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: email@example.com. Phone: 534-6455.
Office: APM 6210.
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
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
- Sam Buss, An introduction to proof
theory, in Handbook
of Proof Theory, edited by S. Buss, Elsevier North-Holland, 1998, pp
See pages 3-9 for a concise introduction to propositional logic, including the
- Sam Buss, Propositional Proof
Complexity: An Introduction, in 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, in 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, Lectures on Proof
Theory, Technical Report number SOCS-96.1, School of Computer Science,
McGill University, 1996, 117 pages.
An older introduction to propositional proof complexity and other topics.
- Stephen A. Cook and Robert A. Reckhow, The relative efficiency of
propositional proof systems, Journal of Symbolic Logic 44 (1979) 36-50.
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, Linear Unification, Journal
of Computer and System Sciences 16 (1978) 158-167.
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, Resolution width size tradeoffs for the pigeonhole
principle, To appear in IEEE Complexity (CCC'2002), May 2002. Available in Postscript and PDF.
- Sam Buss and Toni Pitassi, Resolution
and the weak pigeonhole principle, Proceedings, Computer Science Logic
(CSL'97), Lecture Notes in Computer Science #1414, Springer-Verlag 1998, pp. 149-156.
- Pavel Pudlak, Lower
bounds for resolution and cutting planes proofs and monotone computations,
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.
- Sam Buss and Peter Clote, Cutting
planes, connectivity and threshold logic, Archive for Mathematical Logic
35 (1996) 33-62. Another source for information on cutting planes (the course
will not cover this material).
- Alexander A. Razborov, Lower
bounds for the polynomial calculus, Computational Complexity 7 (1998)
291-324. The original reference for the degree lower bound on polynomial calculus
refutations of the pigeonhole principle.
- R. Impagliazzo, P. Pudlak, J. Sgall, Lower
bounds for the polynomial calculus and the Groebner basis algorithm,
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.