Generation complexity versus distinction complexity

Abstract

Among the several notions of resource-bounded Kolmogorov complexity that suggest themselves, the following one due to Levin [Le] has probably received most attention in the literature. With some appropriate universal machine U understood, let the Kolmogorov complexity of a word w be the minimum of |d|+logt over all pairs of a word d and a natural number t such that U takes time t to check that d determines w. One then differentiates between generation complexity and distinction complexity [A, Sip], where the former asks for a program d such that w can actually be computed from d, whereas the latter asks for a program d that distinguishes w from other words in the sense that given d and any word u, one can effectively check whether u is equal to w. Allender et al. [A] consider a notion of solvability for nondeterministic computations that for a given resource-bounded model of computation amounts to require that for any nondeterministic machine N there is a deterministic machine that exhibits the same acceptance behavior as N on all inputs for which the number of accepting paths of N is not too large. They demonstrate that nondeterminism is solvable for computations restricted to polynomially exponential time if and only if for any word the generation complexity is at most polynomial in the distinction complexity. We extend their work and a related result by Fortnow and Kummer [FK] as follows. First, nondeterminism is solvable for linearly exponential time bounds if and only if generation complexity is at most linear in distinction complexity. Second, nondeterminism is solvable for polynomial time bounds if and only if the conditional generation complexity of a word w given a word y is at most linear in the conditional distinction complexity of w given y; hence, in particular, the latter condition implies that P is equal to UP. Finally, in the setting of space bounds it holds unconditionally that generation complexity is at most linear in distinction complexity. © 2008 Springer-Verlag Berlin Heidelberg.