**Math 262A**

**Seminar:
Introduction to Propositional Proof Complexity**

Winter 2014

**Instructor:**

Sam Buss

Email: sbuss@math.ucsd.edu

Office: APM 6210.

Office phone: 858-534-6455.

**Dates:** Winter 2014 quarter, Fridays 12:00-1:30

**Room:** APM 7218

**First meeting:** Friday, January 10, 12:00pm-1:30pm

**Course number:** Math 268 (Topics in Logic)

**Overview:**
This seminar will cover topics in propositional proof complexity,
especially upper and lower bounds on proof size and other measures
of proof complexity. Proof systems to be discussed include:
resolution, Res(k), Frege systems, extended Frege systems,
mod-k and TC0-Frege, cutting planes, the Nullstellensatz system,
and the Polynomial Calculus. Other topics include:
switching lemmas, Craig interpolation, natural proofs,
cryptographic conjectures, and applications of
monotone circuit complexity. If time permits, we may
also cover related topics from SAT solvers.
Lectures will be mostly presented by the instructor,
but it is hoped there will also be some guest lectures.

Propositional proof complexity addresses open problems in computational complexity --- ultimately, the NP vs coNP question --- from the point of view of mathematical logic: This combines questions from computational complexity (about expressibility of propositional properties) with questions about the provability of propositional properties) and brings a range of tools from mathematical logic to bear on the P versus NP problem.

This is an introductory seminar. Although some background in logic or computational complexity would be helpful, there are no particular prerequisites beyond a certain level of mathematical maturity.

- January 10, 2014. Frege proof system and abstract proof systems. Scribe notes by Marco Carmosino.
- January 17, 2014. Frege and extended Frege proofs. Scribe notes by Marco Carmosino.
- January 31, 2014. Tree-like Frege. Substitution Frege (part 1). Scribe notes by Marco Carmosino.
- February 7, 2014. Substitution Frege and extended Frege (part 2). Scribe notes by Marco Carmosino.
- February 14, 2014. Introduction to resolution. Scribe notes by Marco Carmosino.

- S. Buss, Towards NP-P via Proof Complexity and Search", Annals of Pure and Applied Logic 163, 7 (2012) 906-917.
- S. Buss,
Propositional
Proof Complexity: An Introduction",
In
*Computational Logic*, edited by U. Berger and H. Schwichtenberg, Springer-Verlag, Berlin, 1999, pp. 127-178. - P. Beame and T. Pitassi,
Propositional
Proof Complexity: Past, Present and Future",
In
*Current Trends in Theoretical Computer Science Entering the 21st Century*, World Scientific, 2001, pp. 42-70. - P. Beame with A. Sabharwal,
Propositional
Proof Complexity",
In
*Computational Complexity Theory*, IAS/Park City Clay Mathematics Series 10, 2000, pp. 199-246. - N.Segerlind, The Complexity of Propositional Proofs", Bulletin of Symbolic Logic 13 (2007) 417-481.

These are rough handwritten notes intended for Sam's archival storage. Not recommended for anyone else. Click here.