NP-hard sets are superterse unless NP is small

Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonuniform complexity measure, how easy is NP in the uniform complexity measure? Let P T (SPARSE) be the class of languages that are polynomial time Turing reducible to some sparse sets. Then it is well known that P T (SPARSE) = P=poly. Hence the above question is equivalent to the following question. NP `?PT (SPARSE): It has been shown by Wilson [18] that th

On adaptive versus nonadaptive bounded query machines

AbstractThe polynomial-time adaptive (Turing) and nonadaptive (truth-table) bounded query machines are compared with respect to sparse oracles. A k-query adaptive machine has been found which, relative to a sparse oracle, cannot be simulated by any (2k−2)-query nonadaptive machine, even with a different sparse oracle. Conversely, there is a (3·2k−2)-query nonadaptive machine which, relative to a sparse oracle, cannot be simulated by any k-query adaptive machine, with any sparse oracle