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
- Definition of semantic security

- 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

- Definition of perfect zero knowledge
- 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, Heisenberg uncertainty principle (probably in lecutre we'll just explain the Heisenberg uncertainty principle, and skip its application to secure crypto. Its in the notes if you want to read about it).

- 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
- NP subset PCP(poly, 1)

- Information Theoretic Lower Bounds Chapers 12 and 13 (wishful thinking?)

- Sorting lower bound
- Element Uniqueness lower bound
- Communication complexity and the tiling lower bound for equality

- Yao's minimax technique for sorting