### Syllabus

• 2/12/04:
Diagonalization
Relativization, Baker-Gill-Solovay Theorem: There are $A$ and $B$ such that $\\mathrm{P}^A=\\mathrm{NP}^A$ and $\\mathrm{P}^B \\neq \\mathrm{NP}^B$
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 super-linear lower-bound on formula size
Barrington's construction of small-width Branching Programs
• 2/21/07:
Furst-Saxe-Sipser: Parity is not in $\\mathrm{AC}^0$, Switching Lemma and Random Restrictions
• 2/26/07:
Razborov, Method of Approximations: Replacing deterministic gates with probabilistic polynomials
Simplicity of $\\mathrm{AC}^0$ Lemma by Smolensky: Circuits in $\\mathrm{AC}^0$ can be approximated by low-degree polynomials on large fraction of inputs
DeMillo-Lipton-Schwartz-Zippel Lemma of Computer Science
Parity cannot be approximated by low-degree polynomials
• 2/28/07:
Communication complexity (CC)
Karchmer-Wigderson games: Communication Complexity = Circuit depth of function
Tiling Complexity (Yao), Tiling Lemma: Lower-bound 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 ST-connectivity in $O(\\log^{2-\\epsilon} n)$ space for $\\epsilon > 0$?
OPEN: Does $\\mathrm{NSPACE}(s) \\subseteq \\mathrm{SPACE}(s^{2-\\epsilon})$ for $\\epsilon > 0$?
Fortnow's Theorem
• 3/7/07:
Alternation II: The Polynomial Hierarchy, classes $\\Sigma^{\\mathrm{P}}_i$ and $\\Pi^{\\mathrm{P}}_i$
Karp-Lipton Theorem
• 3/12/07:
OPEN: How to generate a prime number in $[N,2N]$ deterministically?
OPEN: Finding square roots modulo prime deterministically (we know in RP,ZPP)
Algebraic Circuit Identity Testing (ACIT)
Strong Turing-Church Hypothesis
ZPP/RP/co-RP/BPP, $\\mathrm{ZPP}=\\mathrm{RP} \\cap \\text{co-RP}$
Amplification for (co-)RP/BPP: Success probability $1/\\mathrm{poly}(n)$ is equivalent to success probability $1-1/2^{\\mathrm{poly}(n)}$
• 3/14/07:
Amplification: Strong BPP = Weak BPP
[Adleman] $\\mathrm{BPP} \\subseteq \\mathrm{P/poly}$. Implies: $\\mathrm{NP} \\subseteq \\mathrm{BPP} \\Rightarrow \\mathrm{NP} \\subseteq \\mathrm{P/poly} \\subseteq$ Polynomial Hierarchy collapses
[Sipser, Lautemann] $\\mathrm{BPP} \\subseteq \\Sigma_2$
One-way permutations
[Valiant-Vazirani] $\\mathrm{SAT} \\leq_{\\mathrm{RP}} \\text{UNIQUE-SAT}$: Part I
• 3/19/07:
$\\mathrm{SAT} \\leq_{\\mathrm{RP}} \\mathrm{UNIQUE-SAT}$
OPEN: Can we derandomize?
OPEN: Can we even just improve the probability of success in the reduction?
$\\mathrm{SAT} \\leq_{\\mathrm{STRONG-BP}} \\oplus \\mathrm{SAT}$
$\\mathrm{NP} \\leq \\oplus \\mathrm{SAT}$
[Toda] Parity/BP/existential/for-all operators
$\\mathrm{NP} \\subseteq \\mathrm{BP} \\cdot \\oplus \\cdot \\mathrm{P}$ and $\\text{co-NP} \\subseteq \\mathrm{BP} \\cdot \\oplus \\cdot \\mathrm{P}$
• 3/21/07:
$\\forall,\\exists,\\oplus,\\mathrm{BP}$-operators
Classes $\\oplus\\,C$ and $\\mathrm{BP}\\,C$ closed under complementation
$\\oplus\\oplus\\,C \\equiv \\oplus\\,C$ and $\\mathrm{BP}\\,\\mathrm{BP}\\,C \\equiv \\mathrm{BP}\\,C$ and $\\mathrm{BP}\\,\\oplus\\,C\\equiv \\oplus\\,\\mathrm{BP}\\,C$
Toda's Theorem: $\\Sigma_k^{\\mathrm{P}},\\Pi_k^{\\mathrm{P}} \\subseteq\\mathrm{BP}\\oplus\\,\\mathrm{P}$ and $\\mathrm{BP}\\,\\oplus\\,\\mathrm{P} \\subseteq \\mathrm{P}^{\\#\\mathrm{P}}$
• 4/2/07:
Interactive proofs
[Babai-Moran] Arthur-Merlin Proofs
[Goldreich-Micali-Wigderson] GRAPH NON-ISOMORPHISM by IP[2]
Private coins = public coins
One-sided error = two-sided error
$\\text{NP}\\subseteq \\text{MA}$ and $\\text{MA}\\subseteq \\text{AM}$ and $\\text{BPP}\\subseteq \\text{MA}=\\text{AMAMAM ... } \\subseteq \\text{IP}[\\text{poly}]= \\text{PSPACE}$. Last equality by [Lund-Fortnow-Karloff-Nissan, Shamir]
• Private coins and 2-sided error = public coins and 1-sided error
PERMANENT is random-self reducible