# Query complexity, or why is it difficult to separate*NP*^{ A }*∩coNP*^{ A } from*P*^{ A } by random oracles*A*?

- 114 Downloads
- 28 Citations

## Abstract

By the*query-time complexity* of a relativized algorithm we mean the total length of oracle queries made; the*query-space complexity* is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles.

A notable unresolved question in relativized complexity theory is the separation of NP^{A}∩ ∩ co NP^{A} from*P*^{A} under random oracles*A*. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity.

If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages L^{A} and {0, 1}^{*}∖L^{A} under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept L^{A}. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1−ε) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.

## AMS subject classification (1980)

68Q15## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. V.Aho, J. D.Ullman and M.Yannakakis, On Notions of Information Transfer in VLSI Circuits,
*Proc. 15th STOC*,**1983**, 133–139.Google Scholar - [2]
- [3]L.Babai, Random oracles separate
*PSPACE*from the polynomial time hierarchy, Technical Report 86-001 (1986), Dept. Comp. Sci., Univ. of Chicago; to appear in*Inf. Proc. Letters*.Google Scholar - [4]L.Babai, Arthur-Merlin games: a randomized proof system and a short hierarchy of complexity classes,
*JCSS, to appear*.Google Scholar - [5]T. Baker, J. Gill andR. Solovay, Relativizations of the
*P=?NP*question,*SIAM J. Comp.*,**4**(1975), 431–442.Google Scholar - [6]C. H. Bennett andJ. Gill, Relative to a random oracle
*A, P*^{A}≠NP^{A}≠coNP^{A}with probability 1,*SIAM J. Comp.*,**10**(1981), 96–113.Google Scholar - [7]M.Blum and R.Impagliazzo, Generic oracles and oracle classes,
*Proc. 28th FOCS*(**1987**), 118–126. Extended Abstract.Google Scholar - [8]R. V. Book, Bounded query machines: on
*NP*and*PSPACE, Theoretical Computer Science*,**15**(1981), 27–39.Google Scholar - [9]R. V. Book, T. J. Long andA. L. Selman, Quantitative relativization of complexity classes,
*SIAM J. Comp.*,**13**(1984), 461–487.Google Scholar - [10]R. V. Book andC. Wrathall, Bounded query machines: on NP(and NPQUERY),
*Theoretical Computer Science*,**15**(1981), 41–50.Google Scholar - [11]J. Y.Cai, With Probability One A Random Oracle Separates
*PSPACE*from the Polynomial Hierarchy,*Proc. 18th STOC*(**1986**), 21–29.Google Scholar - [12]M. L.Furst, J.Saxe and M.Sipser, Parity, circuits, and the polynomial time hierarchy,
*Proc. 22nd FOCS*(**1981**), 260–270.Google Scholar - [13]S.Goldwasser, S.Micaly and C.Rackoff, The knowledge complexity of interactive proofsystems,
*Proc. 17th STOC*,**1985**, 291–304.Google Scholar - [14]S.Goldwasser and M.Sipser, Private coins versus public coins in interactive proof systems,
*Proc. 18th STOC*(**1986**), 59–68.Google Scholar - [15]J.Hartmanis and L. A.Hemachandra, One-way functions, robustness, and the non-isomorphism of NP-complete sets,
*Proc. 2nd Structurde in Complexity Theory*, (**87)**, 160–173.Google Scholar - [16]
- [17]
- [18]N.Nisan, Probabilistic vc. Deterministic Decision Trees and CREW PRAM Complexity,
*preprint*.Google Scholar - [19]
- [20]
- [21]A. C.-C.Yao, Separating the polynomial-time hierarchy by oracles,
*Proc. 26th FOCS*(**1985**), 1–10.Google Scholar