# Complexity Theory Lecture Notes

There are two graduate-level 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.

## 198:538 -- Complexity of Computation

• Levin's Lower Bound Theorem (These notes present a lovely theorem that should be in all textbooks but isn't. Everyone knows Blum's "speed-up 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 constant-depth 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. Depth-reduction 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 (Valiant-Vazirani 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 3-clique.)
• Notes12 (The general lower bound for monotone circuit size required to compute k-clique.)
• Notes13 (Resolution-based theorem proving, Craig interpolation, related results.)
• Notes14 (Relationships between resolution refutation length and (monotone) circuit size.)
• Notes15 (An introduction to probabilistically-checkable proofs. There are no further class notes on PCP; refer instead to the text by Arora available through ECCC.)