Commutative Queries

Beigel, Richard; Chang, Richard

doi:10.1006/inco.2000.2918

Cited by 4 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The techniques used to prove these lemmas include the census trick and the mind change technique. These techniques are fairly common in the literature on bounded query classes [Hem89,Bei91,BCO93,BC97].…”

Section: Normal Forms For Bounded Query Classesmentioning

confidence: 99%

Bounded Queries, Approximations, and the Boolean Hierarchy

Chang¹

2001

Information and Computation

Self Cite

View full text Add to dashboard Cite

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NP-hard approximation problems. The results in this paper take advantage of this machine-based model to prove that in many cases, NP-approximation problems have the upward collapse property. That is, a reduction between NP-approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if MaxClique reduces to (log n)-approximating MaxClique using many-one reductions, then the Traveling Salesman Problem (TSP) is equivalent to MaxClique under many-one reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean Hierarchy and nondeterministic bounded query classes.

show abstract

Section: Normal Forms For Bounded Query Classesmentioning

confidence: 99%

Bounded Queries, Approximations, and the Boolean Hierarchy

Chang¹

2001

Information and Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…(1) We define FQ* (r\ A \,..., r k A k ) to be the set of all functions Turing reducible to A i © • • • © A k via a Turing machine that on any input makes a total of r t queries to Aj. (2) We define FQ(r\A\,.. .,r k A k ) to be the set of all functions that are in FQ* {r\A\,...,r k A k )viaa.…”

Section: Ai ® • • • (B a K = \^J{ka + J \ A E Aj)mentioning

confidence: 99%

On the commutativity of jumps

McNicholl¹

2000

J. symb. log.

View full text Add to dashboard Cite

We study the following classes:● Q* (r1A1…..rkAk) which is defined to be the collection of all sets that can be computed by a Turing machine that on any input makes a total of ri, queries to Ai, for all i ∈ {1..… k}.● Q(r1A1…..rkAk) which is defined like Q* (r1A1….. rkAk) except that queries to Ai, must be made before queries to Ai+1 for all i ∈ {1….. k – 1}.● QC(r1A1….. rkAk) which is defined like Q{r1A1….. rkAk) except that the Turing machine must halt even if given incorrect answers to some of its queries.We show that if A1 ….. Ak are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r1…..rkAk). This extends a result of Beigel's [1]. Since the second class is not affected by permutations, we say that these sets commute with each other. We also show that jumps that are too close together may not commute. We also characterize the commutative sequences of sets obtained by iterating the jump operation through an ordinal notation.

show abstract

“…Beigel and Chang [BC97] use the techniques in the proof of Theorem 3.2 for some new results on commutative queries.…”

Section: Claim 34 the Algorithm Inmentioning

confidence: 99%