Moser, Philippe
(2008)
Martingale Families and Dimension in P.
Theoretical Computer Science, 400 (13).
pp. 4661.
ISSN 03043975
Abstract
We introduce a new measure notion on small complexity classes (called Fmeasure),
based on martingale families, that gets rid of some drawbacks of previous measure notions:
it can be used to define dimension because martingale families can make money
on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. As
applications to Fmeasure, we answer a question raised in [1] by improving their result to:
for almost every language A decidable in subexponential time, PA = BPPA. We show that
almost all languages in PSPACE do not have small nonuniform complexity. We compare
Fmeasure to previous notions and prove that martingale families are strictly stronger
than Γmeasure [1], we also discuss the limitations of martingale families concerning finite
unions. We observe that all classes closed under polynomial manyone reductions have
measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families
to introduce a natural generalization of Lutz resourcebounded dimension [13] on P, which
meets the intuition behind Lutz’s notion. We show that Pdimension lies between finitestate
dimension and dimension on E. We prove an analogue to the Theorem of Eggleston
in P, i.e. the class of languages whose characteristic sequence contains 1’s with frequency
α, has dimension the Shannon entropy of α in P.
Item Type: 
Article

Additional Information: 
Preprint version of original published article. Moser, P., Martingale families and dimension in P,
Theoretical Computer Science, Volume 400, Issues 1–3, 9 June 2008, Pages 4661, http://www.sciencedirect.com/ http://dx.doi.org/10.1016/j.tcs.2008.02.013 
Keywords: 
Martingale Families; Dimension; P dimension; small complexity classes; 
Academic Unit: 
Faculty of Science and Engineering > Computer Science 
Item ID: 
3502 
Depositing User: 
Philippe Moser

Date Deposited: 
29 Feb 2012 14:26 
Journal or Publication Title: 
Theoretical Computer Science 
Publisher: 
Elsevier 
Refereed: 
No 
URI: 

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