Abstract
We have shown that the logics (±HP) *[FOS] and (±HP) 1[FOS] are of the same expressibility, and both capture P∥NP. This result gives us the weakest possible hint that it might be wiser to try and show that (±STC *[FOS] collapses to (±STC) 1[FOS] as opposed to trying to show that STC 1[FOS] is closed under complementation: an attempt to use the methods of [Imm88] to achieve this latter result has failed (see [BCD89]).
As to achieving the former result, one approach might be to consider logspace DOTM's with oracles in NSYMLOG. In particular, one could try to show that such oracle machines are equivalent to logspace DOTM's which make all their oracle queries in parallel (as is the case when the oracle is in NP) and then to code such computations as formulae of DTC 1[STC1[FOS]]. This is just a suggestion, but it should be clear that a consideration of oracle machines with restricted access to oracles not necessarily in NP should be undertaken.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. BORODIN, S. A. COOK, P. W. DYMOND, W. L. RUZZO, AND M. TOMPA, Two applications of inductive counting for complementation problems, SIAM J. Comput., 18, 3 (1989), 559–578.
R. J. BEIGEL, Bounded queries to SAT and the Boolean Hierarchy, to appear, Theoret. Comput. Sci..
S. R. BUSS AND L. HAY, On truth-table reducibility to SAT and the Difference Hirarchy over NP, Proc. 3rd Symp. on Structure in Complexity Theory, IEEE Press (1988), 224–233.
J. CAI, T. GUNDERMANN, J. HARTMANIS, L. A. HEMACHANDRA, V. SEWELSON, K. W. WAGNER, AND G. WECHSUNG, The Boolean hierarchy I: Structural properties, SIAM J. Comput., 17, 6 (1988), 1232–1252.
J. CAI, T. GUNDERMANN, J. HARTMANIS, L. A. HEMACHANDRA, V. SEWELSON, K. W. WAGNER, AND G. WECHSUNG, The Boolean hierarchy II: Applications, SIAM J. Comput., 18, 1 (1989), 95–111.
N. IMMERMAN, Languages that capture complexity classes, SIAM J. Comput., 16, 4 (1987), 760–778.
N. IMMERMAN, Nondeterministic space is closed under complementation, SIAM J. Comput., 17, 5 (1988), 935–938.
J. KÖBLER, personal communication as cited in [Wag90].
J. KÖBLER, U. SCHÖNING, AND K. W. WAGNER, The difference and the truth-table hierarchies for NP, RAIRO Inform. Theory, 21 (1987), 419–435.
I. A. STEWART, Comparing the expressibility of languages formed using NP-complete operators, J. Logic Computat., 1, 3 (1991), 305–330.
I. A. STEWART, On completeness for NP via projection translations, Math. Systems Theory, to appear.
I. A. STEWART, Complete problems involving Boolean labelled structures and projection translations, J. Logic Computat., 1, 6 (1991), 861–882.
I. A. STEWART, Using the Hamiltonian operator to capture NP, J. Comput. System Sci., 45, 1 (1992), 127–151.
I. A. STEWART, Logical characterizations of bounded query classes I: logspace oracle machines, L.N.C.S. 620 (1992), 470–479.
K. W. WAGNER, Bounded query classes, SIAM J. Comput., 19, 5 (1990), 833–846.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Stewart, I.A. (1993). Logical characterization of bounded query classes II: Polynomial-time oracle machines. In: Börger, E., Jäger, G., Kleine Büning, H., Martini, S., Richter, M.M. (eds) Computer Science Logic. CSL 1992. Lecture Notes in Computer Science, vol 702. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56992-8_25
Download citation
DOI: https://doi.org/10.1007/3-540-56992-8_25
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56992-3
Online ISBN: 978-3-540-47890-4
eBook Packages: Springer Book Archive