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Approximate counting by hashing in bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Emil Jeřábek
Emil Jeřábek*
Affiliation:
Institute of Mathematics of the Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, E-mail: jerabek@math.cas.cz, URL: http://math.cas.cz/~jerabek
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Abstract

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

Keywords

Bounded arithmeticapproximate countinguniversal hashing03F30
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 829 - 860
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Beyersdorff, Olaf and Müller, Sebastian, A tight Karp-Lipton collapse result in bounded arithmetic, ACM Transactions on Computational Logic, to appear.Google Scholar
[2]Samuel R., Buss, Bounded arithmetic, Bibliopolis, Naples, 1986, revision of 1985 Princeton University Ph.D. thesis.Google Scholar
[3]Buss, Samuel R., Relating the bounded arithmetic and polynomial time hierarchies, Annals of Pure and Applied Logic, vol. 75 (1995), no. 1-2, pp. 67–77.CrossRefGoogle Scholar
[4]Buss, Samuel R., First-order proof theory of arithmetic, (Buss, Samuel R., editor), Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, Amsterdam, 1998, pp. 79–147.Google Scholar
[5]Cai, Jin-Yi, , Journal of Computer and System Sciences, vol. 73 (2007), no. 1, pp. 25–35.CrossRefGoogle Scholar
[6]Canetti, Ran, More on BPP and the polynomial-time hierarchy, Information Processing Letters, vol. 57 (1996), no. 5, pp. 237–241.CrossRefGoogle Scholar
[7]Carter, J. Lawrence and Wegman, Mark N., Universal classes of hash functions, Journal of Computer and System Sciences, vol. 18 (1979), no. 2, pp. 143–154.CrossRefGoogle Scholar
[8]Clote, Peter and KrajičEk, Jan (editors), Arithmetic, proof theory, and computational complexity, Oxford Logic Guides, vol. 23, Oxford University Press, 1993.CrossRefGoogle Scholar
[9]Clote, Peter, Open problems, In Arithmetic, proof theory, and computational complexity [8], pp. 1–19.Google Scholar
[10]Cobham, Alan, The intrinsic computational difficulty of functions, Proceedings of the 2nd International Congress of Logic, Methodology and Philosophy of Science (Bar-Hillel, Yehoshua, editor), North-Holland, 1965, pp. 24–30.Google Scholar
[11]Cook, Stephen A., Feasibly constructive proofs and the prepositional calculus, Proceedings of the 7th Annual ACM Symposium on Theory of Computing, ACM Press, 1975, pp. 83–97.Google Scholar
[12]Cook, Stephen A. and Krajíˇek, Jan, Consequences of the provability of NP ⊆ P/poly, this Journal, vol. 72 (2007), no. 4, pp. 1353–1371.Google Scholar
[13]Erdős, Paul, On a problem in graph theory, Mathematical Gazette, vol. 47 (1963), no. 361, pp. 220–223.CrossRefGoogle Scholar
[14]Goldreich, Oded and Wigderson, Avi, Improved derandomization of BPP using a hitting set generator, Proceedings of RANDOM-APPROX '99 (Hochbaum, Dorit S., Jansen, Klaus, Rolim, José D. Р., and Sinclair, Alistair, editors), Lecture Notes in Computer Science, vol. 1671, Springer, 1999, pp. 131–137.Google Scholar
[15]Goldwasser, Shafi and Sipser, Michael, Private coins versus public coins in interactive proof systems, Randomness and computation (Micali, Silvio, editor), Advances in Computing Research, vol. 5, JAI Press, Greenwich, 1989, pp. 73–90.Google Scholar
[16]Graham, Ronald L. and Spencer, Joel H., A constructive solution to a tournament problem, Canadian Mathematical Bulletin, vol. 14 (1971), no. 1, pp. 45–48.CrossRefGoogle Scholar
