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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.

We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.

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. 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.

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We show that there are 5 formulas in the language of the Turing degrees, D,

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 show that there are 5 formulas in the language of the Turing degrees, D,

Abstract. We study low level nondefinability in the Turing degrees. We prove a variety of results, including for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤,REA) where REA stands for the “r.e. in and above ” predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1-theory of (D, ≤,REA) is decidable. 1.