, Volume 9, Issue 4, pp 385-392

Query complexity, or why is it difficult to separateNP A ∩coNP A fromP A by random oraclesA?

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Abstract

By thequery-time complexity of a relativized algorithm we mean the total length of oracle queries made; thequery-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 NPA∩ ∩ co NPA 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 query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages LA and {0, 1}*∖LA under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept LA. 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.