We consider the range avoidance problem (called Avoid): given the description of a circuit $C:\{0, 1\}^n \to \{0, 1\}^\ell$ (where $\ell > n$), find a string $y\in\{0, 1\}^\ell$ that is not in the range of $C$. This problem is complete for the class APEPP that corresponds to explicit constructions of ... more >>>

Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a $y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$, $NC^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ ... more >>>

The range avoidance problem (denoted by Avoid) asks to find a string outside of the range of a given circuit $C:\{0,1\}^n\to\{0,1\}^m$, where $m>n$. Although at least half of the strings of length $m$ are correct answers, it is not clear how to deterministically find one. Recent results of Korten (FOCS'21) ... more >>>

We show that there is a language in $\mathrm{S}_2\mathrm{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds for the classes $\Sigma_2\mathrm{E}$, $(\Sigma_2\mathrm{E}\cap\Pi_2\mathrm{E})/_1$, and $\mathrm{ZPE}^{\mathrm{NP}}/_1$. Previously, only "half-exponential" circuit lower bounds for these ... more >>>

A recent line of research has introduced a systematic approach to explore the complexity of explicit construction problems through the use of meta problems, namely, the range avoidance problem (abbrev. Avoid) and the remote point problem (abbrev. RPP). The upper and lower bounds for these meta problems provide a unified ... 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 >>>

In this work we observe a tight connection between three topics: $NC^0$ cryptography, $NC^0$ range avoidance, and static data structure lower bounds. Using this connection, we leverage techniques from the cryptanalysis of $NC^0$ PRGs to prove state-of-the-art results in the latter two subjects. Our main result is a quadratic improvement ... more >>>

Given a circuit $C : \{0,1\}^n \to \{0,1\}^m$ from a circuit class $F$, with $m > n$, finding a $y \in \{0,1\}^m$ such that $\forall x \in \{0,1\}^n$, $C(x) \ne y$, is the range avoidance problem (denoted by $F$-AVOID). It is known that deterministic polynomial time algorithms (even with access ... more >>>