A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m \ge 2$ is a pair of ordered lists $\mathcal{U} = (\mathbf{u}_1, \cdots, \mathbf{u}_K)$ and $\mathcal{V} = (\mathbf{v}_1, \cdots, \mathbf{v}_K)$ where $\mathbf{u}_i, \mathbf{v}_j \in \mathbb{Z}_m^n$ with the following property: for any $i \in [K]$, the inner product $\langle \mathbf{u}_i, \mathbf{v}_i \rangle ... more >>>

In this work we study oblivious complexity classes. Among our results:
1) For each $k \in \mathbb{N}$, we construct an explicit language $L_k \in O_2P$ that cannot be computed by circuits of size $n^k$.
2) We prove a hierarchy theorem for $O_2TIME$. In particular, for any function $t:\mathbb{N} \rightarrow \mathbb{N}$ ... more >>>

We study the Range Avoidance Problem (Avoid), in which the input is an expanding circuit $C : \{0,1\}^n \to \{0,1\}^{n+1}$, and the goal is to find a $y \in \{0,1\}^{n+1}$ that is not in the image of $C$. We are interested in the randomized complexity of this problem, i.e., in ... more >>>

In a recent breakthrough, Chen, Hirahara and Ren prove that S$_2$E/$_1 \not\subset$ SIZE$[2^n/n]$ by giving a single-valued FS$_2$P algorithm for the Range Avoidance Problem (Avoid) that works for infinitely many input size $n$.

Building on their work, we present a simple single-valued FS$_2$P algorithm for Avoid that works for all ... more >>>