Quantum Bounded Query Complexity

We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: o P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one complete for PP have the property that FP_||^A is included in FEQP^A[1]. In general we prove that for any set A there is a set X such that FP^A is included in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9

Query Order and the Polynomial Hierarchy

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] initiated the field of query order, which studies the ways in which computational power is affected by the order in which information sources are accessed. The present paper studies, for the first time, query order as it applies to the levels of the polynomial hierarchy. We prove that the levels of the polynomial hierarchy are order-oblivious. Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that all leaf language classes - and thus essentially all standard complexity classes - inherit all order-obliviousness results that hold for P.Comment: 14 page

Separating Cook Completeness from Karp-Levin Completeness Under a Worst-Case Hardness Hypothesis

We show that there is a language that is Turing complete for NP but not many-one complete for NP, under a worst-case hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time O(2^2^n^c) (for some c > 1) accepting Sigma^* whose accepting computations cannot be computed by bounded-error, probabilistic machines running in time O(2^2^{beta * 2^n^c) (for some beta > 0). This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis

A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard. The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc