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Tautologies from Pseudo-Random Generators

Published online by Cambridge University Press:  15 January 2014

Jan Krajíček
Jan Krajíček*
Affiliation:
Mathematical Institute, Academy of Sciences, Žitná 25, 11567 Praha 1, Czech Republic Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic, E-mail:[email protected]
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Abstract

We consider tautologies formed from a pseudo-random number generator, defined in Krajíček [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajíček [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 2 , June 2001 , pp. 197 - 212
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1] Ajtai, M., The complexity of the pigeonhole principle, Proceedings of the IEEE 29th annual symposium on foundation of computer science, 1988, pp. 346–355.Google Scholar
[2] Alekhnovich, M., Ben-Sasson, E., Razborov, A. A., and Wigderson, A., Pseudorandom generators in propositional proof complexity, preprint, 03 2000.Google Scholar
[3] Buss, S. R., Bounded arithmetic, Bibliopolis, Naples, 1986, Revision of 1985 Princeton University Ph.D. thesis.Google Scholar
[4] Cobham, A, The intrinsic computational difficulty of functions, Proceedings of logic, methodology and philosophy of science (Bar-Hillel, Y., editor), North-Holland, 1965, pp. 24–30.Google Scholar
[5] Cook, S. A., The complexity of theorem proving procedures, Proceedings of the 3rd annual ACM symposium on theory of computing, ACM Press, 1971, pp. 151–158.Google Scholar
[6] Cook, S. A., Feasibly constructive proofs and the propositional calculus, Proceedings of the 7th annual ACM symposium on theory of computing, ACM Press, 1975, pp. 83–97.Google Scholar
[7] Cook, S. A. and Reckhow, A. R., The relative efficiency of propositional proof systems, The Journal of Symbolic Logic, vol. 44 (1979), no. 1, pp. 36–50.CrossRefGoogle Scholar
[8] Dowd, M., Propositional representations of arithmetic proofs, Ph.D. thesis , University of Toronto, 1979.CrossRefGoogle Scholar
[9] Haken, A., The intractability of resolution, Theoretical Computer Science, vol. 39 (1985), pp. 297–308.CrossRefGoogle Scholar
[10] Impagliazzo, R. and Wigderson, A., P = BPP unless E has sub-exponential circuits: derandomizing the XOR lemma, Proceedings of the 29th annual ACM symposium on theory of computing, 1997, pp. 220–229.Google Scholar
[11] Krajíček, J., On the weak pigeonhole principle, to appear in Fundamenta Mathematicae (preprint on web 08 9 '99).Google Scholar
[12] Krajíček, J., Speed-up for propositional Frege systems via generalizations of proofs, Commentationes Mathematicae Universitatis Carolinae, vol. 30 (1989), no. 1, pp. 137–140.Google Scholar
[13] Krajíček, J., Lower bounds to the size of constant-depth propositional proofs, The Journal of Symbolic Logic, vol. 59 (1994), no. 1, pp. 73–86.CrossRefGoogle Scholar
[14] Krajíček, J., Bounded arithmetic, propositional logic, and complexity theory, Encyclopedia of mathematics and its applications, vol. 60, Cambridge University Press, 1995.Google Scholar
[15] Krajíček, J., On Frege and extended Frege proof systems, “Feasible mathematics II” (Clote, P. and Remmel, J., editors), Birkhauser, 1995, pp. 284–319.CrossRefGoogle Scholar
[16] Krajíček, J., Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, The Journal of Symbolic Logic, vol. 62 (1997), no. 2, pp. 457–486.CrossRefGoogle Scholar
[17] Krajíček, J. and Pudlák, P., Propositional proof systems, the consistency of first order theories and the complexity of computations, The Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 1063–1079.CrossRefGoogle Scholar
[18] Krajíček, J. and Pudlák, P., Propositional provability in models of weak arithmetic, Computer science logic, October 1989 (Kaiserlautern) (Boerger, E., Kleine-Bunning, H., and Richter, M. M., editors), Lecture Notes in Computer Science, vol. 440, Springer-Verlag, 1990, pp. 193–210.Google Scholar
[19] Krajíček, J. and Pudlák, P., Quantified propositional calculi and fragments of bounded arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 29–46.CrossRefGoogle Scholar
[20] Krajíček, J. and Pudlák, P., Some consequences of cryptographical conjectures for and EF, Information and Computation, vol. 140 (1998), no. 1, pp. 82–94.CrossRefGoogle Scholar
[21] Krajíček, J., Pudlák, P., and Sgall, J., Interactive computations of optimal solutions, Mathematical foundations of computer science (Rovan, B., editor), Lecture Notes in Computer Science, vol. 452, Springer-Verlag, B. Bystrica, 08 1990, pp. 48–60.Google Scholar
[22] Krajíček, J., Pudlák, P., and Takeuti, G., Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 143–153.CrossRefGoogle Scholar
[23] Maciel, A., Pitassi, T., and Woods, A., A new proof of the weak pigeonhole principle, preprint, 1999.CrossRefGoogle Scholar
[24] Paris, J. and Wilkie, A. J., Counting problems in bounded arithmetic, Methods in mathematical logic, Lecture Notes in Mathematics, vol. 1130, Springer-Verlag, 1985, pp. 317–340.CrossRefGoogle Scholar
[25] Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, The Journal of Symbolic Logic, vol. 53 (1988), pp. 1235–1244.CrossRefGoogle Scholar
[26] Pudlák, P., Some relations between subsystems of arithmetic and the complexity of computations, Logic from computer science (Moschovakis, Y. N., editor), Mathematical Sciences Research Institute Publication, vol. 21, Springer-Verlag, 1992, proceedings of a workshop held 11 13-17, 1989 in Berkeley, pp. 499–519.CrossRefGoogle Scholar
[27] Pudlák, P., The lengths of proofs, Handbook of proof theory (Buss, S., editor), Elsevier, 1997.Google Scholar
[28] Razborov, A. A., Bounded arithmetic and lower bounds in Boolean complexity, Feasible mathematics (Clote, P. and Remmel, J., editors), Progress in Computer Science and Applied Logic, vol. 13, Birkhauser, 1995, pp. 344–386.CrossRefGoogle Scholar
[29] Razborov, A. A., Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic, Rossiîskaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya, vol. 59 (1995), no. 1, pp. 201–224.Google Scholar
[30] Razborov, A. A. and Rudich, S., Natural proofs, Journal of Computer and System Sciences, vol. 55 (1997), no. 1, pp. 24–35.CrossRefGoogle Scholar
[31] Tseitin, G. C., On the complexity of derivations in propositional calculus, Studies in mathematics and mathematical logic, part II (Slisenko, A. O., editor), 1968, pp. 115–125.Google Scholar
[32] Urquhart, A., The complexity of propositional proofs, this Bulletin, vol. 1 (1995), no. 4, pp. 425–467.Google Scholar
[33] Yao, A., Theory and applications of trapdoor functions, Proceedings of the 23rd annual symposium on foundation of computer science, 1982, pp. 92–99.Google Scholar