Professor: Jin-Yi Cai
Tuesday, Thursday 1:00 pm -- 2:15 pm. Rm 1289 Comp. Sci.
This is a second course in complexity theory.
We will deal with the standard complexity classes, but with
an emphasis on more modern topics, especially concerning
the interaction of Randomness
and Computation.
The students are expected to be reasonably familiar with
the standard complexity
classes: LOGSPACE,
NL, P, NP, co-NP, PH, PSPACE;
simple probability bounds; ZPP, RP, coRP, BPP.
The lecture notes handed out give a brief review.
Depending on knowledge of the class,
we will also present (or briefly review)
Interactive proof systems,
Arthur-Merlin classes,
public coins versus private coins;
Immerman-Szelepcsenyi Theorem;
LFKN protocol, IP=PSPACE.
We start by considerng one-way functions, weak and strong, and
reductions between them. Then we will discuss
random walks on graphs, and application to space bounded
computation. Nisan's geneartor. Deterministic
simulation of randomized logspace.
Time-space trade-offs in space bounded computations.
Then we consider some lower bound proofs;
hardness - randomness;
pseudorandom generators;
amplifications;
expanders and extractors;
some topics in graph isomorphisms;
leftover hash lemma, hardcore bit;
Hartmanis conjectures;
the permanent, PP, and closure properties;
average-case vs worst-case complexity.
The pace and selection of material will depend on class
interests and background.
Some sample material (not exclusive) to be covered can be found in
the following set of
notes.
Also a reference book is "Pseudorandomness and Cryptographic
Applications" by Mike Luby, Princeton University Press.
Some materials can be found in the book
"Computational Complexity" by Christos Papadimitriou,
Addison-Wesley.
Students are expected to take turns to be scribes for the lectures,
and polish and distribute the class notes. They are also expected to
engage in active discussions on research topics and open problems.