Random languages for nonuniform complexity classes

A language A is considered to be random for a class C if for every language B in C the fraction of the strings where A and B coincide is approximately 1/2. We show that there exist languages in DSPACE(f(n)) which are random for the non-uniform class DSPACE(g(n))=h(n), where n, g(n) and h(n) are in o(f(n)). Non-uniform complexity classes were introduced by Karp and Lipton Karp and Lipton 1980] and allow an advice string that depends only on the length of the input as additional information. This paper extends a result by Wilber Wilber 1983] who proved bounds for the existence of random languages for (uniform) time and space classes. Huynh Huynh 1987] provides a result for the special case of P=poly-random languages in EXPSPACE. Here we explore a diierent method using strings with high generalized Kolmogorov complexity Hartmanis 1983]. A characterization of the non-uniform space classes in terms of Kolmogorov complexity is given. This generalizes a result of Balcc azar, D az, and Gabarrr o 1987b] where characterizations of the class PSPACE=poly are given.