.
- Lecture 29 : Final Project Presentations (Sat, Jun 13, 2009, 13:30-17:00)
   
[Contents]
[References]
[Reports]
   
- Lecture 28 (Thu, Jun 11, 2009, 13:30-15:00)
   
[Contents]
[References]
[Scribe notes Not yet in]
[Additional Reading]
   
Barriers in Proving Circuit Lower Bounds : Relativisation(recap), Natural Proofs,
Algebraization(overview).
Suggested Additional Reading:
- Lecture 27 (Wed, Jun 10, 2009, 13:30-15:00) - Scribe: Shiteng Chen
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
Completing the proof of Impagliazzo-Kabanets theorem in detail.
Suggested Additional Reading:
- Lecture 26 (Wed, Jun 3, 2009, 13:30-15:00) - Scribe: Jing He
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
Counting Complexity, Arithmetic Circuits, Permanent, Determinant.
Derandomization implies circuit lower bounds.
Impagliazzo-Kabanet theorem.
Suggested Additional Reading:
- Lecture 25 (Mon, Jun 1, 2009, 13:30-15:00) - Scribe: Xiaohui Bei
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
Interactive Proofs, IP, AM, MA. Structural properties of
these classes. IP=PSPACE. Mayer's theorem(statement).
Suggested Additional Reading:
- Lecture 24 (Wed, May 27, 2009, 13:30-15:00) - Scribe: Youming Qiao
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
From worst case hardness to mild average hardness.
Goldreich-Levin hard bit, Product/XOR Lemma. Aplyifying hardness
using product constructions.
Suggested Additional Reading:
- Lecture 23 (Mon, May 25, 2009, 13:30-15:00) - Scribe: Kai Jin
   
[Contents]
[References]
[Scribe notes Not yet in]
[Additional Reading]
   
Derandomizing BPP based on worst case assumptions. "next-bit" unpredictability.
Nisan-Wigderson Generator.
Suggested Additional Reading:
- Lecture 22 (Sun, May 24, 2009, 14:00-16:00) - Scribe: Shiteng Chen
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
Detailed presentation of Nisan's derandomization result for the space domain.
Suggested Additional Reading:
- Lecture 21 (Wed, May 20, 2009, 13:30-15:00) - Scribe: Jing He
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
   
