Browse Theses

This thesis studies average-case complexity theory and polynomial-time reducibilities. Levin initiated a general study of average-case complexity by defining a robust notion of average polynomial time and the notion of distributional NP-completeness. Cai and Selman gave a general definition of T on average for arbitrary time bounds T . Their definition of average polynomial time slightly modifies Levin's definition to avoid some of the difficulties of Levin's definition. In this thesis we study reasonable distributions, distributionally-hard languages, and reductions among distributional problems.

We prove several results demonstrating that it suffices to restrict one's attention to reasonable distributions. We show that if NP has a DTIME(2 n )-bi-immune language, then every DistNP-complete problem must have a reasonable distribution. We prove that the class, P_pcomp , a class defined by Schuler and Yamakami, remains unchanged when restricted to reasonable distributions. We strengthen and present a simpler proof of a result of Belanger and Wang, which shows that Cai and Selman's definition of average-polynomial time is not closed under many-one reductions.

Cai and Selman showed that every is P-bi-immune language is distributionally-hard. We study the question of whether there exist distributionally-hard languages that are not P-bi-immune. First we show that such languages exist if and only if P contains P-printable-immune sets. Then we extend this characterization significantly to include assertions about several traditional questions about immunity, about finding witnesses for NP-machines, and about the existence of one-way functions.

Next we study polynomial-time reducibilities. Ladner, Lynch, and Selman showed, in the context of worst-case complexity, that various polynomial-time reductions differ in E. We show similar results for reductions between distributional problems and we show that most of the completeness notions for DistEXP are different.

Finally, we turn our attention to the question of whether various notions of NP-completeness are different. Lutz and Mayordomo and Ambos-Spies and Bentzien, under hypotheses about the stochastic properties of NP, showed that various completeness notions in NP are different. We introduce a structural hypothesis not involving stochastic properties and prove that the existence of a Turing complete language for NP that is not truth-table complete follows from our hypothesis.