DDEGREES OF RANDOMIZED COMPUTABILITY
RUPERT H ¨OLZL AND CHRISTOPHER P. PORTER
Abstract.
In this survey we discuss work of Levin and V’yugin on collections of sequencesthat are non-negligible in the sense that they can be computed by a probabilistic algorithmwith positive probability. More precisely, Levin and V’yugin introduced an ordering oncollections of sequences that are closed under Turing equivalence. Roughly speaking, giventwo such collections A and B , A is less than B in this ordering if A \ B is negligible. Thedegree structure associated with this ordering, the
Levin-V’yugin degrees (or
LV-degrees ),can be shown to be a Boolean algebra, and in fact a measure algebra.We demonstrate the interactions of this work with recent results in computability theoryand algorithmic randomness: First, we recall the definition of the Levin-V’yugin algebra andidentify connections between its properties and classical properties from computability theory.In particular, we apply results on the interactions between notions of randomness and Turingreducibility to establish new facts about specific LV-degrees, such as the LV-degree of thecollection of 1-generic sequences, that of the collection of sequences of hyperimmune degree,and those collections corresponding to various notions of effective randomness. Next, weprovide a detailed explanation of a complex technique developed by V’yugin that allows theconstruction of semi-measures into which computability-theoretic properties can be encoded.We provide two examples of the use of this technique by explicating a result of V’yugin’sabout the LV-degree of the collection of Martin-L¨of random sequences and extending theresult to the LV-degree of the collection of sequences of DNC degree.
Contents
1. Introduction 22. Background 32.1. Some notation 32.2. Turing functionals and computable measures 42.3. Notions of algorithmic randomness 43. Negligibility and Non-negligibility 53.1. Left-c.e. semi-measures 53.2. Examples of Negligible and Non-Negligible Collections 84. The Levin-V’yugin Degrees 94.1. Elementary properties of the LV-degrees 104.2. Additional results about the LV-degrees 144.3. Open questions 185. How to build a Semi-measure 185.1. The General Template 205.2. Verification of the General Template 245.3. The Roadmap 276. Implementing the Template 286.1. A first example 286.2. A new application of the technique 32 a r X i v : . [ m a t h . L O ] J u l RUPERT H ¨OLZL AND CHRISTOPHER P. PORTER
7. Applications to Π classes 36Acknowledgments 36References 361. Introduction
The tools of algorithmic randomness have been particularly useful in studying the power ofrandom oracles in the context of Turing reducibility. It is well-known that access to a randomoracle does not aid in the computation of any individual sequence, as Sacks [Sac63] provedthat any sequence that is computable from positive measure many oracles must be computable.However, if instead we attempt to compute some element of a collection of sequences by meansof a random oracle, the situation is quite different.For instance, in unpublished work, Martin proved that the collection of sequences of hyper-immune degree has Lebesgue measure 1 (see Downey and Hirschfeldt [DH10, Theorem 8.21.1]).A careful examination of this proof yields, for any δ >
0, an algorithm which with probabilityat least 1 − δ computes from a random oracle a function not dominated by any computablefunction (as noted by G´acs and reported by Rumyantsev and Shen [RS14]). Other types ofsequences known to be computable from positive measure many sequences are the 1-genericsequences (as shown by Kurtz [Kur81] and Kautz [Kau91]), the sequences of DNC degree(first established by Kuˇcera [Kuˇc85]), and sequences satisfying certain algebraic properties inthe upper semi-lattice of the Turing degrees under Turing reducibility (studied by Barmpalias,Day, and Lewis-Pye [BDLP14]).Collections of sequences C ⊆ ω with the property that only measure 0 many sequencescompute an element of C have been referred to as negligible (for instance, in V’yugin [V’y82]and Levin [Lev84]), and thus those collections C with the property that positive measure manysequences compute an element of C are called non-negligible . The focus of our study here isa Boolean algebra of non-negligible subsets of 2 ω that are closed under Turing equivalenceand where two such subsets are identified with each other if they differ only by a negligibleset. This Boolean algebra, first introduced by Levin and V’yugin [LV77] and systematicallystudied by V’yugin [V’y82], will be referred to as the Levin-V’yugin algebra ; its elements willbe referred to as the
Levin-V’yugin degrees , or LV -degrees for short.A significant portion of this article is a survey of previously established results about theLevin-V’yugin algebra, but we also establish new facts about it as well. Much of our focus willfurthermore be on explicating a technique developed by V’yugin [V’y82] for building left-c.e.semi-measures, which has applications outside of the study of the algebra, such as in the studyof probabilistic computation. We first provide a general schematic account of this techniqueand then use it establish the following result.
Theorem 1.1 (V’yugin [V’y12]) . For any δ > , there is a probabilistic algorithm thatproduces with probability at least − δ a non-computable sequence that does not compute anyMartin-L¨of random sequence. We will then apply V’yugin’s technique to prove the following generalization of Theorem 1.1.
Theorem 1.2.
For any δ > , there is a probabilistic algorithm that produces with probabilityat least − δ a non-computable sequence that does not compute any sequence of DNC degree. EGREES OF RANDOMIZED COMPUTABILITY 3
Theorems 1.1 and 1.2 both follow from a result due to Kurtz [Kur81], namely that forevery δ >
0, there is a probabilistic algorithm that produces a 1-generic sequence withprobability 1 − δ . Since a 1-generic sequence can compute neither a Martin-L¨of randomsequence nor a sequence of DNC degree, the results follow. However, V’yugin’s techniquealso has implications for the study of Π classes, that is, effectively closed subsets of 2 ω : theprobabilistic algorithms whose existence can be shown using V’yugin’s technique are in factTuring functionals on 2 ω with a closed range; and since such a functional is effective, its rangeis even Π . Thus, V’yugin’s proof of Theorem 1.1 establishes the following stronger result. Corollary 1.3.
For every δ > , there is a Turing functional Φ such that (i) the domain of Φ has Lebesgue measure − δ , (ii) the range of Φ is a Π class, and (iii) no sequence in the range of Φ computes a Martin-L¨of random sequence. Similarly, the proof of Theorem 1.2 that we provide here establishes the following result.
Corollary 1.4.
For every δ > , there is a Turing functional Φ such that (i) the domain of Φ has Lebesgue measure − δ , (ii) the range of Φ is a Π class, and (iii) no sequence in the range of Φ computes a sequence of DNC degree. The remainder of this article is structured as follows. In Section 2, we review the necessarybackground. Section 3 introduces the notions of negligibility and non-negligibility and providesa number of examples from classical computability theory and algorithmic randomness. TheLevin-V’yugin degrees, defined in terms of negligibility, are introduced in Section 4. Thegeneral features of V’yugin’s technique for constructing semi-measures are initially laid out inSection 5, while specific examples of the technique are provided in Section 6. Lastly, in Section7 we conclude with a final observation about the connection between V’yugin’s technique andΠ classes. 2. Background
Some notation.
We fix the following notation and terminology. We denote the naturalnumbers by ω , and the set of infinite binary sequences, also known as Cantor space , by 2 ω .We denote the set of finite binary strings by 2 <ω and the empty string by ε . Let Q be the setof non-negative dyadic rationals, that is, rationals of the form m/ n for m, n ∈ ω .Given X ∈ ω and an integer n , X (cid:22) n is the string that consists of the first n bits of X , and X ( n ) is the ( n + 1) st bit of X (so that X (0) is the first bit of X ). If σ and τ are strings, then σ (cid:22) τ means that σ is an initial segment of τ . Similarly, for X ∈ ω , σ ≺ X means that σ isan initial segment of X .Given a string σ , the cylinder (cid:74) σ (cid:75) is the collection of elements of 2 ω having σ as an initialsegment. Similarly, given S ⊆ <ω , (cid:74) S (cid:75) is defined to be the collection (cid:83) σ ∈ S (cid:74) σ (cid:75) . The cylindersform a basis for the usual product topology on Cantor space, and thus the open sets forthis topology are those of the form (cid:74) S (cid:75) for some S . An open set U is said to be effectivelyopen (or Σ ) if U = (cid:74) S (cid:75) for some computably enumerable (hereafter, c.e.) set S ⊆ <ω . An effectively closed (or Π ) set is the complement of an effectively open set. A sequence of opensets ( U n ) n ∈ ω is said to be uniformly effectively open if there exists a sequence ( S n ) n ∈ ω ofuniformly c.e. sets of strings such that U n = (cid:74) S n (cid:75) for all n ∈ ω . RUPERT H ¨OLZL AND CHRISTOPHER P. PORTER
For
A ⊆ ω , we write ( A ) ≡ T for the closure of A under Turing equivalence, that is, we let( A ) ≡ T := { X ∈ ω : ( ∃ Y ∈ A ) X ≡ T Y } . Turing functionals and computable measures.
We assume that the reader is familiarwith the basics of computability theory (for instance, the material covered in Soare [Soa16,Chapters I-IV], Nies [Nie09, Chapter 1], or Downey and Hirschfeldt [DH10, Chapter 2]).
