Commutative queries

Abstract

We consider polynomial-time Turing machines that have access to two oracles and investigate when the order of oracle queries is sign@cant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries does not mattel: This improves upon the previous result of Hemaspaandra, Hemaspaandra and Hempel, who showed that the order of the queries does not matter if the base machine asks only one query to each oracle. On the other hand, we prove that, for computing functions, the order of oracle queries does matter; unless PH collapses.

R SN 1-tt (NP) Distinguishes Robust Many-One and Turing Completeness

“…Part 2 of Definition 2.3 is somewhat related to work of Selivanov [Sel94a]. This fact, and the comments of the rest of this paragraph, were noted independently by an earlier version of the present paper [HHH97b] and by Klaus Wagner ([Wag97], see also [BC97]), whose observations are in a more general form (namely, applying to more than two sets and to more abstract classes). We now discuss the basic facts known about the relationship between the classes of Selivanov (for the case of "△"s of two sets; see Wagner [Wag97] for the case of more than two sets) and the classes discussed in this paper.…”

“…These papers propose and study the effect of the order of database access on the power of database-accessing machines. That line of research has led recently to the counterintuitive downward collapse result that, for each k ≥ 2, BF96], see also [HHH97c]), and to a number of other interesting results [Wag97,BC97].…”

Self-Specifying Machines

“…For any a and b for which ≤ b a is defined and any class C, let R b a (C) = {L | (∃C ∈ C)[L ≤ b a C]}. Hemaspaandra, Hempel, and Wechsung [20] introduced the study of the power of ordered query access (other papers on or related to query order include [18,1,25,3,48,34,19] and the survey [17]). We adopt this notion, and extend it to the function class case and to the positive query case.…”

“…Let us adopt Valiant's [42] standard definition of #NP (informally, #NP = (#P) NP ), and the analogous definition of #PH (informally, #PH = (#P) PH ) [49]. 3 In this paper, we have discussed the sets ≤ #P m -reducible to certain classes. One might naturally wonder whether ≤ #NP m -reductions to the same classes yield even greater computational power.…”

Two queries

“…In a later paper, Hemaspaandra, Hemaspaandra and Hempel [HHH97b] show that our Theorem 3.2 similarly extends to the boolean hierarchy over p 2 . Beigel and Chang [BC97] use the techniques in the proof of Theorem 3.2 for some new results on commutative queries.…”