Applications of recursive operators to randomness and complexity

This dissertation contributes to the ongoing study of the role of randomness in relativized complexity theory. Chapter 1 provides preliminary notational conventions, and enough introductory background preparation in computability, complexity, and randomness to make this thesis generally self-contained. In Chapter 2 properties of recursive operators are developed for the purpose of proving results about random sequences. The method of "proof by prediction" is introduced in accordance with the notion of a predictor and recursive family of functions. The ideas and results in Chapter 2 are then used in Chapter 3 to prove a theorem on deterministic bounded-query machines for which results such as $P(A) \not= NP(A),$ for every $A\in{\bf RAND}$ follow as corollaries. Chapter 4 begins with an inquiry as to whether or not a result similar to Theorem 3.1.5 holds for nondeterministic oracle machines. The concept of autoreducibility is introduced to answer in the negative. Furthermore, some results on the autoreducibility of random sequences are provided. In particular, it is shown that every random sequence is i.o. autoreducible; i.e. $A\in{\bf RAND}$ implies the existence of a deterministic oracle machine M such that $A = L(M, A),$ and infinitely often an input x is decided without the needed query "$x \in A$?". Chapter 5 returns to relativized complexity and studies the problem of separating higher levels of the polynomial hierarchy relative to a random oracle. The results of Chapter 5 intend to provide evidence supporting an infinite relativized hierarchy. For example, it is shown that there exists an "infinite hierarchy" of random oracles$$A = A\sb0 = \Sigma\sp{0A},\ A\sb1 = \Sigma\sp{1A},\ A\sb2 = \Sigma\sp{2A},\...$$where $\Sigma\sp{kA}$ denotes application of a recursive operator $\Sigma:\{0, 1\}\sp\infty\to\{0, 1\}\sp\infty\ k$ times on input A. The sequence will be shown to satisfy the following properties: (1) $A\sb{k}\in NP(A\sb{k-1})\Rightarrow A\sb{k}\in\Sigma\sbsp{k}{p}(A);$ (2) $A\sb{k}\not\in{\rm co} - NP(A\sb{k-1})\subset\Pi\sbsp{k}{p}(A).$ Chapter 5 concludes with a discussion of the application of circuit theory to the problem of separating the second and third levels.