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

On adaptive versus nonadaptive bounded query machines

AbstractThe polynomial-time adaptive (Turing) and nonadaptive (truth-table) bounded query machines are compared with respect to sparse oracles. A k-query adaptive machine has been found which, relative to a sparse oracle, cannot be simulated by any (2k−2)-query nonadaptive machine, even with a different sparse oracle. Conversely, there is a (3·2k−2)-query nonadaptive machine which, relative to a sparse oracle, cannot be simulated by any k-query adaptive machine, with any sparse oracle

Learning Recursive Functions From Approximations

This article investigates algorithmic learning, in the limit, of correct programs for recursive functionsffrom both input/output examples offand several interesting varieties ofapproximateadditional (algorithmic) information aboutf. Specifically considered, as such approximate additional information aboutf, are Rose\u27s frequency computations forfand several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which havefas a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX-learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX-learning (as well as for BC-learning, where a finalsequenceof correct programs is learned), for the cases of providing the types of additional information considered in this paper, the maximal probability is determined such that the entire class of recursive functions is learnable with that probability