A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement $x$ and the proof ${\pi}$ are vectors over a finite field $\mathbb{F}$, and the proof is verified by making a single dot-product query $\langle {q},({x} \| {\pi})\rangle$ jointly to ${x}$ and ${\pi}$. A DPP can ... more >>>

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to the best of our knowledge, the first known family of codes that can be decoded (and encoded) in nearly linear time, even as they ... more >>>

We prove that the most natural low-degree test for polynomials over finite fields is ``robust'' in the high-error regime for linear-sized fields. Specifically we consider the ``local'' agreement of a function $f\colon \mathbb{F}_q^m \to \mathbb{F}_q$ from the space of degree-$d$ polynomials, i.e., the expected agreement of the function from univariate ... more >>>

We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both \emph{exact} and \emph{approximate} versions of the algorithm. The input to the algorithms are (1) coefficients of an $m$-variate polynomial $f$ with degree $d$ in each ... more >>>

Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function $f : \{0, 1\}^n\to \{0, 1\}$ is the minimum $\lambda \geq 1$ such that for all positive integers $t$,
\[Pr_{\rho\sim R_p} [\text{DT}_{\text{depth}}(f|_\rho) \geq t] \leq (p\lambda)^t.\]
.
Håstad’s celebrated switching lemma ... more >>>

A problem is downward self-reducible if it can be solved efficiently given an oracle that returns
solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is
well studied and it is known that all downward self-reducible problems are in PSPACE. In this
paper, we initiate the study of ... more >>>

We study the following natural question on random sets of points in $\mathbb{F}_2^m$:

Given a random set of $k$ points $Z=\{z_1, z_2, \dots, z_k\} \subseteq \mathbb{F}_2^m$, what is the dimension of the space of degree at most $r$ multilinear polynomials that vanish on all points in $Z$?

We ... more >>>

The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [DKSS13], the lemma has found nu- merous applications in both math and computer ... more >>>

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom group, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1, A_2, A_3 \subset G$, we have
\[ \left|\Pr_{x,y\sim G}\left[ x \in ... more >>>

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>>

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these ... more >>>

We construct an explicit family of 3XOR instances which is hard for Omega(sqrt(log n)) levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time.
Our construction is based on the high-dimensional expanders devised by Lubotzky, ... more >>>

We introduce a variant of PCPs, that we refer to as *rectangular* PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the *row* of each query and the other determining the *column*.

We ... more >>>

Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are far from all codewords, by probing a given word only at a very small (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. ... more >>>

Recently, Cohen, Haeupler and Schulman gave an explicit construction of binary tree codes over polylogarithmic-sized output alphabet based on Pudl\'{a}k's construction of maximum-distance-separable (MDS) tree codes using totally-non-singular triangular matrices. In this short note, we give a unified and simpler presentation of Pudl\'{a}k and Cohen-Haeupler-Schulman's constructions.

We prove that for every constant $c$ and $\epsilon = (\log n)^{-c}$, there is no polynomial time algorithm that when given an instance of 3LIN with $n$ variables where an $(1 - \epsilon)$-fraction of the clauses are satisfiable, finds an assignment that satisfies at least $(\frac{1}{2} + \epsilon)$-fraction of clauses ... more >>>

We study the probabilistic degree over reals of the OR function on $n$ variables. For an error parameter $\epsilon$ in (0,1/3), the $\epsilon$-error probabilistic degree of any Boolean function $f$ over reals is the smallest non-negative integer $d$ such that the following holds: there exists a distribution $D$ of polynomials ... more >>>

We develop the notion of double samplers, first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders.

We show how double samplers give a generic way of amplifying distance in a way that enables ... more >>>

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.

Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse ... more >>>

Agreement tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most PCP constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the ... more >>>

Nisan and Szegedy showed that low degree Boolean functions are juntas. Kindler and Safra showed that low degree functions which are *almost* Boolean are close to juntas. Their result holds with respect to $\mu_p$ for every *constant* $p$. When $p$ is allowed to be very small, new phenomena emerge. ... more >>>

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function F$:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be F$_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)}$ (mod $2^k$). We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with ... more >>>

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is the evaluation of a degree-$d$ polynomial over $\mathbb{F}_q$ for $q$ prime.

* $\mathrm{Test}_{e,k}$: Pick $P_1,\ldots,P_k$ independent random degree-$e$ polynomials and accept iff the function $fP_1\cdots P_k$ is the evaluation of a degree-$(d+ek)$ polynomial.

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially ... more >>>

We make progress on some questions related to polynomial approximations of $\mathrm{AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC 1991), that any $\mathrm{AC}^0$ circuit of size $s$ and depth $d$ has an $\varepsilon$-error probabilistic polynomial over the reals ... more >>>

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $\mu$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $${{CC}}^\mu_{0.49}(f) = O\left(\left(\log {{rprt}}_{1/4}(f) \cdot \log \log {{rprt}}_{1/4}(f)\right)^2\right),$$ where ${{CC}^\mu_\varepsilon(f)$ is the distributional communication complexity with error at most $\varepsilon$ under the distribution $\mu$ and ... more >>>

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/poly(n)$, while making at most $O(poly\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/poly\log\log n}$. Previous constructions that ... more >>>

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on ... more >>>

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$
formed by the intersection of $k$ halfspaces, we prove that
$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ... more >>>

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored ... more >>>

We initiate the study of the tradeoff between the {\em length} of a
probabilistically checkable proof of proximity (PCPP) and the
maximal {\em soundness} that can be guaranteed by a $3$-query
verifier with oracle access to the proof. Our main observation is
that a verifier limited to querying a short ... more >>>

We examine the communication required for generating random variables
remotely. One party Alice will be given a distribution D, and she
has to send a message to Bob, who is then required to generate a
value with distribution exactly D. Alice and Bob are allowed
to share random bits generated ... more >>>

We continue the study of the trade-off between the length of PCPs
and their query complexity, establishing the following main results
(which refer to proofs of satisfiability of circuits of size $n$):
We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$
that can be verified by making $o(\log\log n)$ Boolean queries.
more >>>

For a boolean formula \phi on n variables, the associated property
P_\phi is the collection of n-bit strings that satisfy \phi. We prove
that there are 3CNF properties that require a linear number of queries,
even for adaptive tests. This contrasts with 2CNF properties
that are testable with O(\sqrt{n}) ... more >>>

We present a simple proof of the bounded-depth Frege lower bounds of
Pitassi et. al. and Krajicek et. al. for the pigeonhole
principle. Our method uses the interpretation of proofs as two player
games given by Pudlak and Buss. Our lower bound is conceptually
simpler than previous ones, and relies ... more >>>

Most known constructions of probabilistically checkable proofs (PCPs)
either blow up the proof size by a large polynomial, or have a high
(though constant) query complexity. In this paper we give a
transformation with slightly-super-cubic blowup in proof size, with a
low query complexity. Specifically, the verifier probes the proof ... more >>>