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11
An oracle builder’s toolkit
, 2002
"... 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 ..."
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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 SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
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An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom 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 nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom 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 nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
On the computational complexity of Longley's H functional
 Presented at Second International Workshop on Implicit Computational Complexity, UC/Santa Barbara
, 2000
"... Longley [Lon98b] discovered a functional H that, when added to PCF, yields a language that computes exactly SR, the sequentially realizable functionals of van Oosten [vO99]. We show that if P ̸ = NP, then the computational complexity of H (and of similar SRfunctionals) is inherently infeasible. The ..."
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Longley [Lon98b] discovered a functional H that, when added to PCF, yields a language that computes exactly SR, the sequentially realizable functionals of van Oosten [vO99]. We show that if P ̸ = NP, then the computational complexity of H (and of similar SRfunctionals) is inherently infeasible. The sequentially realizable functionals (denoted SR) is a class of “sequentially computable ” highertype functionals. This class, which includes the PCFcomputable functionals along with elements that fail to be Scottcontinuous1, has quite strong and natural mathematical properties [Lon98a, Lon98b, Lon99,
Complexity of the rquery Tautologies in the Presence of a Generic Oracle
, 1997
"... Extending techniques in Dowd (Information and Computation vol. 96 (1992)) and those in Poizat (J. Symbolic Logic vol. 51 (1986)), we study computational complexity of rTAUT [A] in the case when A is a generic oracle, where r is a positive integer and rTAUT [A] denotes the collection of all rquery ..."
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Extending techniques in Dowd (Information and Computation vol. 96 (1992)) and those in Poizat (J. Symbolic Logic vol. 51 (1986)), we study computational complexity of rTAUT [A] in the case when A is a generic oracle, where r is a positive integer and rTAUT [A] denotes the collection of all rquery tautologies with respect to an oracle A. We introduce the notion of ceilinggeneric oracles, as a generalization of Dowd’s notion of tgeneric oracles to arbitrary finitely testable arithmetical predicates. We study how existence of ceilinggeneric oracles affects behavior of a generic oracle, by which we show that {X: coNP [X] is not a subset of NP [rTAUT [X]] } is comeager in the Cantor space. Moreover, using ceilinggeneric oracles, we present an alternative proof of the fact (Dowd) that the class of all tgeneric oracles has Lebesgue measure zero.
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced
The Algebraic Structure of the Isomorphic Types of Tally, Polynomial Time Computable Sets
"... We investigate the polynomial time isomorphic type structure of PT ae fA : A ` f0g g (the class of tally, polynomial time computable sets). We partition PT into six parts: D \Gamma ; D \Gamma ; C; S; F; F , and study their pisomorphic properties separately. The structures of hdeg 1 (F ); i ..."
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We investigate the polynomial time isomorphic type structure of PT ae fA : A ` f0g g (the class of tally, polynomial time computable sets). We partition PT into six parts: D \Gamma ; D \Gamma ; C; S; F; F , and study their pisomorphic properties separately. The structures of hdeg 1 (F ); i, hdeg 1 ( F ); i, and hdeg 1 (C); i are obvious, where F , F , and C are the class of tally finite sets, the class of tally cofinite sets, and the class of tally bidense sets respectively. The following results for the structures of hdeg 1 ( D); i and hdeg 1 (S); i will be proved, where D is the class of tally, codense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor codense), polynomial time computable sets. 1. hdeg 1 ( D); i is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in hdeg 1 ( D); i can be distinguished by first order formulas. 3. There are infinitely many nontrivial automorphisms f...
Lattice initial segments of the hyperdegrees
, 2009
"... 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 isomorph ..."
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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.
Undecidability of local structures of sdegrees and Qdegrees
"... We show that the first order theory of the Σ02 sdegrees is undecidable. Via isomorphism of the sdegrees with the Qdegrees, this also shows that the first order theory of the Π02 Qdegrees is undecidable. Together with a result of Nies, the proof of the undecidability of the Σ02 sdegrees yields ..."
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We show that the first order theory of the Σ02 sdegrees is undecidable. Via isomorphism of the sdegrees with the Qdegrees, this also shows that the first order theory of the Π02 Qdegrees is undecidable. Together with a result of Nies, the proof of the undecidability of the Σ02 sdegrees yields a new proof of the known fact (due to Downey, LaForte and Nies) that the first order theory of the c.e. Qdegrees is undecidable.