Definition 2.1. (i) A Turing functional Φ : ⊆ ω → ω is represented by a c.e. set S Φ of pairs of strings ( σ, τ ) such that if ( σ, τ ) , ( σ (cid:48) , τ (cid:48) ) ∈ S Φ and σ (cid:22) σ (cid:48) , then τ (cid:22) τ (cid:48) or τ (cid:48) (cid:22) τ . (ii) For each σ ∈ <ω , we define Φ σ to be the maximal string (in the order given by (cid:22) )in the set { τ : ( ∃ σ (cid:48) (cid:22) σ )(( σ (cid:48) , τ ) ∈ S Φ ) } ∪ { ε } . Similarly, for each s ∈ ω , Φ σs isthe maximal string in the set { τ : ( ∃ σ (cid:48) (cid:22) σ )(( σ (cid:48) , τ ) ∈ S Φ [ s ]) } ∪ { ε } , where S Φ [ s ] is theapproximation of the c.e. set S Φ at stage s . (iii) Let Φ X be the minimal (in the order given by (cid:22) ) z ∈ <ω ∪ ω such that Φ X (cid:22) n (cid:22) z forall n . (iv) We set dom(Φ) = { X ∈ ω : Φ X ∈ ω } . (v) For τ ∈ <ω , let Φ − ( τ ) be { σ ∈ <ω : ∃ τ (cid:48) (cid:23) τ : ( σ, τ (cid:48) ) ∈ Φ } . (vi) Lastly, for
A ⊆ ω , let Φ − ( A ) be { X ∈ ω : Φ X ∈ A} . When Φ X ∈ ω , we will often write Φ X as Φ( X ) to emphasize that we view the functional Φas a (partial) map from 2 ω to 2 ω .A measure µ on 2 ω is computable if σ (cid:55)→ µ ( (cid:74) σ (cid:75) ) is computable as a real-valued function,that is, if there is a computable function (cid:101) µ : 2 <ω × ω → Q such that | µ ( (cid:74) σ (cid:75) ) − (cid:101) µ ( σ, i ) | ≤ − i for every σ ∈ <ω and i ∈ ω . For all measures appearing in this article we assume that µ (2 ω ) ≤ µ ( (cid:74) σ (cid:75) ) as µ ( σ ). By Carath´eodory’sTheorem, if the values µ ( σ ), for σ ∈ <ω , of a measure µ on 2 ω are fixed, then there is aunique extension of µ to the Borel σ -algebra generated by the sets (cid:74) σ (cid:75) , for σ ∈ <ω . In thisarticle, all measures will be defined in this way, which implies in particular that the same setsare measurable for each of these measures.The uniform (or Lebesgue) measure λ is the probability measure for which each bit of thesequence has value 0 with probability 1 /
2, independently of the values of the other bits. Itcan be defined as the unique Borel measure such that λ ( σ ) = 2 −| σ | for all strings σ . Clearly, λ is a computable measure.2.3. Notions of algorithmic randomness.
The primary notion of algorithmic randomnessthat we will consider in this study is Martin-L¨of randomness.
Definition 2.2. (i) A Martin-L¨of test is a sequence ( U i ) i ∈ ω of uniformly effectively open subsets of ω suchthat for each i , λ ( U i ) ≤ − i . (ii) X ∈ ω passes the Martin-L¨of test ( U i ) i ∈ ω if X / ∈ (cid:84) i ∈ ω U i . (iii) X ∈ ω is Martin-L¨of random , denoted X ∈ MLR , if X passes every Martin-L¨of test. We will also consider relative versions of Martin-L¨of randomness, obtained by relativizingthe above notion of a Martin-L¨of test to some oracle A ∈ ω ; such a class will be written asMLR A . For A = ∅ ( n ) , the resulting notion of randomness is known as ( n + 1)-randomness.Other randomness notions can be obtained as follows. EGREES OF RANDOMIZED COMPUTABILITY 5
Definition 2.3.
