Cook and Reckhow 1979 pointed out that NP is not closed under complementation iff there is no propositional proof system that admits polynomial size proofs of all tautologies. Theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus at a conjecture from K.2004 in foundations of the theory that there is a proof complexity generator hard for all proof systems. This 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 intersects all infinite NP sets. We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), 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.2009 is a good candidate for $g$. We define a new hardness property of generators, the $\bigvee$-hardness, and shows that one specific gadget generator is the $\bigvee$-hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite NP sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite NP sets.

翻译：Cook与Reckhow (1979)指出，NP在补运算下不封闭当且仅当不存在一个命题证明系统能对所有重言式提供多项式大小的证明。证明复杂度生成器理论旨在构造对强证明系统（可能对所有证明系统）困难的重言式集合。我们聚焦于K.2004提出的该理论基础猜想：存在一个对所有证明系统困难的证明复杂度生成器。该猜想可（对p-时间生成器）等价表述为与证明复杂度概念无关的形式：*存在一个p-时间函数$g$，该函数将每个输入拉伸一位，使得其值域与所有无限NP集相交。我们考察该猜想的多个侧面，包括其与有界算术（见证性与独立性结果）、时间有界Kolmogorov复杂度、命题证明系统的可行析取性质以及证明搜索复杂度的联系。我们认为K.2009中提出的特定装置生成器是$g$的优良候选。我们定义生成器的新困难性——$\bigvee$-困难性，并证明该特定装置生成器是（关于任意足够强的证明系统的）$\bigvee$-最困难生成器。我们定义可行无限NP集类，并证明在电路复杂性假设下，该猜想对所有可行无限NP集成立。