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 \cite{Kra-dual} 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 show 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.
This is a rather major revision of the whole paper.
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:
\begin{itemize}
\item There exist a p-time function $g$ extending each input by one bit such that its range $rng(g)$
intersects all infinite NP sets.
\end{itemize}
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'08 is a good candidate for $g$.
The condition in the conjecture is supposed to mean:
|g(x)| = 1 + |x|.
