Syllabus
1) Oracle Turing machines.
2) Polynomial hierarchy (definitions based on oracles and on alternating quantifiers, proof of their equivalence)
3) Quantified boolean formulas QBF and their completenes for PSPACE and Σi.
4) Nondeterministic hierarchy theorems.
5) Log-space reducibility, P-completeness and its consequences.
6) Szelepcsenyi-Immerman theorem, NL=coNL.
7) Non-uniform computational models - advice functions, boolean circuits, classes NC and P/poly, functions with maximum circuit size.
8) Probabilistic algorithms - classes RP, coRP, ZPP, and BPP.
9) Error reduction in BPP, BPP is in P/poly, BPP is in Σ2.
10) NP-completeness of UNIQUE-SAT (probabilistic reduction)
11) PCP theorem (without proof) and its applications in inaproximability.
Annotation
This course extends the basic course on computational complexity. It introduces the students to the concepts of polynomial hierarchy classes, probabilistic computation, oracle computation, non-uniform computational models and the PCP theorem.