We give a general reduction of lengths-of-proofs lower bounds for
constant depth Frege systems in DeMorgan language augmented by
a connective counting modulo a prime $p$
(the so called $AC^0[p]$ Frege systems)
to computational complexity
lower bounds for search tasks involving search trees branching upon
values of linear maps on the vector space of
low degree polynomials over the finite field with $p$ elements.

Clarifying a step in the proof of 5.1.

We give a general reduction of lengths-of-proofs lower bounds for
constant depth Frege systems in DeMorgan language augmented by
a connective counting modulo a prime $p$
(the so called $AC^0[p]$ Frege systems)
to computational complexity
lower bounds for search tasks involving search trees branching upon
values of maps on the vector space of
low degree polynomials over the finite field with $p$ elements.

Section 6 was revised.

We give a general reduction of lengths-of-proofs lower bounds for
constant depth Frege systems in DeMorgan language augmented by
a connective counting modulo a prime $p$
(the so called $AC^0[p]$ Frege systems)
to computational complexity
lower bounds for search tasks involving search trees branching upon
values of linear maps on the vector space of
low degree polynomials over the finite field with $p$ elements.