Material Covered

1. Overview of the history of the subject (0.5-1 lecture)

   Hilbert's 1900 lecture, Gödel, Turing, and Church, etc.

2. Computability theory: What can we compute? (2.5-3 weeks)

   Turing machines, Church-Turing thesis, decidability, halting problem, Reductions between problems, Rice's Theorem. (Sipser text, Chapters 3, 4, and parts of 5 and 6).

3. Complexity theory: How fast can we compute it?
   1. Classic time and space classes (6 weeks)

      Sipser text, Chapters 7-9

      Roughly speaking, the hierarchy
      

      $$\mbox{L} \subseteq \mbox{NL} \subseteq \mbox{P} \subseteq \mbox{NP} \subseteq \mbox{PSPACE}$$
      
      

      Specific topics will include

      • Complexity measures--especially time, also space
      • P and NP, SAT, poly-time reducibility

      • NP-completeness

      • Cook-Levin theorem
      • PSPACE, TQBF, Savitch's theorem

      • Games, Generalized Geography
      • L and NL, NL $=$ coNL
      • Hierarchy theorems

      • Provably intractable problems, oracles

   2. Probabilistic and alternating classes (1.5-2 weeks)

      Sipser text Chapter 10.2-10.3 has a brief overview; Papadimitriou has more.

      • Alternating time and space and the polynomial hierarchy

      • Probabilistic Computation

      • BPP

4. Advanced topics as time allows selected from (hoping for 3-5.5 weeks):
   1. Approximation algorithms.
   2. Intro to computational learning theory.
   3. Interactive proofs.
   4. Intro to modern cryptography.