Bounded Queries to SAT and the Boolean Hierarchy

  title={Bounded Queries to SAT and the Boolean Hierarchy},
  author={Richard Beigel},
  journal={Theor. Comput. Sci.},
  • R. Beigel
  • Published 29 July 1991
  • Computer Science, Mathematics
  • Theor. Comput. Sci.

Advice from Nonadaptive Queries to NP

This paper investigates the classes of languages that are decided by bounded truth-table reductions to an NP set in which evaluators do not have full access to the answers to the queries but get only partial information such as the number of queries that are in the oracle set or even just this number modulo some constant.

Modulo Information from Nonadaptive Queries to NP

The classes of languages accepted by nondeterministic polynomial-time Turing machines that have restricted access to an NP oracle coincide with an appropriate level of the Boolean hierarchy when m is even or k 2m, and the results are generalized to the case when the NP machines are replaced by Turing machines accepting languages of the l th level of.

Bounded Queries, Approximations, and the Boolean Hierarchy

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy and proves that in many cases, NP-approximation problems have the upward collapse property.

Oracles that Compute Values

The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses.

Bounded query functions with limited output bits

  • Richard ChangJ. Squire
  • Computer Science, Mathematics
    Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
The paper explores the difference between parallel and serial queries to an NP-complete oracle, SAT, from the perspective of functions with a limited number of output bits, and shows that there exists a function with 2 bit output that cannot be computed using 3 parallel queries to SAT, unless the polynomial hierarchy collapses.

NP trees and Carnap's modal logic

  • G. Gottlob
  • Computer Science
    Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
  • 1993
This work considers problems and complexity classes definable by interdependent queries to an oracle in NP, and shows that the following problems are all P/sup NP/[O(logn)] complete: validity-checking of formulas in Carnap's modal logic, checking whether a formula is almost surely valid over finite structures in modal logics K, T, and S4, and checkingWhether a formula belongs to the stable set of beliefs generated by a propositional theory.

On Using Oracles That Compute Values

This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable

Commutative queries

  • R. BeigelRichard Chang
  • Computer Science, Mathematics
    Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems
  • 1997
It is proved that the order of oracle queries does matter unless PH collapses, and this improves upon the previous result of Hemaspaandra, HemaspAandra and Hempel, who showed that the orders of the queries did not matter.

Some connections between bounded query classes and non-uniform complexity

1 Commutative Queries

It is proved that, for solving decision problems, the order of oracle queries does not matter and, for computing functions, the orders of oracles queries does matter, unless PH collapses.



Query-limited reducibilities

The Nonspeedup Theorem is proved, which states that 2$\sp{n}$ parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries toAny oracle whatsoever.

The complexity of optimization problems

The central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation, and it quantifies "how much" NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function.

Bounded Query Classes

The Boolean hierarchy is generalized in such a way that it is possible to characterize P and O in terms of the generalization, and the class $P^{\text{NP}}[O(\log n)]$ can be characterized in very different ways.

NP-hard Sets are P-Superterse Unless R = NP

All NP-hard sets (under p tt-reductions) are p-superterse, unless it is possible to distinguish uniquely satissable formulas from satiss-able formulas in polynomial time, which mostly solves the main open question in 4.

Bounded query classes and the difference hierarchy

A hierarchy of sets that are reducible toA based on bounding the number of queries toA that an oracle machine can make is defined, i.e. sets in a logarithmic way.

Trial and error predicates and the solution to a problem of Mostowski

  • H. Putnam
  • Computer Science
    Journal of Symbolic Logic
  • 1965
The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the

Using Self-Reducibilities to Characterize Polynomial Time

It is shown that the extent to which a set is self-reducible can determine whether or not the set is polynomially decidable, and that if sufficiently large deterministic and nondeterministic time classes are separated by tally sets, then polynomial time cannot be characterized by coupling ⊕ P and p-cheatability.

Terse, Superterse, and Verbose Sets

The range of possible query savings is limited by the following theorem: F A n cannot be computed with only ⌊ log n ⌋ queries to a set X unless A is recursive.

On truth-table reducibility to SAT and the difference hierarchy over NP

  • S. BussL. Hay
  • Computer Science
    [1988] Proceedings. Structure in Complexity Theory Third Annual Conference
  • 1988
It is shown that polynomial-time truth-table reducibility by Boolean circuits to SAT is the same as log-space truth-table reducibility via Boolean formulas to SAT and the same as log-space Turing

Polynomial Terse Sets