## An extension of the recursively enumerable Turing degrees (2006)

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Venue: | Journal of the London Mathematical Society |

Citations: | 22 - 16 self |

### BibTeX

@ARTICLE{Simpson06anextension,

author = {Stephen G. Simpson},

title = {An extension of the recursively enumerable Turing degrees},

journal = {Journal of the London Mathematical Society},

year = {2006},

volume = {75},

pages = {2007}

}

### OpenURL

### Abstract

Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with non-empty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally non-recursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is one-to-one, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.

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Citation Context ...D25, 03D30, 68Q30. This research was partially supported by NSF grant DMS-0070718.s288 STEPHEN G. SIMPSON reals. The concept of 1-randomness was already well known from algorithmic information theory =-=[17]-=-. After 1999, we and other authors [2–5, 30–32] continued the study of Pw, using priority arguments to prove structural properties, just as for RT. In addition, we [31] discovered families of specific... |

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Citation Context ...es in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1. Introduction A principal object of study in recursion theory going back to the seminal work of Turing =-=[34]-=- and Post [23] has been the countable upper semilattice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of ... |

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Citation Context ...been the countable upper semilattice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [25], Rogers =-=[24]-=-, Soare [33], and Odifreddi [20, 21]. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich... |

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Citation Context ...countable upper semilattice RT of recursively enumerable Turing degrees, i.e., Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [27], Rogers [26], Soare =-=[35]-=-, and Odifreddi [22, 23]. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich, to this da... |

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Citation Context ... f ∈ � ∞ i=0 U C i .IfC =0(n−1) = the (n−1)th Turing jump of 0, where 0 is recursive and n � 1, then f is said to be n-random. Thus f is 1-random if and only if f is random in the sense of Martin-Löf =-=[18]-=-, and f is 2-random if and only if f is random relative to the halting problem. For a thorough treatment of randomness and n-randomness, see Kautz [13] or Downey and Hirschfeldt [7]. We write Note tha... |

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Citation Context ...tice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [25], Rogers [24], Soare [33], and Odifreddi =-=[20, 21]-=-. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich, to this day no specific, natural e... |

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Citation Context ...nse of Martin-Löf [18], and f is 2-random if and only if f is random relative to the halting problem. For a thorough treatment of randomness and n-randomness, see Kautz [13] or Downey and Hirschfeldt =-=[7]-=-. We write Note that μ(Rn)=1. Rn = {f ∈ 2 ω | f is n-random}. Lemma 3.2. Rn is Σ 0 n+1. In particular, R1 is Σ 0 2, and R2 is Σ 0 3. Proof. It is well known (see, for instance, [31, Theorem 8.3]) that... |

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Citation Context ...t 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1. Introduction A principal object of study in recursion theory going back to the seminal work of Turing [34] and Post =-=[23]-=- has been the countable upper semilattice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [25], Ro... |

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Citation Context ...ntable upper semilattice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [25], Rogers [24], Soare =-=[33]-=-, and Odifreddi [20, 21]. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich, to this da... |

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Citation Context ...ost [23] has been the countable upper semilattice RT of recursively enumerable Turing degrees, that is, Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks =-=[25]-=-, Rogers [24], Soare [33], and Odifreddi [20, 21]. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and stru... |

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Citation Context ...d only if f is random in the sense of Martin-Löf [18], and f is 2-random if and only if f is random relative to the halting problem. For a thorough treatment of randomness and n-randomness, see Kautz =-=[13]-=- or Downey and Hirschfeldt [7]. We write Note that μ(Rn)=1. Rn = {f ∈ 2 ω | f is n-random}. Lemma 3.2. Rn is Σ 0 n+1. In particular, R1 is Σ 0 2, and R2 is Σ 0 3. Proof. It is well known (see, for ins... |

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Citation Context ...y 0 = deg w(2 ω ) is the bottom element of Pw. Let PA be the set of completions of Peano arithmetic. Identifying sentences with their Gödel numbers, we may view PA as a Π 0 1 subset of 2 ω . By Scott =-=[27]-=-, deg w(PA) = 1 is the top element of Pw; see also [31, Section 6]. Remark 2.11. Just like the countable semilattice RT, the countable distributive lattice Pw is known to be structurally rich. Binns a... |

