This article is a continuation of our search for tautologies that are
hard even for strong propositional proof systems like EF, cf.
[Kra-wphp,Kra-tau]. The particular tautologies we study, the
τ-formulas, are obtained from any 𝒫/poly map g; they
express that a string is outside of the range of g. Maps g
considered here are particular pseudorandom generators. The ultimate
goal is to deduce the hardness of the τ-formulas for at least
EF from some general, plausible computational hardness hypothesis.
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In this paper we introduce the notions of pseudo-surjective and
iterable functions (related to free functions of [Kra-tau]).
These two properties imply the hardness of the τ-formulas from
the function but unlike the hardness they are preserved under
composition and iteration. We link the existence of maps with these
two properties to the provability of circuit lower bounds, and we
characterize maps g yielding hard τ-formulas in terms of a
hitting set type property (all relative to a propositional proof
system). We show that a proof system containing EF admits a
pseudo-surjective function unless it simulates a proof system WF
introduced by Jeřábek [Jer], an extension of EF.
We propose a concrete map g as a candidate function possibly
pseudo-surjective or free for strong proof systems. The map is
defined as a Nisan-Wigderson generator based on a random function and
on a random sparse matrix. We prove that it is iterable in a
particular way in resolution, yielding the output/input ratio
n3-ε (that improves upon a direct construction of
Alekhnovich et al. [ABRW]).