[Problem Set 2 (pdf)(tex)]
Extractors, Explicit constructions. Pseudorandom generators. Implications in the
derandomization. Pseudorandom generators for space bounded computations.
Some ideas which are used in derandomizing BPL using O(log^2 n)
space (details postponed to the next lecture).
References:
- Chapter 21 of the textbook by Arora-Barak.
Suggested Additional Reading:
- Lecture 20 (Mon, May 18, 2009, 13:30-15:00) - Scribe: Youming Qiao
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Success probablity amplification of randomized algorithms using expander graphs.
Definitions of imperfect randomness. Basic set up for derandomization.
Concept of Extractors.
Suggested Additional Reading:
- Lecture 19 (Sat, May 16, 2009, 14:00-16:00) - Scribe: Hongyu Liang
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Spectral gap for any regular graph. Equivalence between expansion definitions.
Expander mixing lemma. Graph products and details of Reingold's log-space
algorithm for testing undirected connectivity.
Suggested Additional Reading:
- Lecture 18 (Wed, May 13, 2009, 13:30-14:45) - Scribe: Hao Song
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Random walks on graphs. Mixing lemma.
Randomized log space algorithm for undirected connectivity.
Spectral expansion, combinatorial expansion.
Basic definitions and parameters.
Log space connectivity for expander graphs.
Outline of Reingold's algorithm.
Suggested Additional Reading:
- Lecture 17 (Mon, May 11, 2009, 13:15-14:45) - Scribe: Hao Song
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
BPP in \Sigma_2 \cap \Pi_2. Space bounded randomized classes.
Suggested Additional Reading:
- Lecture 16 (Wed, May 6, 2009, 13:30-15:00) - Scribe: Kai Jin.
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Randomized Computation and Complexity Classes PP, BPP and RP.
Comparison of these classes with the ones we have seen so far like P,
NP, PSPACE etc. Amplifying the success probability.
Complete derandomization. BPP \in P/poly.
RP algorithm for Polynomial identity testing problem.
Suggested Additional Reading:
- Lecture 15 (Wed, Apr 29, 2009, 13:30-15:00) - Scribe: Yu Wu
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Power of Negations. Markov's tight characterisation for number of negation gates
that are needed to compute any Boolean function. Fisher's strengthening to
polynomial size setting. A method to derive lower bounds against a negation
limited circuits from lower bounds against monotone circuits.
Suggested Additional Reading:
- Lecture 14 (Mon, Apr 27, 2009, 13:30-15:00) - Scribe: Jing He
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Monotone Boolean functions and monotone Boolean circuits.
Clique function. Upper bounds on monotone circuit size
computing clique. Almost tight (which is super polynomial)
lower bounds for monotone circuit size for clique.
Suggested Additional Reading:
- Lecture 13 (Wed, Apr 22, 2009, 13:30-15:00) - Scribe: Xiahui Bei
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Representation and Approximation of Boolean functions by polynomials.
Smolensky's polynomial method.
Proof that PARITY is not in AC^0[3] (AC^0 circuits which can also use mod-3 gates).
Suggested Additional Reading:
- Lecture 12 (Mon, Apr 20, 2009, 13:30-15:00) - Scribe: Kai Ren
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Review of last lecture. Rest of the details of the proof of
lower bounds for parity. Proof of Switching Lemma.
Suggested Additional Reading:
- Lecture 5 from
Madhu Sudan's course has a slightly different (possibly simpler to understand) proof
of what we did in class.
- Lecture 11 (Wed, Apr 15, 2009, 13:30-15:00) - Scribe: Kai Ren
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Explicit Lower Bound question. Review of decision tree model, depth/height lower bounds for parity.
Exponential size lower bounds for constant depth circuits computing parity. Statement of switching
Lemma. Completing the proof assuming switching lemma. Proof of switching lemma postponed to next
lecture.
Suggested Additional Reading:
- Lecture 5 from
Madhu Sudan's course has a slightly different (possibly simpler to understand) proof
of what we did in class.
- Lecture 10 (Mon, Apr 13, 2009, 13:30-15:00) - Scribe: Youming Qiao
   
[Contents]
[References]
[Scribe notes
(pdf)
(tex)]
[Additional Reading]
Advice Functions, P/poly, Sparse sets. Turing reduction to Sparse sets.
Circuit Lower bounds.
Conditional Circuit Lower bounds for NP (Karp-Lipton-Sipser Collapse result).
Shannon's theorem (almost all Boolean functions require 2^n/n size circuits).
Lypanov's theorem (all Boolean function are computed by circuits of size 2^n/n+o(2^n/n)), proof
skipped.
References:
- Chapter 1, of Vollmer's book on "Introduction to Circuit Complexity".
- Chapter 4 of Jin Yi Cai's full set of
Lecture Notes.
Suggested Additional Reading:
- Lecture 9 (Sat, Apr 11, 2009, 14:00-16:30) - Scribe: Bangsheng Tang
   
[Contents]
[References]
[Scribe notes (pdf)(tex)]
[Additional Reading]
Poly size Boolean Formulas computes exactly NC^1.
Decision tree codel of computation, upper and lower bounds
for parity function. Deterministic, Non-deterministic
and permutation branching programs. Characterisations for
L, NL. Programs over algebraic structures. Barrington's Theorem.
Suggested Additional Reading:
- Lecture 8 (Wed, Apr 8, 2009, 13:30-15:00) - Scribe: Shiteng Chen
   
[Contents]
[References]
[Scribe notes (pdf)] (tex)]]
[Additional Reading]
   
