Time-bounded conditional Kolmogorov complexity of a string $x$ given $y$, $K^t(x\mid y)$, is the length of a shortest program that, given $y$, prints $x$ within $t$ steps. The Chain Rule for conditional $K^t$ with error $e$ is the following hypothesis: there is a constant $c\in\mathbb{N}$ such that, for any strings ... more >>>

We prove that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded $\gamma_2$-norm and discuss applications in communication complexity, operator theory, spectral ... more >>>

Given an integer-valued function $f:\{0,1\}^n\rightarrow \{0,1,\dots, m-1\}$ that is mildly hard to compute on instances drawn from some distribution $D$ over $\{0,1\}^n$, we show that the function $g(x_1, \dots, x_t) = f(x_1) + \dots + f(x_t)$ is strongly hard to compute on instances $(x_1, \dots, x_t)$ drawn from the product ... more >>>