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, An introduction to proof theory, in Handbook of Proof Theory, edited by S. Buss, Elsevier North-Holland, 1998, pp 1-78.
  See pages 3-9 for a concise introduction to propositional logic, including the completeness theorem.
  
• 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.