Abstract
We introduce balanced t(n)-genericity which is a refinement of the genericity concept of Ambos-Spies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resource-bounded version of Church's stochasticity [6]. By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resource-bounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutz's resource-bounded measure theory [13] based on martingales: We show that t(n)-stochasticity coincides with a weak notion of t(n)-randomness based on so-called simple martingales but that it is strictly weaker than t(n)-randomness in the sense of Lutz.
Supported by the Human Capital and Mobility Program of the European Community under grant CHRX-CT93-0415 (COLORET).
Supported in part by the EC through the Esprit Bra project 7141 (ALCOM II), and by the Spanish government through project DGICYT PB94-0564 and Accion Integrada HA-119.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
K. Ambos-Spies. Resource-bounded genericity. To appear in Computability, Enumerability and Unsolvability: Directions in Recursion Theory (B. Cooper et al., Eds.), Cambridge University Press. Extended abstract in Proc. 10th Conf. on Structure in Complexity Theory, pages 162–181. IEEE Computer Society Press, 1995.
K. Ambos-Spies, H. Fleischhack, and H. Huwig. Diagonalizations over deterministic polynomial time. In Proc. of CSL 87, Lecture Notes in Comput. Sci. 329, pages 1–16. Springer Verlag, 1988.
K. Ambos-Spies, H.-C. Neis, and S. A. Terwijn. Genericity and measure for exponential time. Theoret. Comput. Sci. (to appear). Extended abstract in Proc. MFCS 94, Lecture Notes in Comput. Sci. 841, pages 221–232. Springer Verlag, 1994.
K. Ambos-Spies, S. A. Terwijn, and X. Zheng. Resource-bounded randomness and weakly complete problems. Theoret. Comput. Sci. (to appear). Extended abstract in Proc. 5th ISAAC, Lecture Notes in Comput. Sci. 834, pages 369–377. Springer Verlag, 1994.
J. L. Balcazar and E. Mayordomo. A note on genericity and bi-immunity. In Proc. 10th Conf. on Structure in Complexity Theory, pages 193–196. IEEE Computer Society Press, 1995.
A. Church. On the concept of a random sequence. Bull. Amer. Math. Soc., 45:130–135, 1940.
R. A. Di Paola. Random sets in subrecursive hierarchies. J. Assoc. Comput. Mach., 16:621–630, 1969.
S. A. Fenner. Notions of resource-bounded category and genericity. In Proc. 6th Conf. on Structure in Complexity Theory, pages 196–221. IEEE Computer Society Press, 1991.
S. A. Fenner. Resource-bounded Baire category: A stronger approach. In Proc. 10th Conf. on Structure in Complexity Theory, pages 182–192. IEEE Computer Society Press, 1995.
D. Juedes and J. H. Lutz. Weak completeness in E and E 2. Theoret. Comput. Sci., 143:149–158, 1995.
K. Ko. On the notion of infinite pseudorandom sequences. Theoret. Comput. Sci., 48:9–33, 1986.
J. H. Lutz. Category and measure in complexity classes. SIAM J. Comput., 19:1100–1131, 1990.
J. H. Lutz. Almost everywhere high nonuniform complexity. J. Comput. and System Sci., 44:220–258, 1992.
E. Mayordomo. Contributions to the study of resource-bounded measure. PhD thesis, Barcelona, 1994.
C. P. Schnorr. Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeittheorie. Lecture Notes in Math. 21. Springer Verlag, 1971.
V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121–189, 1990.
J. Ville. Ètude Critique de la Notion de Collectif. Gauthiers-Villars, Paris, 1939.
R. von Mises. Grundlagen der Wahrscheinlichkeitsrechnung. Math. Z., 5:52–89, 1919.
R. Wilber. Randomness and the density of hard problems. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 335–342, 1983.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ambos-Spies, K., Mayordomo, E., Wang, Y., Zheng, X. (1996). Resource-bounded balanced genericity, stochasticity and weak randomness. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_6
Download citation
DOI: https://doi.org/10.1007/3-540-60922-9_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60922-3
Online ISBN: 978-3-540-49723-3
eBook Packages: Springer Book Archive