Abstract The three quantifier theory of (R; ^T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman [1998]. The two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ^T but includes function symbols.

Abstract not found

Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK

Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal

A computably enumerable (c.e.) Turing degree is a diamond base if and only if it is the bottom of a diamond of c.e. degrees with top 0 # . Cooper and Li [3] showed that no low 2 c.e. degree can bound a diamond bases. In the present paper, we show that there exists a nonlow 2 c.e. degree which does not bound a diamond base. Thus, we refute an attractive natural attempt to define the jump class low 2 .

Abstract. The three quantifier theory of (R, ≤T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ≤T but includes function symbols. Theorem. The two quantifier theory of (R, ≤, ∨, ∧), the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on R) is undecidable. The same result holds for various lattices of ideals of R which are natural extensions of R preserving join and infimum when it exits. 1.

We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1-types. subject code classifications: 03D25 (03C65 03D35 06A06)

Each automorphism of the truth-table degrees with order and jump is equal to the identity on a cone with base 0 (4). A degree structure is said to be rigid on a cone if each automorphism of the structure is equal to the identity on the set of degrees above a fixed degree. It is known [4] that the structure of the Turing degrees with jump is rigid on a cone. This is shown by applying a jump inversion theorem and results on initial segments. In this paper it is shown using a weaker jump inversion theorem that also the structure of truth-table degrees with jump is rigid on a cone. For definitions relating to initial segments we refer to [13]. 1 Initial segment construction Lemma 1.1. Suppose for each e, g lies on a tree Te which is e-splitting for some c for some tables with the properties of Proposition 4.9, in the sense of [5]. Then g is hyperimmune-free. Proof. For each e ∈ ω there exists e ∗ ∈ ω such that for all stages s and all oracles g, if {e ∗ } g s(x) ↓ then {e ∗ } g (x) = {e} g (x) and {e} g s(y) ↓ for all y ≤ x. If g lies on Te ∗ then it follows that {e}g is total and {e ∗ } T (σ) (x) ↓ for each σ of length x + 1. Hence {e} g = {e ∗ } g is dominated by the recursive function f(x) = max{{e} T (σ) (x) : |σ | = x + 1}. Proposition 1.2. Let L be a Σ 0 4(y)-presentable upper semilattice with least and greatest element. Then there exist t, i, g such that 1. t: ω → 2 is 0 ′ ′-computable, 2. i is the characteristic function of a set I such that I ≤m y (3), 3. g ′ ′ (e) = t(i(0),..., i(e)) for all e ∈ ω, 4. [0, g] is isomorphic to L, and 5. g is hyperimmune-free 1 Proof. The proof in [5] must be modified to employ the lattice tables of Proposition

We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 |G | registers, and each of the 2 k subsets of the registers contributes some information. 1

. We study the problem of the interpretability of arithmetic in the r.e. degrees in models of fragments of Peano arithmetic. The main result states that there is an interpretation # ## # # such that every formula # of Peano arithmetic corresponds to a formula # # in the language of the partial ordering of r.e. degrees such that for every model N of # 4 -induction, N |= # if and only if RN |= # # , where RN is the structure whose universe is the collection of r.e. degrees in N . This supplies, for example, statements #m about the r.e. degrees which are equivalent (over I# 4 ) to I#m for every m > 4. 1. Introduction. A basic goal of reverse mathematics is to determine the axiom systems needed to prove particular theorems of mathematics by showing that they are equivalent (over a given base theory) to some specific axiom system. (See [12] for a general introduction to reverse mathematics in the setting of second order arithmetic.) In reverse recursion theory our setti...