Sharply Bounded Alternation within P

We deene the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantiication over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Gold-smith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We deene both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL), and present a variety of problems in these classes, including ql m-complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rst-order deenability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

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