Closure of Resource-Bounded Randomness Notions Under Polynomial-Time Permutations

An infinite bit sequence is called recursively random if no computable strategy betting along the sequence has unbounded capital. It is wellknown that the property of recursive randomness is closed under computable permutations. We investigate analogous statements for randomness notions defined by betting strategies that are computable within resource bounds. Suppose that S is a polynomial time computable permutation of the set of strings over the unary alphabet (identified with N). If the inverse of S is not polynomially bounded, it is easy to build a polynomial time random bit sequence Z such that Z ◦ S is not polynomial time random. So one should only consider permutations S satisfying the extra condition that the inverse is polynomially bounded. Now the closure depends on additional assumptions in complexity theory. Our first result, Theorem 3.2, shows that if BPP contains a superpolynomial deterministic time class, such as DTIME(nlog ), then polynomial time randomness is not preserved by some permutation S such that in fact both S and its inverse are in P. Our second result, Theorem 4.5, shows that polynomial space randomness is preserved by polynomial time permutations with polynomially bounded inverse, so if P = PSPACE then polynomial time randomness is preserved.