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Embedding finite lattices into the computably enumerable degrees  a status survey
 In Proceedings of Logic Colloquium
, 2002
"... Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by t ..."
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Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the socalled joinsemidistributive lattices (or lattices without socalled “critical triple”), the collection of embeddable lattices forms a Π0 2set. This paper surveys recent joint work by the authors, concentrating on restricting the number of meets by considering “quasilattices”, i.e., finite upper semilattices in which only some meets of incomparable elements are specified. In particular, we note that all finite quasilattices with one meet specified are embeddable; and that the class of embeddable finite quasilattices with two meets specified, while nontrivial, forms a computable set. On the other hand, more sophisticated techniques may be necessary for finite quasilattices with three meets specified. 1.
Computably Enumerable Sets and QuasiReducibility
 ANNALS OF PURE AND APPLIED LOGIC
, 1998
"... We consider the computably enumerable sets under the relation of Q reducibility. We first give several results comparing the upper semilattice of c.e. Qdegrees, hRQ ; Q i, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding m ..."
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We consider the computably enumerable sets under the relation of Q reducibility. We first give several results comparing the upper semilattice of c.e. Qdegrees, hRQ ; Q i, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of hRQ ; Q i is undecidable.
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 TRANS. AMER. MATH. SOC
, 2004
"... 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 longstanding open question. A negative solution ..."
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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 longstanding 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.
Computational Processes, Observers and Turing Incompleteness
"... We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and comp ..."
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We propose a formal definition of Wolfram’s notion of computational process based on iterated transducers together with a weak observer, a model of computation that captures some aspects of physicslike computation. These processes admit a natural classification into decidable, intermediate and complete, where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting. It is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process.
Universality, Turing Incompleteness and Observers
"... The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics a ..."
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The development of the mathematical theory of computability was motivated in large part by the foundational crisis in mathematics. D. Hilbert suggested an antidote to all the foundational problems that were discovered in the late 19th century: his proposal, in essence, was to formalize mathematics and construct a finite set of axioms that are strong enough to prove all proper theorems, but no more. Thus a proof of consistency and a proof of completeness were required. These proofs should be carried only by strictly finitary means so as to be beyond any reasonable criticism. As Hilbert pointed out [19], to carry out this project one needs to develop a better understanding of proofs as objects of mathematical discourse: To reach our goal, we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves. Furthermore, Hilbert hoped to find a single, mechanical procedure that would, at least in principle, provide correct answers to all welldefined questions
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
Lattice Embeddings below a Nonlow Recursively Enumerable Degree
 Israel J. Math
, 1996
"... We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees. 1 Introduction One of the most basic and important questions concerning the structure of the upper semilattice ..."
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We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees. 1 Introduction One of the most basic and important questions concerning the structure of the upper semilattice R of recursively enumerable degrees is the embedding question: what (finite) lattices can be embedded as lattices into R? This question has a long and rich history. After the proof of the density theorem by Sacks [31], Shoenfield [32] made a conjecture, one consequence of which would be that no lattice embeddings into R were possible. Lachlan [21] and Yates [40] independently refuted Shoenfield's conjecture by proving that the 4 element boolean algebra could be embedded into R (even preserving 0). Using a little lattice representation theory, this result was subsequently extended by LachlanLermanThomason [38], [36] who proved that all countable distributive lattices could be embedded (pre...