Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits
This work addresses fundamental problems in circuit complexity and proof theory, with implications for explicit constructions and cryptographic primitives, though it builds incrementally on recent breakthroughs.
The paper tackles the range avoidance problem and its connection to proof complexity generators, showing that the existence of demi-bits generators implies hardness for nondeterministic algorithms, resolves an open problem, and leads to unprovability results in certain theories, with constructions simplified using randomness extractors.
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.