Let X ∈ ω . (i) X is Schnorr random if and only if X passes every Martin-L¨of test ( U i ) i ∈ ω such that λ ( U i ) is computable uniformly in i ∈ ω . (i) X is Kurtz random (or weakly 1-random ) if and only if X is not contained in any Π class of Lebesgue measure . (ii) X is weakly 2-random if and only if X is not contained in any Π class of Lebesguemeasure . (iii) X is difference random if and only if it is Martin-L¨of random and not Turing complete. Let SR and KR denote the collections of Schnorr random and Kurtz random sequences,respectively.Each of the above notions of tests and randomness can also be formulated for arbitrarycomputable measures µ on 2 ω simply by replacing the Lebesgue measure λ in the respectivedefinitions by µ . Thus, for instance, for a fixed computable measure µ , a sequence X is µ -Martin-L¨of random, denoted X ∈ MLR µ , if and only if X is not contained in any µ -Martin-L¨of test. Significantly, Martin-L¨of randomness with respect to some computable measure isTuring invariant in the following sense. Theorem 2.4 (Levin, Zvonkin [LZ70]; Kautz [Kau91]) . For every computable measure µ andfor every non-computable X ∈ MLR µ , there is some Y ∈ MLR such that X ≡ T Y . The requirement that X be non-computable is necessary since every computable sequence X is random with respect to some computable measures on 2 ω , for example the measure δ X defined for A ⊆ ω via δ X ( A ) = (cid:40) X ∈ A , Negligibility and Non-negligibility
To define the notions of negligibility and non-negligibility, we need to review the definitionof left-c.e. semi-measures, which were initially introduced by Solomonoff [Sol64a, Sol64b] andfirst systematically studied by Levin and Zvonkin [LZ70].3.1.
Left-c.e. semi-measures.Definition 3.1. A semi-measure is a function P : 2 <ω → [0 , that satisfies (i) P ( ε ) = 1 , (ii) P ( σ ) ≥ P ( σ
0) + P ( σ for every σ ∈ <ω .In addition, P is left-c.e. if P ( σ ) is the limit of a computable, non-decreasing sequence ofrationals, uniformly in σ ∈ <ω . Functions satisfying conditions (i) and (ii) above are sometimes referred to in the algorithmicrandomness literature as continuous semi-measures to distinguish them from discrete semi-measures. As we do not consider discrete semi-measures in this study, we will not make thisdistinction below.In Section 6, the support of a semi-measure will play an important role.
Definition 3.2.
The support of a semi-measure P , denoted Supp( P ) is the collection ofsequences { X ∈ ω : ∀ n P ( X (cid:22) n ) > } . RUPERT H ¨OLZL AND CHRISTOPHER P. PORTER
It is not immediately clear how to extend semi-measures to Borel subsets of 2 ω . Levin andV’yugin [LV77] proposed the following way of deriving measures from left-c.e. semi-measures. Definition 3.3.
Given a left-c.e. semi-measure P and σ ∈ <ω we define P ( σ ) = inf n (cid:88) σ (cid:22) τ ∧ | τ | = n P ( τ ) .P can be extended to a measure on 2 ω , which we will also write as P , by letting P ( (cid:74) σ (cid:75) ) = P ( σ ) and then applying Carath´eodory’s theorem. One can show inductivelythat P is the maximal measure such that P ( σ ) ≤ P ( σ ) for every σ ∈ <ω (see, for instance,Bienvenu et al. [BHPS17, Proposition 6.5]). As a consequence, P is typically not a probabilitymeasure.Inversely, given any computable measure µ defined on 2 ω , we can identify it with the left-c.e.semi-measure σ (cid:55)→ µ ( (cid:74) σ (cid:75) ) defined on 2 <ω ; then we have µ = µ .An important property of left-c.e. continuous semi-measures is the following. Theorem 3.4 (Levin, Zvonkin [LZ70]) . (i) For every Turing functional Φ , the function λ Φ defined for every σ ∈ <ω via λ Φ ( σ ) = λ ( (cid:74) Φ − ( σ ) (cid:75) ) = λ ( { X ∈ ω : Φ X (cid:23) σ } ) , where Φ X ∈ ω ∪ <ω , is a left-c.e. semi-measure. (ii) For every left-c.e. semi-measure P , there is a Turing functional Φ such that P = λ Φ . Using Theorem 3.4 one can derive an alternative characterization of P for any left-c.e.semi-measure P . Proposition 3.5.
Let P be a left-c.e. semi-measure. Then P ( σ ) = λ ( { X ∈ ω : Φ X ∈ ω ∧ Φ X (cid:23) σ } ) , where Φ is as in Theorem 3.4 (ii). Moreover, for measurable A ⊆ ω , Carath´eodory’s theoremimplies that P ( A ) = λ (Φ − ( A )) . For a proof of the first part of the proposition, see Bienvenu et al. [BHPS17, Proposition 6.5].
