Query order in the polynomial hierarchy

Abstract

Hemaspaandra, Hempel, and Wechsung [HHW] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. P C:D denotes the class of languages computable by a polynomial-time machine that is allowed one query to C followed by one query to D [HHW]. We prove that the levels of the polynomial hierarchy are order-oblivious: P Σ p j :Σ p k = P Σ p k :Σ pj . Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes-and thus essentially all standard complexity classes-inherit all order-obliviousness results that hold for P.

Reducing the Number of Solutions of NP Functions

“…Certainly, there are many similar cases in the literature where similar complete characterizations have been completed or interestingly posed. Hemaspaandra et al [12] have under the assumption that the polynomial hierarchy does not collapse, completely characterized (for pairs of levels of the boolean hierarchy) when ''query order'' (see [10,12], see also [11,28]) matters. Kosub and Wagner [19] have posed, and made powerful progress toward, a complete characterization regarding their boolean hierarchy of NP-partitions.…”

Lower bounds and the hardness of counting properties

“…Deÿnition 19 (see Hemaspaandra Hemaspaandra, Hemaspaandra and Hempel [17]). For any sets A ⊆ * and B ⊆ * , let M A [1]; B denote oracle DPTM M making at most one query to A and any number of queries to B in arbitrary order (it is even legal for the query to A to be sandwiched between queries to B).…”

“…The following notation, modiÿed slightly from Hemaspaandra, Hemaspaandra and Hempel [17], deals with Turing machines that have access to multiple oracles.…”

Query-Monotonic Turing Reductions

“…(For each Turing machine M and for each Y, Z ⊆ Σ * , M Y,Z represents machine M with Y as its first oracle and Z as its second oracle. See [HHH97,HHH98,HHW99] for earlier uses of double oracles-there in the context of studying the effect of the order in which the two databases are accessed. )…”