Query complexity, or why is it difficult to separateNP ^{ A } ∩coNP ^{ A } fromP ^{ A } by random oraclesA ?
 G. Tardos
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By thequerytime complexity of a relativized algorithm we mean the total length of oracle queries made; thequeryspace complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time or spacebounded 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} fromP ^{A} under random oraclesA. 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 querytimebounded 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 querytimebounded 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 nonuniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.
 Title
 Query complexity, or why is it difficult to separateNP ^{ A } ∩coNP ^{ A } fromP ^{ A } by random oraclesA?
 Journal

Combinatorica
Volume 9, Issue 4 , pp 385392
 Cover Date
 19891201
 DOI
 10.1007/BF02125350
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 68Q15
 Industry Sectors
 Authors

 G. Tardos ^{(1)}
 Author Affiliations

 1. Department of Algebra, Eötvös University, Múzeum krt. 68, H1088, Budapest, Hungary