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(Show Context)
Citation Context ... algorithmic information theory [17]. After 1999, we and other authors [2–5, 30–32] continued the study of Pw, using priority arguments to prove structural properties, just as for RT. In addition, we =-=[31]-=- discovered families of specific, natural, intermediate degrees in Pw related to foundationally interesting topics such as reverse mathematics, Gentzen-style proof theory, and computational complexity... |

36 |
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Citation Context ...w analog of the Sacks Splitting Theorem for RT [27]. Namely, for all p, q > 0 in Pw there exist q1, q2 ∈Pw such that q1, q2 �≥ p and sup(q1, q2) =q. (The Pw analog of the Sacks Density Theorem for RT =-=[28]-=- remains as an open problem.) These structural results for Pw are proved by means of priority arguments. They invite comparison with the older, known results for RT , which were also proved by means o... |

34 | A splitting theorem for the Medvedev and Muchnik lattices
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Citation Context ...Pw is the specific degree r1 associated with the set of 1-random reals. The concept of 1-randomness was already well 2sknown from algorithmic information theory [19]. After 1999, we and other authors =-=[2, 3, 4, 5, 32, 33, 34]-=- continued the study of Pw, using priority arguments to prove structural properties, just as for RT . In addition, we [33] discovered families of specific, natural, intermediate degrees in Pw related ... |

33 | Almost everywhere domination - Dobrinen, Simpson |

24 |
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Citation Context ...The set Dw ofsRECURSIVELY ENUMERABLE TURING DEGREES 289 all weak degrees is partially ordered by putting degw(P) � degw(Q) if and only if P �w Q. The concept of weak reducibility goes back to Muchnik =-=[19]-=- and has sometimes been called Muchnik reducibility. Theorem 2.2. Dw is a complete distributive lattice. Proof. The least upper bound of deg w(P) and deg w(Q) inDw is deg w(P × Q), where P × Q = {f ⊕ ... |

24 |
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Citation Context ...ive if g(n) �= {n}(n) for all n ∈ ω. We put d =deg w(DNR) where DNR = {g ∈ ω ω | g is diagonally nonrecursive}. 9sThe Turing degrees of diagonally nonrecursive functions have been studied by Jockusch =-=[12]-=-. In particular, a Turing degree contains a diagonally nonrecursive function if and only if it contains a fixed point free function, if and only if it contains an effectively immune set, if and only i... |

22 | Embeddings into the Medvedev and Muchnik lattices of Π01 classes, Archive for Mathematical Logic 43
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Citation Context ... = 1 is the top element of Pw; see also [31, Section 6]. Remark 2.11. Just like the countable semilattice RT, the countable distributive lattice Pw is known to be structurally rich. Binns and Simpson =-=[2, 5]-=- have shown that every countable distributive lattice is lattice embeddable in every non-trivial initial segment of Pw. Binns [2, 3] has obtained the Pw analogue of the Sacks splitting theorem for RT ... |

21 |
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Citation Context ...∈ TP, and ρ(n) � max(n, 2) for all n less than the length of ρ.ThusTQ is a recursive subtree of ω <ω . Let Q ⊆ ω ω be the set of paths through TQ. (Compare the construction of T in Jockusch and Soare =-=[11]-=-.) It is straightforward to verify that Q ≡w S ∪ P. Note that Q is Π 0 1 and recursively bounded. By Theorem 2.7, we can find a Π 0 1 set Q ∗ ⊆ 2 ω which is recursively homeomorphic to Q. This complet... |

18 | Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion - Jockusch, Lerman, et al. - 1989 |

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Citation Context ...minimal Turing degree. But if f ∈ 2 ω is 1-random, then f is not of minimal Turing degree, because the functions g and h defined by f = g ⊕ h are Turing incomparable (see, for instance, van Lambalgen =-=[16]-=-). This proves that d � r1. An alternative reference for the conclusion d � r1 is Ambos-Spies et al. [1, Theorems 1.4 and 2.1]. Definition 4.4. Let DNRREC be the set of recursively bounded DNR functio... |

14 | Located sets and reverse mathematics - Giusto, Simpson |

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Citation Context ...S, σ0,σ1,...,σk ∈ TP ,andρ(n) ≤ max(n, 2) for all n<the length of ρ. ThusTQ is a recursive subtree of ω <ω .LetQ ⊆ ω ω be the set of paths through TQ. (Compare the construction of T in Jockusch/Soare =-=[14]-=-.) It is straightforward to verify that Q ≡w S ∪ P . Note that Q is Π 0 1 and recursively bounded. By Theorem 2.7 we can find a Π 0 1 set Q∗ ⊆ 2 ω which is recursively homeomorphic to Q. This complete... |