Theorem 3.6 (Levin, Zvonkin [LZ70]) . There is a universal left-c.e. semi-measure, that is, aleft-c.e. semi-measure M such that for every left-c.e. semi-measure P , there is some constant c such that P ( σ ) ≤ c · M ( σ ) for every σ ∈ <ω .Remark . (i) One way to define a universal semi-measure is via a universal functional. For instance,for an effective enumeration (Φ e ) e ∈ ω of all Turing functionals, we can define Φ : 2 ω → ω via Φ(1 e X ) = Φ e ( X ) for each e ∈ ω and X ∈ ω . It is not hard to verify that λ Φ isuniversal. EGREES OF RANDOMIZED COMPUTABILITY 7 (ii) For every left-c.e. semi-measure P , there is some c such that P ( σ ) ≤ c · M ( σ ) . To see this, observe that for the c appearing in Theorem 3.6 we have P ( σ ) = inf n (cid:88) σ (cid:22) τ ∧ | τ | = n P ( τ ) ≤ inf n (cid:88) σ (cid:22) τ ∧ | τ | = n c · M ( τ ) = c · M ( σ ) . (iii) From (ii) and a straightforward argument using open covers of null sets, we can derivethe conclusion that for every left-c.e. semi-measure P , P is absolutely continuous withrespect to M , that is, if M ( B ) = 0 then P ( B ) = 0 for every measurable set B .Using a universal semi-measure we can provide an alternative characterization of µ -Martin-L¨of randomness for each computable measure µ . Theorem 3.8 (Levin [Lev74]; Schnorr, see Chaitin [Cha75]) . Let µ be a computable measure.Then X ∈ MLR µ if and only if there is some c such that µ ( X (cid:22) n ) ≥ c · M ( X (cid:22) n ) for every n . We can now define the notion of negligibility.
Definition 3.9.
We say that
B ⊆ ω is negligible if M ( B ) = 0 . As a consequence of Remark 3.7 (iii) we obtain the following corollary.
Corollary 3.10.
Let P be a left-c.e. semi-measure and B ⊆ ω a negligible collection ofsequences. Then P ( B ) = 0 . In particular, µ ( B ) = 0 for every computable measure µ . Negligibility of a collection can alternatively be characterized by stipulating that no Turingfunctional produce an element of that collection with positive probability, as the followingproposition shows.
Proposition 3.11.
Let (Φ i ) i ∈ ω be an effective enumeration of all Turing functionals. Then ameasurable B ⊆ ω is negligible if and only if λ (cid:32)(cid:91) i ∈ ω Φ − i ( B ) (cid:33) = 0 . Proof. ( ⇒ ) Suppose that λ (cid:16)(cid:83) i ∈ ω Φ − i ( B ) (cid:17) >
0. Then there is some i such that λ (Φ − i ( B )) > P ( σ ) = λ ( (cid:74) Φ − i ( σ ) (cid:75) ) for σ ∈ <ω , it follows from Theorem 3.4 (i) that P is a left-c.e.semi-measure. Moreover, we have P ( B ) = λ (Φ − i ( B )) by Proposition 3.5 and thus P ( B ) > M ( B ) >
0, so B is not negligible.( ⇐ ) Let Φ be a Turing functional such that M = λ Φ , which exists by Theorem 3.4 (ii). If B isnot negligible, then we have 0 < M ( B ) = λ (Φ − ( B )) by Proposition 3.5, and hence λ (cid:16)(cid:91) i ∈ ω Φ − i ( B ) (cid:17) > . (cid:3) Intuitively, a collection of sequences is negligible if one of its elements cannot be obtainedwith positive probability by any probabilistic algorithm. Indeed, we can see a probabilisticalgorithm as consisting of two steps: First we generate infinitely many random bits, thenwe feed them to some Turing functional to produce the desired output. More formally, wecan think of a probabilistic algorithm as given by applying a Turing functional Φ to somerandom sequence. In this case, we can probabilistically compute an element of some fixed
RUPERT H ¨OLZL AND CHRISTOPHER P. PORTER collection B with positive probability if there are positive measure many sequences X suchthat Φ( X ) ∈ B . Proposition 3.11 tells us that the existence of such a probabilistic algorithmto compute elements of B with positive probability is equivalent to the non-negligibility of B .We conclude this subsection with a brief discussion of the atoms of a semi-measure. Definition 3.12.
Let P be a semi-measure. X ∈ ω is an atom of P if there is some δ > such that P ( X (cid:22) n ) > δ for all n . Lemma 3.13.
