-Maximal Sets

Cholak, Peter; Gerdes, Peter M.; Lange, Karen

doi:10.1017/jsl.2015.3

Cited by 2 publications

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References 19 publications

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“…It turns out that maximal and hemimaximal sets are both D-maximal. Cholak, Gerdes, and Lange [9] have completed a classification of all D-maximal sets. The idea is to look at how D(A) is generated.…”

Section: D-maximal Setsmentioning

confidence: 99%

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Some Recent Research Directions in the Computably Enumerable Sets

Cholak

2017

The Incomputable

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Abstract. As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current research.There are many equivalent ways to definite a computably enumerable or c.e. set. The one that we prefer is the domain of a Turing machine or the set of balls accepted by a Turing machine. Perhaps this definition is the main reason that this paper is included in this volume and the corresponding talk in the "Incomputable" conference. The c.e. sets are also the sets which are Σ 0 1 definable in arithmetic. There is a computable or effective listing, {M e |e ∈ ω}, of all Turing machines. This gives us a listing of all c.e. sets, x in W e at stage s iff M e with input x accepts by stage s. This enumeration of all c.e. sets is very dynamic. We can think of balls x as flowing from one c.e. set into another. Since they are sets, we can partially order them by inclusion, ⊆ and consider them as model, E = {W e |e ∈ ω}, ⊆ . All sets (not just c.e. sets) are partially ordered by Turing reducibility, where A ≤ T B iff there is a Turing machine that can compute A given an oracle for B.Broadly, our goal is to study the structure E and learn what we can about the interactions between definability (in the language of inclusion ⊆), the dynamic properties of c.e. sets and their Turing degrees. A very rich relationship between these three notions has been discovered over the years. We cannot hope to completely cover this history in this short paper. But, we hope that we will cover enough of it to show the reader that the interplay between these three notions on c.e. sets is, and will continue to be, an very interesting subject of research.We are assuming that the reader has a background in computability theory as found in the first few chapters of Soare [26]. All unknown notation also follows [26].2000 Mathematics Subject Classification. Primary 03D25. Friedberg SplitsThe first result in this vein was Friedberg [15], every noncomputable c.e. set has a Friedberg split. Let us first understand the result then explore why we feel this result relates to the interplay of definability, Turing degrees and dynamic properties of c.e. sets. The following definition depends on the chosen enumeration of all c.e. sets. We use the enumeration given to us in the second paragraph of this paper, x ∈ W e,s iff M e with input x accepts by stage s, but with the convention that if x ∈ W e,s then e, x < s and, for all stages s, there is at most one pair e, x where x enters W e at stage s. Some details on how we can effectively achieve this type of enumeration can be found in Soare [26, Exercise I.3.11]. Moreover, when given a c.e. set, we are given the index of this c.e. set in terms of our enumeration of all c.e. sets. At times we will have to appeal to Kleene's Recursion Theorem to get this index. By the above definition, A\B is a c.e. set. A\B is the set of balls that enter A before they enter B. If x ∈...

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Section: D-maximal Setsmentioning

confidence: 99%

“…Cholak, Gerdes, and Lange [9] have completed a classification of all D-maximal sets. The idea is to look at how D(A) is generated.…”

Section: D-maximal Setsmentioning

confidence: 99%

Some Recent Research Directions in the Computably Enumerable Sets

Cholak

2017

The Incomputable

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On Splits of Computably Enumerable Sets

Cholak

2016

Computability and Complexity

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Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly. PETER CHOLAK Friedberg SplitsMost of this section is known but we wanted to provide an explicit proof of Corollary 2.7. This corollary will be useful later. One of the focuses of this paper is splitting procedures that always produce a non-trivial split when possible.At this point we will fix the standard uniform enumeration W e,s of all c.e. sets with the convention that at stage s, there is at most one pair e, x where x enters W e at stage s. Some details on how we can effectively achieve this enumeration can be found in Soare [12, Exercise I.3.11].Every c.e. set has an index according to this fixed enumeration. For the sets that we construct we have to appeal to Kleene's Recursion Theorem to find this index. Moreover, by the standard trick of slowing down or pausing our construction, we can assume the enumerations of our fixed point W e and our constructed set A are the same. Our construction, at times, will construct sets other than A. While we will focus on the constructed sets, the actual outcome of our constructions will be a uniform enumeration of all constructed sets. We will be using Kleene's Recursion Theorem with parameters to get a function from each constructed set to an index with the same enumeration for that set in the above enumeration.By the Padding Lemma, we know that each c.e. set A has infinitely many indices. By Rice's Theorem, we know that for a given c.e. set A the set of indices (in this fixed enumeration) for A is not computable.Also at this point we will fix the convention that A, B, W , X and Y always refer to c.e. sets with some fixed index in our given enumeration. Now we need the following.for all W , if W − A is not c.e. then both W − A 0 and W − A 1 are not c.e. sets. Lemma 2.2. If A is not computable and A 0 ⊔ A 1 is a Friedberg split then the split is not trivial. Proof. N − A is not c.e. so N − A 0 and N − A 1 are not c.e. and hence A 0 and A 1 are not computable. Definition 2.3. For A = W e and B = W i , A\B = {x|∃s[x ∈ (W e,s − W i,s )]}and A ց B = A\B ∩B. (This is with respect to our given enumeration and hence this definition depends on our chosen enumeration.)

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-Maximal Sets

Cited by 2 publications

References 19 publications

Some Recent Research Directions in the Computably Enumerable Sets

Some Recent Research Directions in the Computably Enumerable Sets

On Splits of Computably Enumerable Sets

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