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Lower bounds for resolution and cutting plane proofs and monotone computations

Published online by Cambridge University Press:  12 March 2014

Pavel Pudlák
Pavel Pudlák*
Affiliation:
Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha 1, Czech Republic, E-mail: pudlak@beba.cesnet.cz
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Abstract

We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 981 - 998
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Ajtai, M., The complexity of the pigeonhole principle, Proceedings of the 29-th FOCS, 1988, pp. 346–355.Google Scholar
[2]Alon, N. and Boppana, R.B., The monotone circuit complexity of Boolean functions, Combinatorica, vol. 7 (1987), no. 1, pp. 1–22.CrossRefGoogle Scholar
[3]Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T., Pudlák, P., and Woods, A., Exponential lower bounds for the pigeonhole principle, Proc. 24-th STOC, 1992, pp. 200–221.Google Scholar
[4]Bonet, M., Pitassi, T., and Raz, R., Lower bounds for cutting planes proofs with small coefficients, Proc. 27-th STOC, 1995, pp. 575–584.Google Scholar
[5]Buss, S. R. and Clote, P., Cutting planes, connectivity, and threshold logic, Archive for Mathematical Logic, to appear.Google Scholar
[6]Chvátal, V., Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics, vol. 4 (1973), pp. 305–337.CrossRefGoogle Scholar
[7]Chvátal, V., Some linear programming aspects of combinatorics, Proceedings of the Conference on Algebraic Aspects of Combinatorics, Toronto 1975, Utilitas Mathematical Publishing Inc., Winnipeg, 1975, pp. 2–30.Google Scholar
[8]Chvátal, V., Cook, W., and Martmann, M., On cutting plane proofs in combinatorial optimization, Linear Algebra and Its Applications, vol. 114/115 (1989), pp. 455–499.CrossRefGoogle Scholar
[9]Cook, W., Coullard, C. R., and Turán, Gy., On the complexity of cutting plane proofs, Discrete Applied Mathematics, vol. 18 (1987), pp. 25–38.CrossRefGoogle Scholar
[10]Craig, W., Linear reasoning: A new form of the Herbrand-Gentzen theorem, this Journal, vol. 22 (1957), no. 3, pp. 250–287.Google Scholar
[11]Craig, W., Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, this Journal, vol. 22 (1957), no. 3, pp. 269–285.Google Scholar
[12]Gomory, R. E., An algorithm for integer solutions of linear programs, Recent advances in mathematical programming (Graves, R. L. and Wolfe, P., editors), McGraw-Hill, 1963, pp. 269–302.Google Scholar
[13]Haken, A., The intractability of resolution, Theoretical Computer Science, vol. 39 (1985), pp. 297–308.CrossRefGoogle Scholar
[14]Haken, A., Counting bottlenecks to show monotone P ≠ NP, Proc. 36-th Annual IEEE Symp. on Foundations of Computer Science (Milwaukee), 1995, pp. 36–40.CrossRefGoogle Scholar
[15]Impagliazzo, R., Pitassi, T., and Urquhart, A., Upper and lower bounds for tree-like cutting planes proofs, Proc. 9-th Annual IEEE Symp. on Logic in Computer Science, 1994, pp. 220–228.Google Scholar
[16]Krajíček, J., Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, this Journal, to appear.Google Scholar
[17]Krajíček, J., Lower bounds to the size of constant-depth propositional proofs, this Journal, vol. 59 (1994), no. 1, pp. 73–86.Google Scholar
[18]Krajíček, J., Bounded arithmetic, propositional logic and complexity theory, Cambridge University Press, 1995.CrossRefGoogle Scholar
[19]Krajíček, J. and pudlák, P., Some consequences of cryptographical conjectures for S21 and EF, Logic and computational complexity, Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995, pp. 210–220.CrossRefGoogle Scholar
[20]Papadimitriou, Ch. H. and Steiglitz, K., Combinatorial optimization: Algorithms and complexity, Prentice-Hall, 1982.Google Scholar
[21]Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 1235–1244.Google Scholar
[22]Razborov, A. A., Lower bounds on the monotone complexity of some Boolean functions, Doklady Akad. Nauk SSSR, vol. 282 (1985), pp. 1033–1037.Google Scholar
[23]Razborov, A. A., Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic, Izvestiya of the R.A.N., vol. 59 (1995), no. 1, pp. 201–222, Izvestiya: Mathematics, vol. 59, no. 1, pp. 205–227.Google Scholar
[24]Schnorr, C. P., A lower bound on the number of additions in monotone computations, Theoretical Computer Science, vol. 2 (1976), pp. 305–315.CrossRefGoogle Scholar
[25]Schrijver, A., On cutting planes, Combinatorics, vol. 79 (1980).Google Scholar
[26]Schrijver, A., On cutting planes, part II, Annals of Discrete Math., vol. 9, North-Holland, 1980, pp. 291–296.Google Scholar
[27]Wegener, I., The complexity of Boolean functions, Teubner and Wiley, 1987.Google Scholar