We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

An explicit recursion-theoretic definition of a random sequence or random set of natural numbers was given by Martin-Löf in 1966. Other approaches leading to the notions of n-randomness and weak n-randomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of n-random and weakly n-random sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of n-randomness and weak n-randomness including a new definition in terms of a forcing relation analogous to the characterization of n-generic sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than n-randomness. Chapter III is concerned with intrinsic properties of n-random sequences. The main results are that an (n + 1)-random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that n-random sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is n-random then A is n-random relative to B. It follows that any countable distributive lattice can be embedded

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We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, Dh. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of Dh. Corollaries include the decidability of the two quantifier theory of Dh and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of! CK 1. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve!1. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of Dh.

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