CS 1511/2011 Theory of Computation

TENTATIVE Topical Schedule

• Historical (pre 1970) Results Chapter 1
  

• Quantifying Information (notes)
  
•  Entropy and information theory (source 1, source 2, Source 3)
  
• Source coding theorem (Shannon circa 1940's)  (source)
  
•  Kolmogorov complexity (circa 1950's)
  
• Noncomputability of Kolmogorov complexity
• Axiomatic Systems, Logic, Proofs and Computation (notes)
  
• Axiomatic Systems  (Source 1, Source 2)
  
• Difference between truth and provability
• Provably statements are enumerable
• A sound and complete axiomization implies provable statements are decidable/computable
• Godel's Incompleteness Theorems (1930's)
  

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

• Time and Space (notes)
  
• Definition of LogSpace, P, PSpace, ExpTime
• LogSpace in P and PSpace is in ExpTime
• Time and space hiearchy theorems
• Machine based complete problems for P, PSpace, EXPTIME
• Circuit Value Problem is log space complete for P
  
• TQBF is polynomial time complete for PSpace
• PH and Alternation  (notes)
  
• Definition of PH
• PH in PSPACE
  

• Circuits Chapter 6 (1970's) (notes)
  

• Definition of P/Poly
• P subset P/poly
• Unary Halting in P/poly
• Most functions are not in P/poly (Shannon)
  
• Karp-Lipton Theorem

• Randomization Chapter 7 (1970's) (notes)
  

• Problems which have easy randomized algorithms but its not so clear how to solve them detetermnistically
  
• 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) (notes)
  

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

• Energy

• Billiard ball circuits
• Reversable computation and  Landauer's principle
  
• Minimum energy computation

• Quantum Computation Chapter 10 (Mostly 1990's) (notes)
  

• Two 1/2 silvered mirror experiment and bomb testing experiment
• Entanglement
  
• EPR experiment and the parity game
• Simon's Algorithm
• Quantum teleportation
• No cloning theorem (notes)
• Provably secure quantum cyrptography using Quantum Indeterminancy
  
• A brief overview of Shor's algorithm (popular description)
  

• Approximation Algorithms Chapter 11 (Mostly 1990's)(notes)

• Statement of PCP theorem
• Hardness of approximation of MAXSAT
  
• How to use PCP to prove hardness of approximation by reduction
• Proof that NP subset PCP(poly, 1)

• Cryptography Chapter 9 (Mostly 1980's) (notes)
  

• One time pad private key
• Public key cryptography
• Definition of one-way function
• Definition of Pseudo-random generators. Equivalent to unpredictable generator.
  
• One way functions imply pseudo-random generators which imply private key cryptography with smallish keys
  
• Pseudo-random generators imply derandomization of BPP and BPP subset subexponential time
• Bit commitment protocol
  
• Definition of perfect zero knowledge. Definition of semantic security.
  
• Perfect zero knowledge proof of graph isomorphism