18.405 –
Advanced Complexity with Prof. Madhu Sudan
(Return to Petar Maymounkov's homepage.)
Related links:
Problem Sets:
Syllabus
 2/12/04:
 Diagonalization
 Relativization,
BakerGillSolovay Theorem: There are and
such that
and
 Ladner's Theorem:
Between any two languages (one not polynomially reducible to the
other) there exist incomparable languages
 2/20/07:
 Neciporuk's Theorem: An explicit function having a superlinear
lowerbound on formula size
 Barrington's construction of smallwidth Branching Programs
 2/21/07:
 FurstSaxeSipser: Parity is not in
, Switching Lemma and Random
Restrictions
 2/26/07:
 Razborov, Method of Approximations: Replacing
deterministic gates with probabilistic polynomials
 Simplicity of
Lemma by Smolensky: Circuits in
can be approximated by lowdegree
polynomials on large fraction of inputs
 DeMilloLiptonSchwartzZippel Lemma of Computer Science
 Parity cannot be approximated by lowdegree polynomials
 2/28/07:
 Communication complexity (CC)
 KarchmerWigderson
games: Communication Complexity = Circuit depth of function
 Tiling Complexity (Yao), Tiling Lemma: Lowerbound on CC
 The rank lower bound on CC (Mehlhorn and Schmidt, STOC'82)
 3/5/07:
 Alternation I: Alternating Machines, Space/Time/Alternation equivalences
 OPEN: Can you solve directed STconnectivity in
space for ?
 OPEN: Does
for
?
 Fortnow's Theorem
 3/7/07:
 Alternation II: The Polynomial Hierarchy,
classes and
 KarpLipton Theorem
 3/12/07:
 OPEN: How to generate a prime number in
deterministically?
 OPEN: Finding square
roots modulo prime deterministically (we know in RP,ZPP)
 Algebraic
Circuit Identity Testing (ACIT)
 Strong TuringChurch Hypothesis
 ZPP/RP/coRP/BPP,
 Amplification for
(co)RP/BPP: Success probability
is equivalent to success
probability
 3/14/07:
 Amplification: Strong BPP = Weak BPP
 [Adleman] .
Implies:
Polynomial Hierarchy collapses
 [Sipser, Lautemann]
 Oneway permutations
 [ValiantVazirani]
: Part I
 3/19/07:

 OPEN: Can we derandomize?
 OPEN: Can we even just improve the probability of success in the reduction?


 [Toda] Parity/BP/existential/forall operators
 and
 3/21/07:
 operators
 Classes and
closed
under complementation
 and
and
 Toda's Theorem:
and
 4/2/07:
 Interactive proofs
 [BabaiMoran] ArthurMerlin Proofs
 [GoldreichMicaliWigderson] GRAPH NONISOMORPHISM by IP[2]
 Private coins = public coins
 Onesided error = twosided error
 and
and
. Last equality by
[LundFortnowKarloffNissan, Shamir]
 Private coins and 2sided error = public coins and 1sided error
 PERMANENT is randomself reducible