[17]Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer, 1993, second edition 1998.CrossRefGoogle Scholar
[18]Jeřábek, Emil, Dual weak pigeonhole principle, Boolean complexity, and derandomization, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 1–37.CrossRefGoogle Scholar
[19]Jeřábek, Emil, The strength of sharply bounded induction, Mathematical Logic Quarterly, vol. 52 (2006), no. 6, pp. 613–624.CrossRefGoogle Scholar
[20]Jeřábek, Emil, On independence of variants of the weak pigeonhole principle, Journal of Logic and Computation, vol. 17 (2007), no. 3, pp. 587–604.CrossRefGoogle Scholar
[21]Jeřábek, Emil, Approximate counting in bounded arithmetic, this Journal, vol. 72 (2007), no. 3, pp. 959–993.Google Scholar
[22]Krajíček, Jan, NO counter-example interpretation and interactive computation, Logic from Computer Science, Proceedings of a workshop held November 13–17, 1989 in Berkeley (Moschovakis, Y N., editor), Mathematical Sciences Research Institute Publications, vol. 21, Springer, 1992, pp. 287–293.Google Scholar
[23]Krajíček, Jan, Bounded arithmetic, prepositional logic, and complexity theory, Encyclopedia of Mathematics and Its Applications, vol. 60, Cambridge University Press, 1995.CrossRefGoogle Scholar
[24]Krajíček, Jan, Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets, Proceedings of the London Mathematical Society, vol. 81 (2000), no. 3, pp. 257–284.CrossRefGoogle Scholar
[25]Krajíček, Jan, Approximate Euler characteristic, dimension, and weak pigeonhole principles, this Journal, vol. 69 (2004), no. 1, pp. 201–214.Google Scholar
[26]Krajíček, Jan, Pudlák, Pavel, and Takeuti, Gaisi, Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, vol. 52 (1991), no. 1-2, pp. 143–153.CrossRefGoogle Scholar
[27]Maciel, Alexis, Pitassi, Toniann, and Woods, Alan R., A new proof of the weak pigeonhole principle, Journal of Computer and System Sciences, vol. 64 (2002), no. 4, pp. 843–872.CrossRefGoogle Scholar
[28]Ojakian, Kerry E., Combinatorics in bounded arithmetic, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 2004.Google Scholar
[29]Paris, Jeff B., Wilkie, Alex J., and Woods, Alan R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), no. 4, pp. 1235–1244.Google Scholar
[30]Pudlák, Pavel, Ramsey's theorem in bounded arithmetic, Proceedings of Computer Science Logic '90 (Börger, Egon, Büning, Hans Kleine, Richter, Michael M., and Schönfeld, Wolfgang, editors), Lecture Notes in Computer Science, vol. 533, Springer, 1991, pp. 308–317.CrossRefGoogle Scholar
[31]РАЗБОРОВ, АЛЕКСАНДР А., , (Адян, Серγей И., editor), ΒοПрοϲь κИбеРΗеΤИκИ, vol. 134, VINITI, Moscow, 1988, pp. 149–166 (Russian).Google Scholar
[32]Riis, Søren M., Making infinite structures finite in models of second order bounded arithmetic, In Clote, and Krajíček, [8], pp. 289–319.Google Scholar
[33]Russell, Alexander and Sundaram, Ravi, Symmetric alternation captures BPP, Computational Complexity, vol. 7 (1998), no. 2, pp. 152–162.CrossRefGoogle Scholar
[34]Sipser, Michael, A complexity theoretic approach to randomness, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, ACM Press, 1983, pp. 330–335.Google Scholar
[35]Szekeres, Esther and SZEKERES, GEORGE, On a problem of Schütte and Erdös, Mathematical Gazette, vol. 49 (1965), no. 369, pp. 290–293.CrossRefGoogle Scholar
[36]Toda, Seinosuke, On the computational power of PP and ⊕P, Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, 1989, pp. 514–519.CrossRefGoogle Scholar
[37]Zambella, Domenico, Notes on polynomially bounded arithmetic, this Journal, vol. 61 (1996), no. 3, pp. 942–966.Google Scholar