The structure of a logarithmic advice class

The complexity class Full-P / log, corresponding to a form of logarithmic advice for polynomial time, is studied. In order to understand the inner structure of this class, we characterize Full-P /log in terms of Turing reducibility to a special family of sparse sets. Other characterizations of Full-P / log, relating it to sets with small information content, were already known. These used tally sets whose words follow a given regular pattern and tally sets that are regular in a resource-bounded Kolmogorov complexity sense. We obtain here relationships between the equivalence classes of the mentioned tally and sparse sets under various reducibiities, which provide new knowledge about the logarithmic advice class. Another way to measure the amount of information encoded in a language in a nonuniform class, is to study the relative complexity of computing advice functions for this language. We prove bounds on the complexity of ad vice functions for Full-P / log and for other subclasses of it. As a consequence, Full-P / log is located in the Extended Low Hierarchy