Logarithmic advice classes

Abstract

Karp and Lipton [9] introduced the notion of non-uniform complexity classes where a certain amount of "side information", the advice, is given for free. The advice only depends on the length of the input. Karp and Lipton (and also later researchers [22,17,2,12]) concentrated on the study of classes of the form C/poly where C is P, NP, or PSPACE, and poly denotes a polynomial size advice. This paper starts a study of classes of the form C/log. As a main result it is shown that in the context of an NP/log computation a log-bounded advice is equivalent to a sparse oracle in NP. In contrast, it has been shown that a poly-bounded advice corresponds to an arbitrary sparse oracle set. Furthermore, a general theorem is presented that generalizes Karp and Lipton's "round-robin tournament" method.