Summary: Lower Bounds for the Low Hierarchy1
and Lane A. Hemachandra3
Department of Computer Science Department of Computer Science
Rutgers University University of Rochester
New Brunswick, NJ 08903 Rochester, NY 14627
The low hierarchy in NP [Sc-83] and the extended low hierarchy [BBS-86] have been
useful in characterizing the complexity of certain interesting classes of sets. However,
until now, there have been no results establishing whether a given lowness result is the
We prove absolute lower bounds on the location of classes in the extended low hierarchy,
and relativized lower bounds on the location of classes in the low hierarchy in NP. In some
cases, we are able to show that the classes are lower in the hierarchies than was known
previously. In almost all cases, we are able to prove that our results are essentially optimal.
We also examine the interrelationships among the levels of the low hierarchies and the
classes of sets reducible to or equivalent to sparse and tally sets under different notions of
reducibility. We feel that these results clarify the structure underlying the low hierarchies.
The low hierarchy within NP (with levels ^L1, L1, ^L2, L2, . . . ) was defined and studied by