Martingale Families and Dimension in P

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 [1] by improving their result to: for almost every language A decidable in subexponential time, P = BPP. 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 [1], 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 [13] 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.