The University of Chicago Hierarchy Theorems and Resource Tradeoffs for Semantic Classes a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Computer Science By

Computational complexity theory studies the minimum resources (time, space, randomness etc.) to solve computational problems. Two fundamental questions in this area are: 1. Can more problems be solved given more of a given resource? A positive answer to this question is known as a hierarchy theorem. 2. Can one resource be traded off with another when solving a given problem? Such a result is known as a resource tradeoff. We study these questions in the context of semantic classes, which are classes for which no efficient enumeration of machines is known. Several natural classes such as BPP (probabilistic polynomial time with two sided error), RP, MA and NP ∩ coNP are semantic classes. No hierarchy theorems are known for uniform polynomial timebounded semantic classes. We investigate the existence of hierarchies for natural variants of semantic classes, focussing mainly on semantic classes with small advice. Complexity classes with advice are defined by Turing machines receiving some auxiliary information depending only on the input length. We prove the following results: 1. A hierarchy theorem for polynomial-time semantic classes with O(log(n) log(log(n))) bits of advice. 2. A hierarchy theorem for quasipolynomial-time semantic classes with 1 bit of advice. 3. Hierarchy theorems for BPP and RP with 1 bit of advice. 4. A hierarchy theorem for average-case BPP.