Hanlin Ren

Jiawei Li, Yuhao Li, Hanlin Ren

We study the *refuter* problems for proof complexity lower bounds. Suppose $φ$ is a hard tautology that does not admit any length-$s$ proof in some proof system $P$. In the corresponding refuter problem, we are given (query access to) a purported length-$s$ proof $π$ in $P$ that claims to have proved $φ$, and our goal is to find an invalid derivation step within $π$. As suggested by witnessing theorems in bounded arithmetic, the *computational complexity* of these refuter problems is closely tied to the *metamathematics* of the underlying lower bounds. We focus on refuter problems corresponding to lower bounds for *resolution*, which is arguably the single most studied system in proof complexity. To capture the complexity of refuter problems for resolution *size* lower bounds, we introduce a new class $\mathrm{rwPHP}(\mathsf{PLS})$ in decision-tree $\mathsf{TFNP}$, which can be seen as a randomized version of $\mathsf{PLS}$. Interpreted in bounded arithmetic, our results show that the theory $\mathsf{T}^1_2(α) + \mathrm{dwPHP}(\mathsf{PV}(α))$ characterizes the "reasoning power" required to prove (the "easiest") resolution size lower bounds. As a corollary, we obtain surprisingly efficient proofs of resolution lower bounds. In particular, we show that many resolution size lower bounds can be proved in low-width *random resolution* [Pudlák--Thapen, CCC'17].

Hanlin Ren, Yichuan Wang, Yan Zhong

Given a circuit $G: \{0, 1\}^n \to \{0, 1\}^m$ with $m > n$, the *range avoidance* problem ($\text{Avoid}$) asks to output a string $y\in \{0, 1\}^m$ that is not in the range of $G$. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of *proof complexity generators* -- circuits $G: \{0, 1\}^n \to \{0, 1\}^m$ where $m > n$ but for every $y\in \{0, 1\}^m$, it is infeasible to prove the statement "$y\not\in\mathrm{Range}(G)$" in a given propositional proof system. This paper connects these two problems with the existence of *demi-bits generators*, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). $\bullet$ We show that the existence of demi-bits generators implies $\text{Avoid}$ is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich' PRG, we prove the hardness of $\text{Avoid}$ even when the instances are constant-degree polynomials over $\mathbb{F}_2$. $\bullet$ We show that the dual weak pigeonhole principle is unprovable in Cook's theory $\mathsf{PV}_1$ under the existence of demi-bits generators secure against $\mathbf{AM}$, thereby separating Jerabek's theory $\mathsf{APC}_1$ from $\mathsf{PV}_1$. $\bullet$ We transform demi-bits generators to proof complexity generators that are *pseudo-surjective* with nearly optimal parameters. Our constructions build on the recent breakthroughs on the hardness of $\text{Avoid}$ by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use *randomness extractors* to significantly simplify the construction and the proof.