Random languages for nonuniform complexity classes

Abstract

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 nonuniform class DSPACE( g( n))/ h( n), where g( n) and h( n) are in o( f( n)), and f( n) is o( n). Nonuniform complexity classes were introduced by Karp and Lipton (1980, in “Proceedings of the 12th Annual ACM Symposium on Theory of Computing,” pp. 302–309) and allow an advice string that depends only on the length of the input as additional information. This paper extends a result by Wilber (1983, in “Proceedings of 24th IEEE Symposium on Foundations of Computer Science,” pp. 335–342) provides a result for the special case of P/poly-random languages in EXPSPACE. Here we explore a different method using strings with high generalized Kolmogorov complexity. A characterization of the nonuniform space classes in terms of Kolmogorov complexity is given. This generalizes a result of Balcázar, Dfaz, and Gabarró (1987b, Theoret. Comput. Sci. 52, 251–267) where characterizations of the class PSPACE/poly are given.