Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits

2025-11-17

AI Summary

This paper studies the *Avoid* problem: given a circuit (G) with input length (n) and output length (m > n), construct a string (y) outside its image (mathrm{Im}(G)). The core challenge is to simultaneously ensure computational hardness of *Avoid* and the existence of a *proof-complexity generator*—i.e., no efficient proof in standard propositional proof systems can certify that any particular (y) lies outside (mathrm{Im}(G)). To this end, the authors first establish a novel connection between *Avoid* hardness and *demi-bit generators* resilient against nondeterministic attacks. Assuming demi-hardness of LPN-style and Goldreich-type PRGs, they prove *Avoid* is hard for constant-depth polynomial-size circuits. They then construct a *pseudosurjective* proof-complexity generator, resolving the Chen–Li open problem optimally. Finally, they show that if demi-bit generators satisfy AM security, then PV(_1) cannot prove the dual weak pigeonhole principle—thereby separating APC(_1) from PV(_1) with significantly simpler constructions and proofs than prior work.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing

Abstract

Given a circuit $G: {0, 1}^n o {0, 1}^m$ with $m > n$, the *range avoidance* problem ($ ext{Avoid}$) asks to output a string $yin {0, 1}^m$ 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 $G: {0, 1}^n o {0, 1}^m$ where $m > n$ but for every $yin {0, 1}^m$, it is infeasible to prove the statement "$y otinmathrm{Range}(G)$" 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). $ullet$ We show that the existence of demi-bits generators implies $ ext{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 $ ext{Avoid}$ even when the instances are constant-degree polynomials over $mathbb{F}_2$. $ullet$ 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 $mathsf{PV}_1$. $ullet$ 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 $ ext{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.

Problem

• Studies hardness of range avoidance problem for nondeterministic algorithms
• Connects demi-bits generators to proof complexity generators construction
• Shows unprovability of dual weak pigeonhole principle in PV₁

Innovation

• Connecting range avoidance hardness to demi-bits generators
• Proving unprovability in Cook's theory using demi-bits
• Transforming demi-bits generators into pseudo-surjective proof generators