CS 2110 Theory of Computation

Fall 2008

Instructor

Kirk Pruhs

Email: kirk@cs.pitt.edu
Phone : 412-624-8844

TA

Christine Chung
Email: chung@cs.pitt.edu



Class Time and Location

Monday, Wednesday, Friday: 11:30 - 12:45 in 5313 Sennott Square. The course is scheduled three times per week to allow me to miss some weeks to travel for academic conferences/collaborations. The total number of meetings will be the same as a normal two lecture per week course.

Announcements

• Class is canceled Monday September 15, Wednesday September 17 and Friday September 19 as I will be at the European algorithms conference.
• Everyone should join the Google Group http://groups.google.com/group/cs2110/ which you should use for questions etc. To get an invitation, send Christine your gmail address.
  
• Homeworks assignments can be found here. These will be update frequently, so check often.
  

Course Description

The initial plan is to spend a few weeks covering the classic (pre 1970) results that are the foundations of theoretical computer science, and then loosely follow the first eleven chapters of the text by Arora and Boaz. If there is time left after this, I will pick an advanced topic or two from the text to cover. I have not taught theory of computation in a while, so all plans are fluid and subject to change based on how class goes.

Course Format

I will give all of the lectures. There will be homework assignments and a final exam, that will count equally in the final grade. For computer science PhD students, the preliminary exam will be the final exam.

Tentative Schedule

Historical (pre 1970) Results  (7 lectures)

• Formalizing Computation (Church, Turing 1930's)
  
• Finite state machines cannot count
• Church-Turing Thesis
•  Evidence: Simulation
• Undecidable Problems  (Church, Turing 1930's)
•  Halting, proof diagonization
• Term rewriting, proof reduction
• Logic, Proofs and Computation
• Goedel's Incompleteness Theorems (1930's)
  
• Formalizing Information
•  Entropy
• Source coding theorem (Shannon circa 1940's)
  
• Noisy channel coding theorem (Shannon circa 1940's)
•  Kolmogorov complexity (circa 1950's)
  
• Noncomputability of Kolmogorov complexity
• Formalizing games and competition
• Existence of mixed Nash equilibrium (von Neumann and Nash circa 1940's and 1950's)
• Introduction to mechnaism design and inefficiency of equilibrium (circa 2000's)
  

P, PH and PSPACE Chapters 2, 3, 4 and 5  (1970's)  (5 lectures)

• Definition of P, NP, Co-NP, PSPACE
• Time and space hiearchy theorems
  
• Definition of PSPACE and the polynomial time hierarchy
• PSPACE is in EXPTIME
• Machine based complete problems for EXPTIME, PSPACE
• Machine based complete problem for classes in PH and alternating Turing Machines

• Why no obvious complete problems for NP intersect CoNP

• Cook-Levin Theorem: First give completeness for circuits and then reduction from circuits to 3SAT

• NP-completeness of THEOREMS
• NP-completeness proofs for subset sum
• NP-completeness proofs for deciding if there is a Nash equilibrium that is good for both players
• TQBF is PSPACE-complete
• PH is in PSPACE
• PSPACE = APTIME 
• PSPACE hardness of some game, see for example this list?
• Ladner's Theorem ?
  
• Baker, Gill, Soloway

Circuits Chapter 6 (1970's) (1.5 lectures)

• Definition of (uniform) NC
• Circuit Value Problem NC hard for P
• Parallel Computation Thesis (Wikepedia)
• NLogSpace and LogSpace in NC
• Definition of P/Poly
• Karp-Lipton Theorem

Randomization Chapter 7 (1970's)  (1.5 lectures)

• Random polynomial time algorithm for primality (Soloway and Strassen 1977)
• Definition of BPP, RP, co-RP, ZPP
• Why these classes seem to not have complete problems
  
• ZPP in BPP
  
• BPP in P/Poly
• BPP in polynomial time hierarchy

Interactive Proofs Chapter 8 (1980's)  (3 lectures)

• Examples: Uno card color, graph non isomorphism
• Definitions of AM and IP
• AM protocol for approximate set size
  
• GNI in AM[2] and IP[k] in AM[k+2]
  
• If GI is NP-complete then the polynomial time hierachy collapses
• IP=PSPACE
  
• History of IP=PSPACE result, great reading about how research happens in the real world
  

Cryptography Chapter 9 (Mostly 1980's) (2 lectures)

• One time pad private key
• Public key cryptography
• Definition of one-way function
• Definition of Pseudo-random generators
• Statement of results that one way functions imply pseudo-random generators which implie private key cryptography with smallish keys and derandomization of BPP
  
• Definition of perfect zero knowledge
• Zero knowledge proof of graph isomorphism
  

Energy  (probably .5 lecture)

• Billiard ball circuits
• Reversable computation
• Minimum energy computation
• 
  

Quantum Computation Chapter 10 (Mostly 1990's) ( 4 lectures)

• Two 1/2 silver mirror experiment
• EPR and the parity game
• Provably secure quantum cyrptography, Heisenberg uncertainty principle
  
• Simon's algorithm
• Grover's algorithm
• Just a few hints at Shor's algorithm (popular description)

Approximation Algorithms Chapter 11 (Mostly 1990's) (1.5 lectures)

• Statement of PCP theorem
• Hardness of approximation of MAXSAT
  
• How to use PCP to prove hardness of approximation by reduction