Sharply Bounded Alternation and Quasilinear Time

Abstract.

We define the sharply bounded hierarchy, SBH(QL)}, a hierarchy of classes within P , using quasilinear-time computation and quantification over strings of length log n . It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations.

We define both SBH(QL) and its corresponding hierarchy of function classes, 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 determining its precise relationship to deterministic time classes can imply P≠ PSPACE . We present characterizations of SBH(QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH(QL).