Abstract
In [AS], we defined a notion of measure on the complexity class P (in the spirit of the work of Lutz [L92] that provides a notion of measure on complexity classes at least as large as E, and the work of Mayordomo [M] that provides a measure on PSPACE). In this paper, we show that several other ways of defining measure in terms of covers and martingales yield precisely the same notion as in [AS]. (Similar “robustness” results have been obtained previously for the notions of measure defined by [L92] and [M], but — for reasons that will become apparent below — different proofs are required in our setting.)
To our surprise, and in contrast to the measures of Lutz [L92] and Mayordomo.
Research supported by NSF grant CCR-9204874.
Preview
Unable to display preview. Download preview PDF.
References
E. Allender and M. Strauss, Measure on small complexity classes, with applications for BPP, Proc. 35th FOCS conference, 1994 pp. 807–818.
D. Juedes and J. Lutz, The complexity and distribution of hard problems, Proc. 34th FOCS Conference, pp. 177–185, 1993.
D. Juedes, J. Lutz and E. Mayordomo, private communication, 1993–94.
J. Lutz, Almost Everywhere High Nonuniform Complexity, Journal of Computer and System Sciences 44 (1992), pp. 220–258.
J. Lutz, The quantitative structure of exponential time, Proc. 8th Structure in Complexity Theory Conference, pp. 158–175, 1993.
E. Mayordomo, Contributions to the Study of Resource-Bounded Measure, PhD Thesis, Universitat Politècnica de Catalunya, Barcelona, 1994. See also [M2], in which a preliminary version of the PSPACE measure appears.
E. Mayordomo, Measuring in PSPACE, to appear in Proc. International Meeting of Young Computer Scientists '92, Topics in Computer Science series, Gordon and Breach.
K. Regan and D. Sivakumar. Improved resource-bounded Borel-Cantelli and stochasticity theorems. Technical Report UB-CS-TR 95-08, Computer Science Dept., University at Buffalo, February 1995.
K. Regan, D. Sivakumar, and J.-Y. Cai. Pseudorandom generators, measure theory, and natural proofs. Technical Report UB-CS-TR 95-02, Computer Science Dept., University at Buffalo, January 1995.
C. P. Schnorr, Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics 218 (1971).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Allender, E., Strauss, M. (1995). Measure on P: Robustness of the notion. In: Wiedermann, J., Hájek, P. (eds) Mathematical Foundations of Computer Science 1995. MFCS 1995. Lecture Notes in Computer Science, vol 969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60246-1_119
Download citation
DOI: https://doi.org/10.1007/3-540-60246-1_119
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60246-0
Online ISBN: 978-3-540-44768-9
eBook Packages: Springer Book Archive