Martingale Families and Dimension in P.
Theoretical Computer Science, 400 (1-3).
We introduce a new measure notion on small complexity classes (called F-measure),
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 F-measure, we answer a question raised in  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 non-uniform complexity. We compare
F-measure to previous notions and prove that martingale families are strictly stronger
than Γ-measure , we also discuss the limitations of martingale families concerning finite
unions. We observe that all classes closed under polynomial many-one reductions have
measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families
to introduce a natural generalization of Lutz resource-bounded dimension  on P, which
meets the intuition behind Lutz’s notion. We show that P-dimension 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.
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