Let P be a semi-measure. X ∈ ω is an atom of P if and only if P ( { X } ) > .Proof. ( ⇒ ) If there is some δ > P ( X (cid:22) n ) > δ for all n , then for each n and each m ≥ n , (cid:88) X (cid:22) n (cid:22) τ ∧ | τ | = m P ( τ ) ≥ P ( X (cid:22) m ) > δ. It follows from the definition of P that P ( X (cid:22) n ) > δ for all n .( ⇐ ) P ( { X } ) > δ > P ( X (cid:22) n ) > δ for all n . Then, forall n , P ( X (cid:22) n ) ≥ P ( X (cid:22) n ) > δ. (cid:3) Proposition 3.14 (Bienvenu et al. [BHPS17]) . Let P be a left-c.e. semi-measure. If X is anatom of P , then X is computable. Examples of Negligible and Non-Negligible Collections.
We now provide a num-ber of examples of negligible and non-negligible collections of sequences, where the first set ofexamples is given by a classical theorem of Sacks.
Theorem 3.15 (Sacks [Sac63]) . For X ∈ ω , λ ( { Y ∈ ω : Y ≥ T X } ) > if and only if X iscomputable. That is, { X } is non-negligible if and only if X is computable. Arbitrary subsets of 2 ω of positive Lebesgue measure are further trivial examples of non-negligible collections. Thus, each of the notions of randomness defined above in Subsection 2.3forms a non-negligible collection.We can find more interesting examples by considering naturally occurring collections ofTuring degrees. We briefly review some of these collections. First, a sequence has PA degree if it computes a consistent completion of Peano arithmetic. A sequence X ∈ ω is high (orhas high Turing degree ) if and only if { X ∈ ω : X (cid:48)(cid:48) ≥ T ∅ (cid:48) } . A sequence X ∈ ω is iffor every c.e. S ⊆ <ω , there is some σ ≺ X such that either σ ∈ S or for all τ (cid:23) σ , τ / ∈ S .Similarly, X ∈ ω is if for every ∅ (cid:48) -c.e. S ⊆ <ω , there is some σ ≺ X such thateither σ ∈ S or for all τ (cid:23) σ , τ / ∈ S . Next, X ∈ ω has hyperimmune-free degree if and onlyif every X -computable function is dominated by some computable function. Accordingly, X has hyperimmune degree if and only if X computes a function that is not dominated byany computable function. X ∈ ω has DNC degree if and only if there is some f ≤ T X suchthat f ( e ) (cid:54) = ϕ e ( e ) for all e ∈ ω . Lastly, X is generalized low (or is in GL ) if and only if X (cid:48) ≡ T X ⊕ ∅ (cid:48) .To establish the negligibility or non-negligibility of the various collections given above, wewill use the following heuristic principles, which are justified by Proposition 3.11.( P ) If every sufficiently random sequence computes an element of some measurable
B ⊆ ω ,then B is non-negligible. ( P ) If no sufficiently random sequence computes an element of some measurable
B ⊆ ω ,then B is negligible. EGREES OF RANDOMIZED COMPUTABILITY 9
Proposition 3.16.
The following collections are non-negligible: (i) the collection of sequences of DNC degree, (ii) the collection of 1-generic sequences, (iii) the collection of sequences of hyperimmune degree, and (iv) the collection of generalized low sequences.Proof.
To show that each of the above collections is non-negligible, we apply ( P ) by identifyinga notion of randomness such that every sequence that is random in the respective sense computesan element of the given collection. For (i), Kuˇcera [Kuˇc85] proved that every Martin-L¨ofrandom sequence has DNC degree. For (ii), Kautz [Kau91] established that every 2-randomsequence computes a 1-generic. Since every 1-generic sequence has hyperimmune degree, itfurther follows that every 2-random sequence computes a sequence of hyperimmune degree,yielding (iii). Lastly, for (iv), Kautz [Kau91] also proved that every 2-random sequence isgeneralized low. (cid:3) Proposition 3.17.
The following collections are negligible: (i) the collection of sequences of PA degree, (ii) the collection of sequences of high degree, (iii) the collection of 2-generic sequences, and (iv) the collection of noncomputable sequences of hyperimmune-free degree.Proof.
