Functions that preserve p-randomness

• Authors: Stephen A. Fenner
• Venue: FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
• Year: 2011

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 is an open interval, f : I → R is a function, and r ∈ I is p-random, then f(r) is p-random provided 1. f is p-computable on the dyadic rational points in I, and 2. f varies sufficiently at r, i.e., there exists a real constant C 0 such that either (∀x ∈ I - {r})[f(x)-f(r)/x-r ≥ C] or (∀x ∈ I - {r})[f(x)-f(r)/x-r ≥ -C]. 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.