__Math 267A - Topics in Logic - Proof Complexity
__Instructor: Sam Buss

Winter 2002, Univ. of California, San Diego

** Online Scribe Notes.**
Whole course available except one lecture.

**Professor: Sam Buss. **
Email: sbuss@ucsd.edu. Phone: 534-6455.
Office: APM 6210.

**Schedule:
Lectures: **MW 3:25-4:40 APM 7314. (Please do not
disturb the previous class until they finish.)

**Office hours:** (Math 160A has priority, email for separate appointments
if desired.)**
** Hours: Monday 12:00-12:50, Wednesday
1:00-1:50, Friday 9:00-9:50.

__ Reading materials:__ Follow the links to download (some of) the
papers.

- Sam Buss,
, in*An introduction to proof theory*edited by S. Buss, Elsevier North-Holland, 1998, pp 1-78.**Handbook of Proof Theory**,

See pages 3-9 for a concise introduction to propositional logic, including the completeness theorem.

- Sam Buss,
**Propositional Proof Complexity: An Introduction***Computational Logic,*edited by U. Berger and H. Schwichtenberg, Springer-Verlag, Berlin, 1999, pp. 127-178.

A general introduction to the complexity of propositional proofs. A good supplement to the course material.

- Sam Buss,
**Bounded Arithmetic and Propositional Proof Complexity**,*Logic of Computation,*edited by H. Schwichtenberg, Springer-Verlag, Berlin, 1997, pp. 67-122.

Another introduction to propositional proof complexity, along with bounded arithmetic and its connections to propositional proof complexity.

- Sam Buss,
Technical Report number SOCS-96.1, School of Computer Science, McGill University, 1996, 117 pages.*Lectures on Proof Theory,*

An older introduction to propositional proof complexity and other topics.

- Stephen A. Cook and Robert A. Reckhow,
Journal of Symbolic Logic 44 (1979) 36-50.**The relative efficiency of propositional proof systems**,

Available from UCSD addresses from JSTOR at http://www.jstor.com.

Direct URL: http://links.jstor.org/sici?sici=0022-4812%28197903%2944%3A1%3C36%3ATREOPP%3E2.0.CO%3B2-G

A founding paper for propositional proof complexity. One of the original sources for some of the material on Frege systems covered in lecture.

- M. S. Paterson and M. N. Wegman,
Journal of Computer and System Sciences 16 (1978) 158-167.**Linear Unification**,

A linear time algorithm for finding a most general unifier. (This is better than we need: polynomial time is OK for our applications.)

- Stefan Dantchev,
To appear in IEEE Complexity (CCC'2002), May 2002. Available in Postscript and PDF.**Resolution width size tradeoffs for the pigeonhole principle**,

- Sam Buss and Toni Pitassi,
Proceedings, Computer Science Logic (CSL'97), Lecture Notes in Computer Science #1414, Springer-Verlag 1998, pp. 149-156.**Resolution and the weak pigeonhole principle**,

- Pavel Pudlak,
, Journal of Symbolic Logic 62(3), 1997, pp.981-998. Contains the Craig interpolation for cutting planes, and the results on monotone real circuits for clique and coloring.**Lower bounds for resolution and cutting planes proofs and monotone computations**

- Sam Buss and Peter Clote,
, Archive for Mathematical Logic 35 (1996) 33-62. Another source for information on cutting planes (the course will not cover this material).**Cutting planes, connectivity and threshold logic**

- Alexander A. Razborov,
Computational Complexity 7 (1998) 291-324. The original reference for the degree lower bound on polynomial calculus refutations of the pigeonhole principle.*Lower bounds for the polynomial calculus,*

- R. Impagliazzo, P. Pudlak, J. Sgall,
ECCC 1997 and Computational Complexity, 8(2) (1999), pp. 127-144 This paper has the version of the proof which will be presented in class of the lower bounds for polynomial calculus refutations of the pigeonhole principle.**Lower bounds for the polynomial calculus and the Groebner basis algorithm**,