The working conjecture from K'04 that there is a proof complexity generator hard for all proof systems can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows: * There exist a p-time function $g$ stretching each input by one bit such that its range $rng(g)$ intersects all infinite NP sets. We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results) and the range avoidance problem, to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from K'09 is a good candidate for $g$.

翻译：K'04提出的工作猜想认为存在一个对所有证明系统都困难的证明复杂性生成元，可等价地（针对p-时间生成元）在不涉及证明复杂性概念的情况下表述如下：* 存在一个p-时间函数$g$，该函数将每个输入拉伸一位，使得其值域$rng(g)$与所有无限NP集相交。我们考察了这一猜想的多个方面，包括其与有界算术（见证与独立性结果）和范围回避问题的联系、与时间有界柯尔莫哥洛夫复杂性的关联、与命题证明系统的可行析取性质的关系以及与证明搜索复杂性的联系。我们论证了K'09中的特定小工具生成元是$g$的良好候选者。