Logarithmic Advice Classes

Karp and Lipton [11] introduced the notion of non-uniform complexity classes where a certain amount of additional information, the advice, is given for free. The advice only depends on the length of the input. Karp and Lipton initiated the study of classes with either logarithmic or polynomial advice; however later researchers [24, 19, 2, 14] 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 considers classes of the form C/log. As a main result it is shown that in the context of an NP/log computation, 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. ∗This work was supported by the Deutsche Forschungsgemeinschaft, grant No. Scho 3/2, and by a grant of CIRIT (Generalitat de Catalunya).