 
Summary: IAS/Park City Mathematics Series
Volume 00, 0000
Exploring Complexity through reductions
Sanjeev Arora
Since we have few techniques for proving strong lowerbounds on Turing ma
chine computations, an interim approach is to study interrelationships between
various complexity questions. We usually prove interrelationships using reductions.
Examples of such e#orts include (in addition to the ones studied in the next three
lectures) the following.
(1) Most of cryptography, like message authentication, digital signatures,
pseudorandom generation, etc. is based on reductions.
(2) Papadimitriou's complexity class based on nonconstructive math proofs [Pap94].
(3) SahaiVadhan's complete problems for zeroknowledge [SV97].
(4) Levin's theory of average case complexity [Lev].
(5) Fixed parameter tractability theory introduced by Downey and Fellows [DF95].
The next three lectures show this reductionbased methodology in action for
three important issues.
We note that there are many examples of complexity classes for which the
reductionbased e#ort has not proved as successful, such as BPP, NP
# coNP, and
