Diagonalizing over deterministic polynomial time

Feasible Analysis, Randomness, and Base Invariance

“…Analogously, one can define Champernowne's sequences in any base r by concatenating all the natural numbers represented in base r. As expected, one ends up with a sequence which is normal in base r. For the specific case of the sequence X defined in (2), it is unknown if its normality is base invariant. That is, let x be the real which in base 10 is represented as 0.X, and in base s is represented as 0.Y (for some infinite sequence over Σ s ).…”

“…(By a result of Schnorr, restricting to rational values is immaterial.) We say that Z is polynomial time random if Z is n c -random for every c, that is, no polynomial time martingale succeeds on Z. Polynomial time random sequences have been studied for instance in [3], where some connections to Lutz's polynomial time bounded measure [16,17] and polynomial genericity [1,2] are discussed.…”

Effective Baire Category Concepts

“…Resource-bounded Baire Category concepts have b e e n i n troduced and studied e.g. in [AFH88], [Lu90], [Fe91], [Fe95] and [Am96]. Most of these concepts can be dened in terms of (partial) extension functions though some of them were originally dened in terms of condition sets.…”

“…The papers [AFH88] and [Am96], where resource-bounded category concepts based on partial extensions were introduced, focus on generic sets, i.e. sets such that the singleton is not nowhere dense or, equivalently, not meager (in the corresponding setting).Though in [Am96] i t i s p o i n ted out how meagerness can be dened in terms of genericity, meager classes were not explicitely studied.…”

“…In particular no game theoretical characterization was given. Reimann [Re97] has shown, however, that the extended Banach-Mazur game introduced here can be adapted to describe the category concepts of [AFH88] and [Am96].…”