CS830:Abstract and Concrete Complexity

--- Randomness and Complexity

Professor: Jin-Yi Cai

TA: Denis Charles

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. We will start with a review of the standard complexity classes: LOGSPACE, NL, P, NP, co-NP, PH, PSPACE; simple probability bounds; ZPP, RP, coRP, BPP. Interactive proof systems, Arthur-Merlin classes, public coins versus private coins; Immerman-Szelepcsenyi Theorem; LFKN protocol, IP=PSPACE. Then we consider some lower bound proofs; hardness - randomness; pseudorandom generators; amplifications; expanders and extractors; some topics in graph isomorphisms; random walks on graphs; Nisan's generator for space bounded computations; simulations and trade-offs in space bounded computations; 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.