To show that each of the above collections is negligible, we apply ( P ) by identifying anotion of randomness such that no sequence that is random in the respective sense computesan element of the given collection. For (i), Franklin and Ng [FN11] extended work ofStephan [Ste02] to show that no difference random sequence computes a completion of PA.For (ii), Kautz [Kau91] established that no 3-random has high degree. As the high degreesare closed upwards under Turing reducibility, this implies that no 3-random computes asequence of high degree. For (iii), Nies, Stephan, and Terwijn [NST05] proved that every2-random sequence forms a minimal pair with every 2-generic in the Turing degrees, and sono 2-random computes a 2-generic. Lastly, for (iv), Lewis, Day, and Barmpalias [BDLP14,Theorem 5.1] showed that for every 2-random sequence X , every noncomputable Y ≤ T X computes a 1-generic sequence and therefore in particular a sequence of hyperimmune degree.So if any 2-random could compute a sequence of hyperimmune-free degree, then this sequencecould in turn compute a sequence of hyperimmune degree; a contradiction with the fact thathyperimmune-freeness is closed downwards under Turing reducibility. (cid:3) The Levin-V’yugin Degrees
Using the notion of negligibility, we can define a degree structure whose elements are givenby Turing invariant subsets of 2 ω . Recall that A ⊆ ω is Turing invariant if X ∈ A and Y ≡ T X imply Y ∈ A . Let I denote the set of measurable Turing invariant subsets of 2 ω .In what follows, all Turing invariant collections of sets that we consider are Borel and thusmeasurable. One can routinely verify that ( I , ∩ , ∪ , c ) is a Boolean algebra. We now define a reducibility ≤ LV on I . Definition 4.1.
Let A , B ∈ I . (i) A ≤ LV B if and only if A \ B is negligible. (ii)
A ≡ LV B if and only if A ≤ LV B and B ≤ LV A . Given A , B ∈ I , A ≤ LV B says that, for any probabilistic algorithm, the probability thatit produces an element of A that is not in B is 0. The stronger statement A < LV B saysin addition that there is some probabilistic algorithm such that the probability that it producesan element of B that is not in A is strictly positive. In this sense, the larger a collection of setsis with regards to the given order, the easier it is to probabilistically produce an element of it.It is well-known that a Boolean algebra modulo an equivalence relation is still a Booleanalgebra. Thus, D LV = I / ≡ LV is a Boolean algebra, which we refer to as the Levin-V’yuginalgebra . In fact, D LV is a measure algebra, since it is a Boolean algebra of measurable setsmodulo M -null sets. Individual elements of D LV will be referred to as LV -degrees . We willwrite LV-degrees as a , b , . . . and so on. For A ∈ I , deg LV ( A ) denotes the LV-degree of A .Given LV-degrees a and b and any A ∈ a and B ∈ b , we define a ∧ b := deg LV ( A ∩ B ), a ∨ b := deg LV ( A ∪ B ), and a c := deg LV (2 ω \ A ).It is straightforward to verify that these are well-defined. In addition, the following isimmediate. Proposition 4.2. (i)
The bottom element of D LV consists of the Turing invariant negligible subsets of ω . (ii) The top element of D LV consists of all Turing invariant A ⊆ ω such that ω \ A isnegligible. Elementary properties of the LV -degrees. Recall that A is an atom of a Booleanalgebra B if there are no elements A , A ∈ B \ { } such that A = A ∨ A and A ∧ A = 0. Toavoid confusion with the atoms of a semi-measure, we will hereafter refer to atoms of D LV as D LV -atoms . As reported by V’yugin [V’y82] in results attributed to Levin, two D LV -atoms arereadily identifiable: the LV-degree of the computable sequences, denoted c , and the LV-degreeof the Martin-L¨of random sequences, denoted r . We provide the proofs of these results here.For A ⊆ ω , let Spec T ( A ) = { deg T ( X ) : X ∈ A} be the Turing degree spectrum of A . Thefollowing basic fact will be useful. Lemma 4.3.
