Complexity Theory Lecture Notes
There are two graduatelevel courses in complexity theory that I have
taught here at Rutgers. Notes that were prepared for some of the
material covered in those courses are available for your reading
pleasure.

Levin's Lower Bound Theorem (These notes present a
lovely theorem that should be in all textbooks but isn't.
Everyone knows Blum's "speedup theorem" that shows that
there are certain problems that have nothing at all like an
optimal algorithm. At first glance, this might indicate that
some problems have no tight lower bound on
their complexity. However this result of Levin's shows
that every computable function does
have a tight lower bound.)
198:540  Combinatorial Methods in Complexity Theory
 Notes1
(Introduction. Proof of the Parity lower bound for constantdepth
circuits, assuming the switching lemma.)
 Notes2
(Start of the proof of the switching lemma, using the
argument based on Kolmogorov complexity.)
 Notes3
(End of the proof of the switching lemma.)
 Notes4
(Bounds on the number of inputs on which an AC^0 circuit
can compute parity correctly. Depthreduction for (probabilistic) AC^0
circuits with mod gates.)
 Notes5
(Constructing deterministic circuits with adequate performance
from probabilistic circuits.)
 Notes6
(AC^0 with mod p gates can't compute mod q.)
 Notes7
(Normal forms for ACC circuits.)
 Notes8
(ACC can be done by depth 2 probabilistic circuits with a
symmetric gate at the root.)
 Notes9
(ValiantVazirani construction to reduce the number of
probabilistic bits, allowing the ACC result to go through
with deterministic circuits.)
 Notes10
(The "fusion method" for proving circuit lower bounds.)
 Notes11
(Application of the "fusion method" to prove a lower
bound on monotone circuit size required to compute 3clique.)
 Notes12
(The general lower bound for monotone circuit size required
to compute kclique.)
 Notes13
(Resolutionbased theorem proving, Craig interpolation, related
results.)
 Notes14
(Relationships between resolution refutation length and (monotone)
circuit size.)
 Notes15
(An introduction to probabilisticallycheckable proofs. There are
no further class notes on PCP; refer instead to the text by
Arora available through
ECCC.)
Other Excellent Sets of Notes
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