
distributed and independent of each other, and the remaining $n-k$ variables
are fixed. A deterministic bit-fixing source extractor is a function $E:\B^n
\ar \B^m$ which on ... more >>>

An $(n,k)$-affine source over a finite field $F$ is a random
variable $X=(X_1,...,X_n) \in F^n$, which is uniformly
distributed over an (unknown) $k$-dimensional affine subspace of $
F^n$. We show how to (deterministically) extract practically all
the randomness from affine sources, for any field of size larger
than $n^c$ (where ... more >>>

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
over F. The tester is usually given access to an oracle
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$
by a single quantum state $|\Psi \rangle$ of size $O(n)$ qubits,
such that:
for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$,
the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved
from $|\Psi \rangle$ by a one-round Arthur-Merlin interactive ... more >>>

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>

We show how to extract random bits from two or more independent
weak random sources, in cases where only one source is of linear
min-entropy and all other sources are of logarithmic min-entropy.
We also give improved constructions of mergers and condensers.
In all that comes below, $\delta$ is an ... more >>>

An arithmetic circuit or formula is multilinear if the polynomial
computed at each of its wires is multilinear.
We give an explicit example for a polynomial $f(x_1,...,x_n)$,
with coefficients in $\{0,1\}$, such that over any field:
1) $f$ can be computed by a polynomial-size multilinear circuit
of depth $O(\log^2 ... more >>>

An arithmetic formula is multi-linear if the polynomial computed
by each of its sub-formulas is multi-linear. We prove that any
multi-linear arithmetic formula for the permanent or the
determinant of an $n \times n$ matrix is of size super-polynomial
in $n$.

We prove a quasi-polynomial lower bound on the size of bounded-depth
Frege proofs of the pigeonhole principle $PHP^{m}_n$ where
$m= (1+1/{\polylog n})n$.
This lower bound qualitatively matches the known quasi-polynomial-size
bounded-depth Frege proofs for these principles.
Our technique, which uses a switching lemma argument like other lower bounds
for ... more >>>

We prove a lower bound of $\Omega(m^2 \log m)$ for the size of
any arithmetic circuit for the product of two matrices,
over the real or complex numbers, as long as the circuit doesn't
use products with field elements of absolute value larger than 1
(where $m \times m$ is ... more >>>

We prove that any Resolution proof for the weak
pigeon hole principle, with $n$ holes and any number of
pigeons, is of length $\Omega(2^{n^{\epsilon}})$,
(for some global constant $\epsilon > 0$).

Weak designs were defined by Raz, Reingold and Vadhan (1999) and are
used in constructions of extractors. Roughly speaking, a weak design
is a collection of subsets satisfying some near-disjointness
properties. Constructions of weak designs with certain parameters are
given in [RRV99]. These constructions are explicit in the sense that
more >>>

We prove super-linear lower bounds for the number of edges
in constant depth circuits with $n$ inputs and up to $n$ outputs.
Our lower bounds are proved for all types of constant depth
circuits, e.g., constant depth arithmetic circuits, constant depth
threshold circuits ... more >>>

We give explicit constructions of extractors which work for a source of
any min-entropy on strings of length n. These extractors can extract any
constant fraction of the min-entropy using O(log^2 n) additional random
bits, and can extract all the min-entropy using O(log^3 n) additional
random bits. Both of these ... more >>>

This paper strengthens the low-error PCP characterization of NP, coming
closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for
membership in any NP language can be verified with a constant
number of accesses, and with an error probability exponentially
small in the ... more >>>

It is well known that probabilistic boolean decision trees
cannot be much more powerful than deterministic ones (N.~Nisan, SIAM
Journal on Computing, 20(6):999--1007, 1991). Motivated by a question
if randomization can significantly speed up a nondeterministic
computation via a boolean decision tree, we address structural
properties of Arthur-Merlin games ... more >>>