 
Summary: On Basing LowerBounds for Learning on WorstCase Assumptions
Benny Applebaum
Boaz Barak
David Xiao
Abstract
We consider the question of whether P = NP im
plies that there exists some concept class that is effi
ciently representable but is still hard to learn in the
PAC model of Valiant (CACM '84), where the learner
is allowed to output any efficient hypothesis approxi
mating the concept, including an "improper" hypoth
esis that is not itself in the concept class. We show
that unless the Polynomial Hierarchy collapses, such a
statement cannot be proven via a large class of reduc
tions including Karp reductions, truthtable reductions,
and a restricted form of nonadaptive Turing reduc
tions. Also, a proof that uses a Turing reduction of
constant levels of adaptivity would imply an important
consequence in cryptography as it yields a transforma
tion from any averagecase hard problem in NP to a