[Problem Set 1 (pdf)
(tex)]
Many-one reductions, Turing reductions, Truth-table
reductions. Completness, NP-Complete problems, Isomorphism of
NP-complete sets, Berman-Hartmanis isomorphism conjecture.
Sparse sets, P/poly, and collapse results.
References:
- Lecture 10 in Jin Yi Cai's course.
- Parts of Chapter 2,3 and 7 of the book on "Theory of Computational Complexity" - Du & Ko
Suggested Additional Reading:
- Check out Manindra Agarwal's work
on isomorphism conjecture and likely structure of
NP-complete sets.
- Five lectures CANCELLED ! ... Mar 23, 25, 29 and Apr 1, 6 2009.
We will compensate for four of them. No lecture on Apr 6, due to QingMing festival.
In the mean time, here are some interesting videos from the research channel
(you don't watch *the* research channel? ! :D)
-
lecture on expanders graphs & eigenvalues by Nati Linial
-
lecture on coding theory by Madhu Sudan.
We will be using these concepts later in the course.
- Lecture 7 (Wed, Mar 18, 2009, 13:30-15:00) - Scribe : Hao Song
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Back to Structural Complexity : Formal treatment of Polynomial Heirarchy, Oracle Turing Machines.
Alternating Turing Machines, Basic Simulations. Circuit characterisation.
References:
- Lecture in Jin Yi Cai's course.
- Chapter 3 of the book on "Theory of Computational Complexity" - Du & Ko
Suggested Additional Reading:
- Lecture 6 (Mon, Mar 16, 2009, 13:30-15:00) - Scribe: Hongyu Liang
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
TC^0 circuits for symmetric functions. Adding n numbers of length n
each (ITADD - Iterated Addition) in NC^1.
Adding log n numbers of length n each (LOGITADD - Log Iterated Addition)
in AC^0. NC^1 circuit for Majority. AC^0 circuits for log Threshold
computation.
References:
- Chapter 1, of Vollmer's book on "Introduction to Circuit Complexity".
Suggested Additional Reading:
- Lecture 5 (Sat, Mar 14, 2009, 14:00-15:00) - Scribe: Hongyu Liang
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Majority function. Definitions of AC, NC, ACC, TC
heirarchy.
Computing threshold in NC^1 (outline) and
logarithmic threshold in AC^0 (outline).
Suggested Additional Reading:
- Lecture 4 (Wed, Mar 11, 2009, 13:30-15:10) - Scribe: Kai Jin
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Nonuniform models of computation. Boolean Circuit Complexity. Resources in the circuit model.
Computing partity function in O(log n) depth and bounded fan-in.
Definition of NC^1. Addition of two n-bit numbers in constant depth and
unbounded fan-in. Definition of AC^0, ACC^0. Uniformity constraints.
Comparison with Turing machine model.
Suggested Additional Reading:
- Attend ITCS Seminar by Kristoffer Hansen on Thrusday at 4pm. The topic is "Constant Width
Characterizations of Small Depth Circuit Classes" and will talk about characterisations of
NC^1, AC^0, ACC^0 that we mentioned in the class.
- Lecture 3 (Mon, Mar 9, 2009, 13:30-15:00) - Scribe: Xiahui Bei
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Immerman-Szelepcsenyi Theorem. A short description of oracle turing machines and notion of relativisation,
and the metalevel argument about relativising proof techniques.
Polynomial Hierarchy (postponed to next lecture).
References: Chapter 2 of the textbook by Du and Ko.
Suggested Additional Reading:
- Two applications of inductive counting for complementation problems -
Borodin, Cook, Dymond, Ruzzo, Tompa - SIAM J. Comput. Volume 18, Issue 3, pp. 559-578 (1989)
- Relativizations of the P vs NP Question.
T. P. Baker, J. Gill, R. Solovay. - SIAM Journal on Computing, 4(4):
431-442 (1975)
- Relative to a Random Oracle A, P^A
!= NP^A != co-NP^A with Probability 1. C. H. Bennett, J. Gill.
SIAM Journal on Computing, 10(1): 96-113 (1981)
- Lecture 2 (Wed, Mar 4, 2009, 13:30-15:00)
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Space Heirarchy Theorem, Time Heirarchy theorem,
Non-determinism, Basic Complexity Classes P,NP,PSPACE,L,NL. Savitch's
Theorem. PSPACE=NPSPACE.
References: Chapter 1 of the textbook by Du and Ko.
Suggested Additional Reading:
- Lecture 1 (Mon, Mar 2, 2009, 13:30-14:45)
   
[Contents]
[References]
[Scribe notes (pdf)
(tex)]
[Additional Reading]
Contents : Introduction to the course,
administrative details.
Recap of Turing machine model, Configurations.
Notion of a resource, Blum axioms.
Tape reduction.
Basic theorems for time and space, Linear Speedup theorem, Tape
Compression Theorem.
References:
- Chapter 1 of Book on "Theory of Computational Complexity" - Du and Ko
Suggested Additional Reading: