CS 1511/2011 Theory of Computation

__TENTATIVE
__Topical Schedule

- Historical (pre 1970) Results Chapter 1

- Formalizing Computation (Church, Turing 1930's) (notes)

- Finite state machines cannot count

- Church-Turing Thesis
- Undecidable Problems (Church, Turing 1930's) (notes)

- Halting, proof diagonization (nice video illustrating proof)

- Logic, Proofs and Computation (notes)

- Godel's Incompleteness Theorems (1930's)

- Formalizing Information (notes)

- Entropy

- Source coding theorem (Shannon circa 1940's)

- Kolmogorov complexity (circa 1950's)

- Noncomputability of Kolmogorov complexity

- 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

- Machine based complete problem for NP
- Cook-Levin Theorem, SAT is complete for NP

- NP-completeness of THEOREMS

- 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 and IP
- AM protocol for approximate set size
- GNI in AM[2] 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

- 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

- 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

- EPR and the parity game
- Simon's Algorithm

- Provably secure quantum cyrptography, Quantum Indeterminancy

- A brief overview of Shor's algorithm (popular description) (Skipping in lecture, leaving reading this as homework)

- 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)

- Information Theoretic Lower Bounds Chapers 12 and 13

- Adversarial lower bound for sorting

- Communication complexity and the tiling lower bound for equality
- Using communication complexity to get an Omega(n^2) lower bound for recognizing palindromes on a 1 tape Turing machine