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Revision #1 to TR22-120 | 10th July 2023 09:07
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#### On the existence of strong proof complexity generators

**Abstract:**
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.

**Changes to previous version:**
This is a rather major revision of the whole paper.

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TR22-120 | 24th August 2022 13:23
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#### On the existence of strong proof complexity generators

**Abstract:**
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 $g$.