Albrecht Hoene

NONDETERMINISTICALLY SELECTIVE SETS

In this note, we study NP-selective sets (formally, sets that are selective via NPSVt functions) as a natural generalization of P-selective sets. We show that, assuming P≠NP∩coNP, the class of NP-selective sets properly contains the class of P-selective sets. We study several properties of NP-selective sets such as self-reducibility, hardness under various reductions, lowness, and nonuniform complexity. We prove many of our results via a “relativization technique,” by using the known properties of P-selective sets. Using this technique, we strengthen a result of Longpre and Selman on hard promise problems and show that the result “NP⊆(NP∩coNP)/poly⇒PH=NPNP” is implicit in Karp and Lipton’s seminal result on nonuniform classes.

Counting, Selecting, adn Sorting by Query-Bounded Machines

We study the query-complexity of counting, selecting, and sorting functions. That is, for a given set A and a positive integer k, we ask, how many queries to an arbitrary oracle does a polynomial-time machine on input (x1, x2,..., x k ) need to determine how many strings of the input are in A. We also ask how many queries are necessary to select a string in A from the input (x1, x2,..., x k ) if such a string exists and to sort the input (x1, x2,..., x k ) with respect to the ordering x ≼ y if and only if x ∈ A ⇒ y ∈ A. We obtain optimal query-bounds for these problems, and show that sets for which these functions have a low query-complexity must be easy in some sense. For such sets we obtain optimal placements in the extended low hierarchy. We also show that in the case of NP-complete sets the lower bounds for counting and selecting hold unless P=NP. Finally, we relate these notions to cheatability and p-superterseness. Our results yield as corollaries extensions of previously know results.

Splittings, Robustness, and Structure of Complete Sets

We investigate the structure of EXP-complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. We study for various types of reductions the question of whether the set difference A-S for a hard set A and a sparse set S is still hard. We also address the question of which complete sets A can be split into sets A1 and A2 such that

On sets with efficient implicit membership tests

A\equiv^P_r A_1\equiv^P_r A_2

The Degree Structure of 1-L Reductions

for reduction type r, i.e., which complete sets are mitotic. We obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set.

On sets polynomially enumerable by iteration

This paper completely characterizes the complexity of implicit membership testing in terms of the well-known complexity class OptP, optimization polynomial time, and concludes that many complex sets have polynomial-time implicit membership tests.

Polynomial-Time Functions Generate SAT: On P-Splinters

A 1-L function is one that is computable by a logspace Turing machine that moves its input head only in one direction. We show that there exist 1-L complete sets for PSPACE that are not 1-L isomorphic. In other words, the 1-L complete degree for PSPACE does not collapse. This contrasts a result of Allender who showed that all 1-L complete sets for PSPACE are polynomial-time isomorphic. Since all 1-L complete sets for PSPACE are equivalent under 1-L reductions that are one-one and quadratically length-increasing this also provides an example of a ≤ 1,qli 1−L -degree that does not collapse to a single 1-L isomorphism type.

On checking versus evaluation of multiple queries

Abstract Sets whose members are enumerated by some Turing machine are called recursively enumerable. We define a set to be polynomially enumerable by iteration if its members are efficiently enumerated by iterated application of some Turing machine. We prove that many complex sets—including all exponential-time complete sets, all NP-complete sets yet obtained by direct construction, and the complements of all such sets—are polynomially enumerable by iteration. These results follow from more general results. In fact, we show that all recursively enumerable sets that are ⪯ p 1 si - self-reducible are polynomially enumerable by iterations, and that all recursive sets that are p 1 si - self-reducible are bi -enumerable. We also show that when the ⪯ p 1 si - self-reduction is via a function whose inverse is computable in polynomial time, then the above results hold with the polynomial enumeration given by a function whose inverse is computable in polynomial time. In the final section of the paper we show that no NP-complete set can be iteratively enumerated in lexicographically increasing order unless the polynomial time hierarchy collapses to NP. We also show that the sets that are monotonically bi-enumerable are “essentially” the same as the sets in parity polynomial time.

Reducibility Classes of P-Selective Sets

We ask if there is a polynomial-time computable function f and an initial element x0 such that