Logical characterization of bounded query classes II: Polynomial-time oracle machines
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 STC1[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 DTC1[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.
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- [BCD89]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.Google Scholar
- [Bei88]R. J. BEIGEL, Bounded queries to SAT and the Boolean Hierarchy, to appear, Theoret. Comput. Sci..Google Scholar
- [BH88]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.Google Scholar
- [CGH88]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.Google Scholar
- [CGH89]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.Google Scholar
- [Imm87]N. IMMERMAN, Languages that capture complexity classes, SIAM J. Comput., 16, 4 (1987), 760–778.Google Scholar
- [Imm88]N. IMMERMAN, Nondeterministic space is closed under complementation, SIAM J. Comput., 17, 5 (1988), 935–938.Google Scholar
- [Köb87]J. KÖBLER, personal communication as cited in [Wag90].Google Scholar
- [KSW87]J. KÖBLER, U. SCHÖNING, AND K. W. WAGNER, The difference and the truth-table hierarchies for NP, RAIRO Inform. Theory, 21 (1987), 419–435.Google Scholar
- [Ste91a]I. A. STEWART, Comparing the expressibility of languages formed using NP-complete operators, J. Logic Computat., 1, 3 (1991), 305–330.Google Scholar
- [Ste91b]I. A. STEWART, On completeness for NP via projection translations, Math. Systems Theory, to appear.Google Scholar
- [Ste91c]I. A. STEWART, Complete problems involving Boolean labelled structures and projection translations, J. Logic Computat., 1, 6 (1991), 861–882.Google Scholar
- [Ste92a]I. A. STEWART, Using the Hamiltonian operator to capture NP, J. Comput. System Sci., 45, 1 (1992), 127–151.Google Scholar
- [Ste92b]I. A. STEWART, Logical characterizations of bounded query classes I: logspace oracle machines, L.N.C.S. 620 (1992), 470–479.Google Scholar
- [Wag90]K. W. WAGNER, Bounded query classes, SIAM J. Comput., 19, 5 (1990), 833–846.Google Scholar