11 | Recursive Function Theory - Dekker, editor |

9 |
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Citation Context ... = 1 is the top element of Pw; see also [31, Section 6]. Remark 2.11. Just like the countable semilattice RT, the countable distributive lattice Pw is known to be structurally rich. Binns and Simpson =-=[2, 5]-=- have shown that every countable distributive lattice is lattice embeddable in every non-trivial initial segment of Pw. Binns [2, 3] has obtained the Pw analogue of the Sacks splitting theorem for RT ... |

8 |
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Citation Context ...or all n ∈ ω. Weputd = deg w(DNR), where DNR = {g ∈ ω ω | g is diagonally non-recursive}.s294 STEPHEN G. SIMPSON The Turing degrees of diagonally non-recursive functions have been studied by Jockusch =-=[9]-=-. In particular, a Turing degree contains a diagonally non-recursive function if and only if it contains a fixed point free function and an effectively immune and biimmune set. Thus, we see that the w... |

8 | A fixed point free minimal degree
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Citation Context ...DNR has no recursive member, i.e., d > 0. By Giusto/Simpson [11, Lemma 6.18], for all f ∈ R1 there exists g ≤T f such that g ∈ DNR. Thus we have 0 < d ≤ r1. It remains to show that d �≥ r1. By Kumabe =-=[18]-=- there is a diagonally nonrecursive function which is of minimal Turing degree. But if f ∈ 2ω is 1-random, then f is not of minimal Turing degree, because the functions g and h defined by f = g ⊕ h ar... |

6 |
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Citation Context ...w analogue of the Sacks splitting theorem for RT [25]. Namely, for all p, q > 0 in Pw there exist q1, q2 ∈Pw such that q1, q2 � p and sup(q1, q2)=q. (The Pw analog of the Sacks density theorem for RT =-=[26]-=- remains as an open problem.) These structural results for Pw are proved by means of priority arguments. They invite comparison with the older, known results for RT, which were also proved by means of... |

6 |
FOM: natural r.e. degrees; Pi01 classes. FOM e-mail list [10
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Citation Context ...zed that most other branches of mathematics are motivated and nurtured by a rich stock of examples. Clearly, it ought to be of interest to somehow overcome this deficiency in the study of RT. In 1999 =-=[28, 29]-=-, we defined a degree structure, here denoted by Pw, which is closely related to RT, but superior to RT in at least two respects. First, Pw exhibits better structural behaviour than RT, in the sense t... |

6 |
classes. Archive for Mathematical Logic
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Citation Context ...Pw is the specific degree r1 associated with the set of 1-random reals. The concept of 1-randomness was already well 2sknown from algorithmic information theory [19]. After 1999, we and other authors =-=[2, 3, 4, 5, 32, 33, 34]-=- continued the study of Pw, using priority arguments to prove structural properties, just as for RT . In addition, we [33] discovered families of specific, natural, intermediate degrees in Pw related ... |

5 | Comparing DNR and WWKL - Ambos-Spies, Kjos-Hanssen, et al. - 2004 |

5 |
Classical Recursion Theory. Number 125
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Citation Context ...lattice RT of recursively enumerable Turing degrees, i.e., Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [27], Rogers [26], Soare [35], and Odifreddi =-=[22, 23]-=-. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich, to this day no specific, natural e... |

5 |
Classical Recursion Theory, Volume 2. Number 143
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(Show Context)
Citation Context ...lattice RT of recursively enumerable Turing degrees, i.e., Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks [27], Rogers [26], Soare [35], and Odifreddi =-=[22, 23]-=-. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and structurally rich, to this day no specific, natural e... |

5 |
Degrees of Unsolvability. Number 55 in Annals of Mathematics Studies
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Citation Context ...d Post [25] has been the countable upper semilattice RT of recursively enumerable Turing degrees, i.e., Turing degrees of recursively enumerable sets of positive integers. See the monographs of Sacks =-=[27]-=-, Rogers [26], Soare [35], and Odifreddi [22, 23]. A major difficulty or obstacle in the study of RT has been the lack of natural examples. Although it has long been known that RT is infinite and stru... |

