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Codable Sets and Orbits of Computably Enumerable Sets
 J. Symbolic Logic
, 1995
"... A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order ..."
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Cited by 10 (5 self)
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A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order Edefinable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness " property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 automorphism method we introduced earli...
Definability and Automorphisms of the Computably Enumerable Sets
, 2010
"... The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. ..."
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The computably enumerable (c.e.) sets have been central to computability theory since its inception. We study the structure of the c.e. sets, which forms a lattice E under set inclusion. Jump classes, such as the low degrees, allow us to classify the c.e. sets according to their information content. The upward closed jump classes Ln and Hn have all been shown to be definable by a latticetheoretic formula, except for L1, the nonlow degrees, which is the only jump class whose definability was unknown. We say a class of c.e. degrees is invariant if it is the set of degrees of a class of c.e. sets that is invariant under automorphisms of E. All definable classes of degrees are invariant. We show that L1 is in fact noninvariant, thus proving a 1996 conjecture of Harrington and Soare in [3] that the nonlow degrees are not definable, and completing the problem of determining the definability of each jump class. 1
Review of Peter Cholak, “Automorphisms of the Lattice of Recursively Enumerable Sets ” Memoirs of the American Math. Soc. (1995) viii+151 pp and
"... Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an ..."
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Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an very influential paper [Po44], Post articulated some of the fundamental notions at the heart of most undecidability proofs. He observed that these proofs worked by coding some “noncomputability ” into the theory at hand thereby arguing that the relevant structures could emulate computation. One key concept was that of effective enumeration which leads to the notion of a computably (recursively) enumerable set. A computably enumerable set is a subset of N which is the range of a computable (total) function. The intuitive idea is that if f is computable, then I can “effectively list ” (not necessarily in order) the range of f as {f(0), f(1),...}. Think of consequences of a computably enumerable set of axioms for a formal system. The other key concept discovered by Turing [1939] and further developed in Post’s paper