Abstract
This article is an abridged and revised version of a 1996 McGill University technical report [15]. The technical report was based on lectures delivered by the author at a workshop in Holetown, Barbados and on the authors prepared overhead transparencies. The audience at this workshop wrote scribe notes which then formed the technical report [15]. The material selected for the present article corresponds roughly to the content of the author’s lectures at the NATO summer school held in Marktoberdorf, Germany in July-August 1997.
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References
M. AjtaiThe complexity of the pigeonhole principlein Proceedings of the 29-th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 346–355.
B. Alon and R. BoppanaThe monotone circuit complexity of Boolean functionsCombinatorica, 7 (1987), pp. 1–22.
M. L. BonetThe Lengths of Propositional Proofs and the Deduction RulePhD thesis, U.C. Berkeley, 1991.
M. L. Bonet and S. R. BussSize-depth tradeoffs for Boolean formulasInformation Processing Letters, 29 (1994), pp. 151–155.
M. L. Bonet, T. Pitassi, and R. RazLower bounds for cutting planes proofs with small coefficientsJournal of Symbolic Logic, 62 (1997), pp. 708–728. An earlier version appeared in Proc. Twenty-Seventh Annual ACM Symposium on the Theory of Computing 1995, pp. 575–584.
R. Boppana and M. SipserThe Complexity of Finite FunctionsMIT Press/Elsevier, 1990, pp. 757–804.
A. Borodin, S. Cook, and N. PippengerParallel computation for well-endowed rings and space-bounded probabilistic machinesInformation and Control, 58 (1992), pp. 113–136.
R. P. BrentThe parallel evaluation of general arithmetic expressions, J. Assoc. Comput. Mach., 21(1974), pp. 201–206.
N. H. Bshouty, R. Cleve, and W. EberlySize-depth tradeoffs for algebraic formulaein Proceedings of the 32th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1991, pp. 334–341.
S. R. BussThe Boolean formula value problem is in ALOGTIMEin Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123–131.
S. R. BussPolynomial size proofs of the propositional pigeonhole principleJournal of Symbolic Logic, 52 (1987), pp. 916–927.
S. R. BussAn introduction to proof theory.Typeset manuscript, to appear in S.R. Buss (ed.) Handbook of Proof Theory, North-Holland, 199?
S. R. BussAlgorithms for Boolean formula evaluation and for tree con- traction, in Arithmetic, Proof Theory and Computational Complexity, P. Clote and J. Krajícek, eds., Oxford University Press, 1993, pp. 96–115.
S. R. BussSome remarks on lengths of propositional proofsArchive for Mathematical Logic, 34 (1995), pp. 377–394.
S. R. BussLectures in proof theoryTech. Rep. SOCS 96.1, School of Computer Science, McGill University, 1996. Notes from a series of lectures given at the at the McGill University Bellair’s Research Institute, Holetown, Barbados, March 1995. Scribe note written by E. Allender, M.L. Bonet, P. Clote, A. Maciel, P. McKenzie, T. Pitassi, R. Raz, K. Regan, J. Torán and C. Zamora.
S. R.. Buss and P. CloteCutting planes, connectivity and threshold logicArchive for Mathematical Logic, 35 (1996), pp. 33–62.
S. R. Buss, S. A. Cook, A. Gupta, and V. RamachandranAn optimal parallel algorithm for formula evaluationSIAM J. Comput., 21 (1992), pp. 755–780.
V. Chvátal and E. SzemerédiMany hard examples for resolutionJournal of the ACM, 35 (1988), pp. 759–768.
S. A. CookThe complexity of theorem proving techniquesin Proceedings of the 3-rd Annual ACM Symposium on Theory of Computing, 1971, pp. 151–158.
S. A. Cook and R. A. ReckhowThe relative efficiency of propositional proof systemsJournal of Symbolic Logic, 44 (1979), pp. 36–50.
W. Cook, C. R. Coullard, and G. TuránOn the complexity of cutting plane proofsDiscrete Applied Mathematics, 18 (1987), pp. 25–38.