5 |
Medvedev degrees of Π 0 1 subsets of 2 ω
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(Show Context)
Citation Context ...Pw is the specific degree r1 associated with the set of 1-random reals. The concept of 1-randomness was already well 2sknown from algorithmic information theory [19]. After 1999, we and other authors =-=[2, 3, 4, 5, 32, 33, 34]-=- continued the study of Pw, using priority arguments to prove structural properties, just as for RT . In addition, we [33] discovered families of specific, natural, intermediate degrees in Pw related ... |

4 |
FOM: priority arguments; Kleene-r.e. degrees; Pi01 classes. FOM e-mail list [10
- Simpson
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(Show Context)
Citation Context ...zed that most other branches of mathematics are motivated and nurtured by a rich stock of examples. Clearly, it ought to be of interest to somehow overcome this deficiency in the study of RT. In 1999 =-=[28, 29]-=-, we defined a degree structure, here denoted by Pw, which is closely related to RT, but superior to RT in at least two respects. First, Pw exhibits better structural behaviour than RT, in the sense t... |

3 | Π 0 1 classes - Binns, Small |

3 |
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Citation Context ...of T2. This follows from Theorem 3.6 plus the well-known, general relationship between Π 0 1 subsets of 2 ω and finitely axiomatizable theories; see [30, Theorem 3.18 and Remark 3.19] and Peretyatkin =-=[22]-=-. Theorem 3.8. We can characterize r1 as the maximum weak degree of a Π 0 1 subset of 2 ω of positive measure. We can characterize r ∗ 2 as the maximum weak degree of a Π 0 1 subset of 2 ω whose Turin... |

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Citation Context ...Pw is the specific degree r1 associated with the set of 1-random reals. The concept of 1-randomness was already well 2known from algorithmic information theory [19]. After 1999, we and other authors =-=[2, 3, 4, 5, 32, 33, 34]-=- continued the study of Pw, using priority arguments to prove structural properties, just as for RT. In addition, we [33] discovered families of specific, natural, intermediate degrees in Pw related t... |

2 |
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(Show Context)
Citation Context ...as no recursive member, that is, d > 0. By Giusto and Simpson [8, Lemma 6.18], for all f ∈ R1 there exists g �T f such that g ∈ DNR. Thus we have 0 < d � r1. It remains to show that d � r1. By Kumabe =-=[15]-=-, there is a diagonally non-recursive function which is of minimal Turing degree. But if f ∈ 2 ω is 1-random, then f is not of minimal Turing degree, because the functions g and h defined by f = g ⊕ h... |

2 |
Finitely Axiomatizable Theories. Siberian School of Algebra and Logic
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Citation Context ... of T2. This follows from Theorem 3.6 plus the well known, general relationship between Π 0 1 subsets of 2ω and finitely axiomatizable theories. See [32, Theorem 3.18 and Remark 3.19] and Peretyatkin =-=[24]-=-. Theorem 3.8. We can characterize r1 as the maximum weak degree of a Π 0 1 subset of 2 ω of positive measure. We can characterize r ∗ 2 as the maximum weak degree of a Π 0 1 subset of 2ω whose Turing... |

2 |
The Medvedev and Muchnik
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Citation Context ...Pw is the specific degree r1 associated with the set of 1-random reals. The concept of 1-randomness was already well 2known from algorithmic information theory [19]. After 1999, we and other authors =-=[2, 3, 4, 5, 32, 33, 34]-=- continued the study of Pw, using priority arguments to prove structural properties, just as for RT. In addition, we [33] discovered families of specific, natural, intermediate degrees in Pw related t... |

1 |
A splitting theorem for the Medvedev and Muchnik
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Citation Context ...tive lattice Pw is known to be structurally rich. Binns and Simpson [2, 5] have shown that every countable distributive lattice is lattice embeddable in every non-trivial initial segment of Pw. Binns =-=[2, 3]-=- has obtained the Pw analogue of the Sacks splitting theorem for RT [25]. Namely, for all p, q > 0 in Pw there exist q1, q2 ∈Pw such that q1, q2 � p and sup(q1, q2)=q. (The Pw analog of the Sacks dens... |

1 | Medvedev degrees of Π0 1 subsets of 2ω ’, preprint, 2001, in preparation - Simpson, Slaman |

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