Complexity-Restricted Advice Functions

We consider uniform subclasses of the nonuniform complexity classes de ned by Karp and Lipton via the notion of advice functions These subclasses are obtained by restricting the complexity of computing correct advice We also investigate the e ect of allowing advice functions of limited complexity to depend on the input rather than on the input s length Among other results using the notions described above we give new characterizations of a NP SPARSE b NP with a restricted access to an NP oracle and c the odd levels of the boolean hierarchy As a consequence we show that every set that is nondeterministically truth table reducible to SAT in the sense of Rich is already deterministically truth table reducible to SAT Furthermore it turns out that the NP reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy