Abstract
We look at the problem of counting the exact number of accepting computation paths of a given nondeterministic Turing machine (NTM). We give a deterministic algorithm that runs in time \(\widetilde{O}(\sqrt{S})\), where S is the size (number of vertices) of the configuration graph of the NTM, and prove its correctness. Our result implies a deterministic simulation of probabilistic time classes like \(\mathsf {PP}\), \(\mathsf {BPP}\), and \(\mathsf {BQP}\) in the same running time. This is an improvement over the currently best known simulation by van Melkebeek and Santhanam [SIAM J. Comput., 35(1), 2006], which uses time \(\widetilde{O}(S^{1 - \delta })\). It also implies a faster deterministic simulation of the complexity classes \(\mathsf {\oplus P}\) and \(\mathsf {Mod_k P}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
There are other multiplicative factors in the running time of the simulation, but \(a^{kt/2}\) is the fastest growing factor.
- 2.
We require \(\log (q a^{kt}t^k)\) time to even read a configuration.
- 3.
In fact, for values of \(a \ge 4\), we can set \(\alpha _a = \sqrt{(27/20) \log a}\) and for \(a \in \{2, 3\}\), setting \(\alpha _a\) to be slightly greater than \(\sqrt{(27/20) \log a}\) works.
References
Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, New York (2009)
Beigel, R., Gill, J., Hertramp, U.: Counting classes: thresholds, parity, mods, and fewness. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 49–57. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52282-4_31
Dyer, M., Goldberg, L.A., Greenhill, C., Jerrum, M.: On the relative complexity of approximate counting problems. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 108–119. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44436-X_12
Gill III, J.T.: Computational complexity of probabilistic Turing machines. In: Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, STOC 1974, New York, pp. 91–95. ACM (1974)
Hennie, F.C., Stearns, R.E.: Two-tape simulation of multitape Turing machines. J. ACM 13(4), 533–546 (1966)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. Lectures in Mathematics. ETH Zürich. Birkhäuser, Basel (2003)
Kalyanasundaram, S., Lipton, R.J., Regan, K.W., Shokrieh, F.: Improved simulation of nondeterministic Turing machines. Theor. Comput. Sci. 417, 66–73 (2012). Earlier version in Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science, 2010
Pippenger, N.: Probabilistic simulations (preliminary version). In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC 1982, New York, pp. 17–26. ACM (1982)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley Publishing Company, Reading (1994)
Schöning, U.: The power of counting. In: Selman, A.L. (ed.) Complexity Theory Retrospective: In Honor of Juris Hartmanison the Occasion of His Sixtieth Birthday, July 5, 1988, pp. 204–223. Springer, New York (1990)
Simon, J.: On some central problems in computational complexity. Technical report, Cornell University, Ithaca (1975)
Toda, S.: On the computational power of \({\sf PP}\) and \({\oplus }{\sf P}\). In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS 1989, pp. 514–519. IEEE (1989)
Torán, J.: Counting the number of solutions. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 121–134. Springer, Heidelberg (1990). https://doi.org/10.1007/BFb0029600
van Melkebeek, D., Santhanam, R.: Holographic proofs and derandomization. SIAM J. Comput. 35(1), 59–90 (2005). Earlier version in Proceedings of the 18th Annual IEEEConference on Computational Complexity, 2003
Acknowledgement
We thank Richard Lipton for helpful discussions, and the referees for comments that improved the presentation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Kalyanasundaram, S., Regan, K.W. (2018). Exact Computation of the Number of Accepting Paths of an NTM. In: Panda, B., Goswami, P. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2018. Lecture Notes in Computer Science(), vol 10743. Springer, Cham. https://doi.org/10.1007/978-3-319-74180-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-74180-2_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74179-6
Online ISBN: 978-3-319-74180-2
eBook Packages: Computer ScienceComputer Science (R0)