**Math 262A**

**Combinatorics: Circuit Complexity**

Fall 2013

**Instructor:**

Sam Buss

Email: sbuss@math.ucsd.edu

Office: APM 6210.

Office phone: 858-534-6455.

Voice mail: 858-792-9674.

Cell phone: 858 then 442 then 2877.

Skype ID:
SamBussSkype.

**Extra class meeting:** Monday, November 25, 3:30-5:00pm in
**APM 5829**

**Office hours:**
Monday 1:00-1:50, Wednesday 2:00-2:50; or by appointment, or just drop by.

**Overview:**
This topics course will be an introduction to circuit complexity, especially
lower bounds on circuit complexity. Circuit complexity studies the
minimum size of circuits required to compute Boolean functions,
and has close connections with many open problems in computational
course will briefly discuss connections to uniform (Turing machine)
computation, but will focus primarily on the combinatorial aspects of
circuit complexity both lower bounds and upper bounds. Initial topics to
be covered include: Shannon-Riordan-Lupanov upper and lower bounds,
formula shrinkage (Subbotovskaya, Andreev, Hastad, etc.),
switching lemmas (Hastad, etc.). The later part of the course will
cover some selected upper bounds, algebraic methods, and
probabilistic circuits.

There are no particular prerequisites beyond a certain level of
mathematical maturity. It is expected that students will have
graduate standing in mathematics or computer science or a related field.

**Course schedule:** Wednesday and Friday, 3:30-4:50, APM B412.

**Scribe notes:**

(Latex sources available by using *.tex
instead of *.pdf in URL filename.)

- September 28, 2013. Scribe notes by Christian Woods. Introduction and Shannon's Theorem.
- October 2, 2013. Scribe notes by Marco Carmosino. Riordan-Shannon formula lower bound; Lupanov formula and circuit upper bounds.
- October 4, 2013. Scribe notes by Jiapeng Zhang. Formula size, Krapchencko's Theorem, Spira-Brent rebalancing.
- October 9, 2013. Scribe notes by Stefan Schneider. Nechiporuk's Theorem.
- October 11, 2013. Scribe notes by Udbhav Singh. Shrinkage Exponents for deMorgan Formulas.
- October 16, 2013. Scribe notes by Benjy Greenberg. Schnorr's circuit lower bound. Monotone functions, slice functions.
- October 18 & 23, 2013. Scribe notes by James Aisenberg. Lower bounds for monotone circuits for Clique.
- October 25, 2013. Scribe notes by Radheshyan Balasundaram. Collapsing circuit depth. October 30, 2013. Scribe notes by Bob Chen. The Switching Lemma.
- November 1, 2013. Scribe notes by Christian Woods. Lower bounds for constant depth circuits for Parity.
- November 6, 2013. Scribe notes by Marco Carmosino. Razborov-Smolensky Theorem.
- November 13, 2013. Scribe notes by Udbhav Singh. Counting, Threshold and Vector Addition.
- November 15, 2013. Scribe notes by Jiapeng Zhang.
- November 20, 2013. Scribe notes by Stefan Schneider. Batcher sorting networks. AKS sorting networks introduction.
- November 22 & 25, 2013. Scribe notes by Radheshyan Balasundaram. AKS sorting networks.
- November 27, 2013. Scribe notes by Bob Chen. Counting in constant depth (BPP in PH).
- December 4, 2013. Scribe notes by TBD. COMING SOON. Log depth division circuits, part I.
- December 6, 2013. Scribe notes by TBD. COMING SOON. Log depth division circuits, part II.
- Raw handwritten notes by the instructor.

**Supplemental reading and other resources:**

**Articles:**- O.B. Lupanov, On the realization of functions for logical algebra by formulae of finite classes (formulae of limited depth) in the basis &,v,-,*. Problems in Cybernetics VI, 1965, pp1-14. (English translation by G. Kiss.)

**Books:**, The Complexity of Computing, 1976.**J. Savage**, The Complexity of Boolean Functions, 1987.**I. Wegener**, The Complexity of Boolean Networks, 1988.**P. Dunne**, Boolean Function Complexity, 2012.**S. Jukna**, Boolean Functions and Computation Models, 2002.**P. Clote and E. Kranakis**