The notion of “lowness” was introduced in computational complexity theory by Schöning [J, Comput. Systems Sci., 27 (1983), pp. 14–28] who studied sets in the class NP. This notion may be interpreted as setting an upper bound on the amount of information that can be encoded by a set. Here ideas from previous studies are incorporated in order to capture the notion of a set being exponentially low. The main result asserts the existence of a sparse set E such that ${\operatorname{DEXT}}(E) = {\operatorname{DEXT}}$, i.e., E is “exponentially low,” but E is not in the class P. In contrast, any set with small generalized Kolmogorov complexity that is exponentially low must be in the class P. In addition, we show that for each $k \geqq 2$, any sparse set S that is low with respect to the class $\Sigma _k^E $ of the exponential-time hierarchy (i.e., $\Sigma _k^E {(S) = \Sigma _k^E } $ ) must be in the class $\Sigma _k^P $ of the polynomial-time hierarchy. Similarly, for each $k \geqq 4$, any set with polynomial-size circuits that is low with respect to the class $\Sigma _k^E $ must be in the class $\Sigma _k^P $ .

A set A not in P is polynomially levelable if any algorithm for A has speedup to a polynomial infinitely often in A: precisely, if given any algorithm M for A and polynomial p, it is possible to find another algorithm $M'$ for A and polynomial $p'$, such that $M'$ runs in time $p'(|x|)$ on infinitely many inputs x in A, on which the running time of M exceeds $p(|x|)$. Intuitively, this condition states that among polynomial time computable approximations to A there is no optimal one, or one giving correct answers on a maximally large subset of A. It appears that most naturally occurring intractable sets are polynomially levelable. We prove this for sets not in P that are either “paddable,” “self-reducible,” or complete for a deterministic time class. We also discuss levelability preserving reductions, and give a simple reducibility characterization of nonlevelable sets.

A software based signal processing subsystem, which performs speech compression and expansion, dtmf detection and generation, and pause detection and reconstruction, is developed as a key component of a voice messaging system. The processing is performed with a TMS32010 single chip signal processor. The signal processing software is organized as part of the VoiceFile software package, which runs on the DSC-2000 VoiceServer hardware to provide all necessary services for the voice messaging application software. The architecture of the voice messaging system and the functionality, architecture, and performance of the speech and signal processing software are discussed.