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Approximate counting by hashing in bounded arithmetic

  • Emil Jeřábek (a1)

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.

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[1]Beyersdorff, Olaf and Müller, Sebastian, A tight Karp-Lipton collapse result in bounded arithmetic, ACM Transactions on Computational Logic, to appear.
[2]Samuel R., Buss, Bounded arithmetic, Bibliopolis, Naples, 1986, revision of 1985 Princeton University Ph.D. thesis.
[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. 6777.
[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. 79147.
[5]Cai, Jin-Yi, , Journal of Computer and System Sciences, vol. 73 (2007), no. 1, pp. 2535.
[6]Canetti, Ran, More on BPP and the polynomial-time hierarchy, Information Processing Letters, vol. 57 (1996), no. 5, pp. 237241.
[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. 143154.
[8]Clote, Peter and KrajičEk, Jan (editors), Arithmetic, proof theory, and computational complexity, Oxford Logic Guides, vol. 23, Oxford University Press, 1993.
[9]Clote, Peter, Open problems, In Arithmetic, proof theory, and computational complexity [8], pp. 119.
[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. 2430.
[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. 8397.
[12]Cook, Stephen A. and Krajíˇek, Jan, Consequences of the provability of NP ⊆ P/poly, this Journal, vol. 72 (2007), no. 4, pp. 13531371.
[13]Erdős, Paul, On a problem in graph theory, Mathematical Gazette, vol. 47 (1963), no. 361, pp. 220223.
[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. 131137.
[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. 7390.
[16]Graham, Ronald L. and Spencer, Joel H., A constructive solution to a tournament problem, Canadian Mathematical Bulletin, vol. 14 (1971), no. 1, pp. 4548.
[17]Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer, 1993, second edition 1998.
[18]Jeřábek, Emil, Dual weak pigeonhole principle, Boolean complexity, and derandomization, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 137.
[19]Jeřábek, Emil, The strength of sharply bounded induction, Mathematical Logic Quarterly, vol. 52 (2006), no. 6, pp. 613624.
[20]Jeřábek, Emil, On independence of variants of the weak pigeonhole principle, Journal of Logic and Computation, vol. 17 (2007), no. 3, pp. 587604.
[21]Jeřábek, Emil, Approximate counting in bounded arithmetic, this Journal, vol. 72 (2007), no. 3, pp. 959993.
[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. 287293.
[23]Krajíček, Jan, Bounded arithmetic, prepositional logic, and complexity theory, Encyclopedia of Mathematics and Its Applications, vol. 60, Cambridge University Press, 1995.
[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. 257284.
[25]Krajíček, Jan, Approximate Euler characteristic, dimension, and weak pigeonhole principles, this Journal, vol. 69 (2004), no. 1, pp. 201214.
[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. 143153.
[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. 843872.
[28]Ojakian, Kerry E., Combinatorics in bounded arithmetic, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 2004.
[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. 12351244.
[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. 308317.
[31]РАЗБОРОВ, АЛЕКСАНДР А., , (Адян, Серγей И., editor), ΒοПрοϲь κИбеРΗеΤИκИ, vol. 134, VINITI, Moscow, 1988, pp. 149166 (Russian).
[32]Riis, Søren M., Making infinite structures finite in models of second order bounded arithmetic, In Clote, and Krajíček, [8], pp. 289319.
[33]Russell, Alexander and Sundaram, Ravi, Symmetric alternation captures BPP, Computational Complexity, vol. 7 (1998), no. 2, pp. 152162.
[34]Sipser, Michael, A complexity theoretic approach to randomness, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, ACM Press, 1983, pp. 330335.
[35]Szekeres, Esther and SZEKERES, GEORGE, On a problem of Schütte and Erdös, Mathematical Gazette, vol. 49 (1965), no. 369, pp. 290293.
[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. 514519.
[37]Zambella, Domenico, Notes on polynomially bounded arithmetic, this Journal, vol. 61 (1996), no. 3, pp. 942966.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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