W. CraigLinear reasoning. A new form of the Herbrand-Gentzen theoremJournal of Symbolic Logic, 22 (1957), pp. 250–268.
M. DowdModel-theoretic aspects of P≠ NP.Typewritten manuscript, 1985.
A. GoerdtCutting plane versus frege proof systemsin Computer Science Logic 1990, E. Börger, ed., Lecture Notes in Computer Science #552, 1992, pp. 174–194.
A. HakenThe intractability of resolutionTheoretical Computer Science, 39 (1985), pp. 297–308.
J. Hartmanis and R.. StearnsOn the computational complexity of algorithmsTransactions of the American Mathematical Society, 117 (1965), pp. 285–306.
S. Kosaraju and A. DelcherA tree-partitioning technique with applications to expression evaluation and term matchingin Proceedings of the Thirteenth Annual Symposium on Foundations of Computing, IEEE Computer Society, 1990, pp. 163–172.
J. KrajícekLower bounds to the size of constant-depth propositional proofsJournal of Symbolic Logic, 59 (1994), pp. 73–85.
J. KrajícekInterpolation theorems lower bounds for proof systems and independence results for bounded arithmeticJournal of Symbolic Logic, 62 (1997), pp. 457–486.
J. Krajíček and P. PudlákThe number of proof lines and the size of proofs in first-order logicArchive for Mathematical Logic, 27 (1988), pp. 69–84.
J. Krajícek and P. PudlakPropositional proof systems the consistency of first-order theories and the complexity of computationsJournal of Symbolic Logic, 54 (1989), pp. 1063–1079.
J. Krajícek, P. Pudlak, and A. WoodsExponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principleRandom Structures and Algorithms, 7 (1995), pp. 15–39.
R. E. LadnerThe circuitvalue problem is log space complete for P, SIGACT News, 7 (1975), pp. 18–20.
D. MundiciTautologies with a unique Craig interpolantAnnals of Pure and Applied Logic, 27 (1984), pp. 265–273.
Y. OfmanOn the algorithmic complexity of discete functionsDoklady Akad. Nauk SSSR, 145 (1962), pp. 48–51. English translation in Soviet Physics - Doklady 7 (1963)589–591.
T. Pitassi, P. Beame, and R. ImpagliazzoExponential lower bounds for the pigeonhole principleComputational Complexity, 3 (1993), pp. 97–140.
P. PudlakLower bounds for resolution and cutting planes proofs and monotone computationsJournal of Symbolic Logic, 62 (1997), pp. 981–998.
A. A. RazborovLower bounds on the monotone complexity of some Boolean functionsSoviet Mathematics Doklady, 31 (1985), pp. 354–357.
R. A. ReckhowOn the Lengths of Proofs in the Propositional CalculusPhD thesis, Department of Computer Science, University of Toronto, 1976. Technical Report #87.
J. Siekmann and G. WrightsonAutomation of Reasoningvol. 1&2, Springer-Verlag, Berlin, 1983.
P. M. SpiraOn time hardware complexity tradeoffs for Boolean functionsin Proceedings of the Fourth Hawaii International Symposium on System Sciences, 1971, pp. 525–527.
R. StatmanComplexity of derivations from quantifier-free Horn formulae,mechanical introduction of explicit definitions and refinent of completeness theorems, in Logic Colloquium ‘76, R. Gandy and M.Hyland, eds., Amsterdam, 1977, North-Holland, pp. 505–517.
G. S. TsejtinOn the complexity of derivation in propositional logicStudies in Constructive Mathematics and Mathematical Logic, 2 (1968), pp. 115–125. Reprinted in: [40, vol 2].
C. S. WallaceA suggestions for a fast multiplierIEEE Transactions on Computers, EC-13 (1964), pp. 14–17.
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Buss, S.R. (1999). Propositional Proof Complexity an Introduction. In: Berger, U., Schwichtenberg, H. (eds) Computational Logic. NATO ASI Series, vol 165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58622-4_5
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