We continue the study of $k$-monotone Boolean functions in the property testing model, initiated by Canonne et al. (ITCS 2017). A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone if it alternates between $0$ and $1$ at most $k$ times on every ascending chain. Such functions represent a natural generalization ... more >>>

A merger is a probabilistic procedure which extracts the
randomness out of any (arbitrarily correlated) set of random
variables, as long as one of them is uniform. Our main result is
an efficient, simple, optimal (to constant factors) merger, which,
for $k$ random vairables on $n$ bits each, uses a ... more >>>

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.

Using this game, we ... more >>>

A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of $n$ cards (labeled $1, ..., n$). For $n$ turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then ... more >>>

The domination number of a graph $G=(V,E)$ is the minimum size of
a dominating set $U \subseteq V$, which satisfies that every
vertex in $V \setminus U$ is adjacent to at least one vertex in
$U$. The notion of a problem kernel refers to a polynomial time
algorithm that achieves ... more >>>

Assume that Alice and Bob, given an authentic channel, have a protocol where they end up with a bit S_A and S_B, respectively, such that with probability (1+eps)/2 these bits are equal. Further assume that conditioned on the event S_A = S_B no polynomial time bounded algorithm can predict the ... more >>>

In the Random Oracle Model (ROM) all parties have oracle access to a common random function, and the parties are limited in the number of queries they can make to the oracle. The Merkle’s Puzzles protocol, introduced by Merkle [CACM ’78], is a key-agreement protocol in the ROM with a ... more >>>

Only a handful candidates for computational assumptions that imply secure key-agreement protocols (KA) are known, and even fewer are believed to be quantum safe. In this paper, we present a new hardness assumption---the worst-case hardness of a promise problem related to an interactive version of Kolmogorov Complexity.
Roughly speaking, the ... more >>>

We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results ... more >>>

A very important problem in quantum communication complexity is to show that there is, or isn?t, an exponential gap between randomized and quantum complexity for a total function. There are currently no clear candidate functions for such a separation; and there are fewer and fewer randomized lower bound techniques that ... more >>>

We show that a decidable promise problem has a non-interactive statistical zero-knowledge proof system if and only if it is randomly reducible to a promise problem for Kolmogorov-random strings, with a superlogarithmic additive approximation term. This extends recent work by Saks and Santhanam (CCC 2022). We build on this to ... more >>>

Assume that for almost all n Kolmogorov complexity
of a string x conditional to n is less than m.
We prove that in this case
there is a program of size m+O(1) that given any sufficiently large
n outputs x.

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity
extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction
more >>>

Solovay has proven that
the minimal length of a program enumerating a set A
is upper bounded by 3 times the absolute value of the
logarithm of the
probability that a random program will enumerate A.
It is unknown whether one can replace the constant
3 by a smaller constant.
more >>>

We introduce the study of Kolmogorov complexity with error. For a
metric d, we define C_a(x) to be the length of a shortest
program p which prints a string y such that d(x,y) \le a. We
also study a conditional version of this measure C_{a,b}(x|y)
where the task is, given ... more >>>

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems $C$ defined ... more >>>

A square matrix $V$ is called rigid if every matrix $V^\prime$ obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major ... more >>>

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions $f ... more >>>