Functions that preserve p-randomness ?
We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general theorem: If I?R I ? R is an open interval, f:I?R f : I ? R is a function, and r?I r ? I is p-random, then f(r) f ( r ) is p-random provided f is p-computable on the dyadic rational points in I, and f varies sufficiently at r , i.e., there exists a real constant C>0 C > 0 such that either Our theorem implies in particular that any analytic function about a p-computable point whose power series has uniformly p-computable coefficients preserves p-randomness in its open interval of absolute convergence. Such functions include all the familiar functions from first-year calculus.
p-randomness randomness polynomial-time complexity measure martingale