  For an NP intersect coNP function g of the Nisan-Wigderson type and a string
b outside its range we consider a two player game on a common input a to the
function. One player, a computationally limited Student, tries to find a bit of
g(a) that differs from the corresponding bit of b. He can query a
computationally unlimited Teacher for the witnesses of the values of constantly
many bits of g(a). The Student computes the queries from a and from Teacher's
answers to his previous queries. It was proved by Krajicek (2011) that if g is
based on a hard bit of a one-way permutation then no Student computed by a
polynomial size circuit can succeed on all a. In this paper we give a lower
bound on the number of inputs a any such Student must fail on. Using that we
show that there is a pseudo-finite set of hard instances on which all uniform
students must fail. The hard-core set is defined in a non-standard model of
true arithmetic and has applications in a forcing construction relevant to
proof complexity.

Computational Complexity;Logic