Given a , a ∈ D LV such that a ∧ a = , there are A , A ∈ I such that (i) Spec T ( A ) ∩ Spec T ( A ) = ∅ and (ii) deg LV ( A ) = a and deg LV ( A ) = a .Furthermore, for any given A ∈ I satisfying deg LV ( A ) = a ∨ a , we can w.l.o.g. assume that (iii) A i ⊆ A for i = 0 , .Proof. The statement a ∧ a = says that if we pick any element B ∈ I of the equivalenceclass a and any element B ∈ I of the equivalence class a , then B ∩ B is negligible. Then A := B \ B ≡ LV B is in the equivalence class a , A := B \ B ≡ LV B is in a , and since B and B are closed under Turing equivalence we also have Spec T ( A ) ∩ Spec T ( A ) = ∅ .To verify (iii), suppose that deg LV ( A ) = a ∨ a for some A ∈ I and let A (cid:48) and A (cid:48) satisfy conditions (i) and (ii) above. Then deg LV ( A ) = deg LV ( A (cid:48) ∪ A (cid:48) ), which implies that EGREES OF RANDOMIZED COMPUTABILITY 11 A ∆( A (cid:48) ∪ A (cid:48) ) is negligible. As A (cid:48) and A (cid:48) are disjoint, this implies that A (cid:48) i \ A is negligiblefor i = 0 ,
1. For i = 0 ,
1, setting A i = A (cid:48) i ∩ A , we have A (cid:48) i = ( A (cid:48) i ∩ A ) ∪ ( A (cid:48) i \ A ) = A i ∪ ( A (cid:48) i \ A ) . Thus, A (cid:48) i and A i differ only by a negligible set for i = 0 ,
1, and thus A and A satisfy (ii).Moreover, since A i ⊆ A (cid:48) i for i = 0 , A and A also satisfy (i). Thus, (iii) holds. (cid:3) Proposition 4.4. c is a D LV -atom.Proof. Suppose that c is not a D LV -atom. Then there are LV-degrees a , a > such that a ∧ a = and a ∨ a = c . Then, if we choose A in condition (iii) of Lemma 4.3 as thecollection of all computable sequences, there are A , A ∈ I satisfying all three conditions ofthat lemma. But clearly, conditions (i) and (iii) are in contradiction with each other in thiscase. (cid:3) Theorem 4.5. r is a D LV -atom. To prove Theorem 4.5, we will need to draw upon several classical results from measuretheory, as well as several auxiliary lemmata. Here we follow V’yugin’s general proof strategywhile filling in more details, especially in isolating and proving Lemma 4.6 below.As noted in Remark 3.7 (iii), for any left-c.e. semi-measure P , P is absolutely continuouswith respect to M . It follows by the Radon-Nikodym Theorem that there is an measurablefunction dPdM such that, for all measurable X ⊆ ω , P ( X ) = (cid:90) X dPdM ( X ) dM ( X ) . The Radon-Nikodym Theorem further guarantees that for any measurable f : 2 ω → R suchthat for all measurable X ⊆ ω the property P ( X ) = (cid:90) X f ( X ) dM ( X )holds, we have f ( X ) = dPdM ( X ) for M -almost every X ∈ ω . Lemma 4.6. dPdM ( X ) = lim n →∞ P ( X (cid:22) n ) M ( X (cid:22) n ) for M -almost every X ∈ ω .Proof. First, recall that for a measure µ on 2 ω , a µ -martingale is a function d : 2 <ω → R ≥ such that µ ( σ ) d ( σ ) = µ ( σ d ( σ
1) + µ ( σ d ( σ σ ∈ <ω (see, for instance, [Nie09, Chapter 7] or [DH10, Section 6.3] for a discussionof the role of martingales in the theory of algorithmic randomness).Now, observe that PM is an M -martingale. Indeed, for every σ ∈ <ω , M ( σ ) P ( σ ) M ( σ ) = P ( σ ) = P ( σ
0) + P ( σ
1) = M ( σ P ( σ M ( σ
0) + M ( σ P ( σ M ( σ . Thus lim n →∞ P ( X (cid:22) n ) M ( X (cid:22) n ) exists for M -almost every X ∈ ω by the martingale convergencetheorem. Thus, by the Radon-Nikodym theorem, we just need to show that P ( A ) = (cid:90) A lim n →∞ P ( X (cid:22) n ) M ( X (cid:22) n ) dM ( X ) ( † )for every clopen A ⊆ ω (which can then can be extended to every measurable A ⊆ ω ). Sincethere is some c such that P ( σ ) ≤ c · M ( σ ) for every σ ∈ <ω , we have for every n that P ( X (cid:22) n ) M ( X (cid:22) n ) ≤ c, and hence by the dominated convergence theorem,lim n →∞ (cid:90) A P ( X (cid:22) n ) M ( X (cid:22) n ) dM ( X ) = (cid:90) A lim n →∞ P ( X (cid:22) n ) M ( X (cid:22) n ) dM ( X ) . ( ‡ )Using ( † ), it now suffices to show that P ( A ) is equal to the left-hand side of ( ‡ ). For eachsufficiently large N , let A = (cid:83) ki =1 (cid:74) σ i (cid:75) for distinct σ , . . . , σ k ∈ N . Thenlim n →∞ (cid:90) A P ( X (cid:22) n ) M ( X (cid:22) n ) dM ( X ) = (cid:90) A P ( X (cid:22) N ) M ( X (cid:22) N ) dM ( X ) (1)= k (cid:88) i =1 (cid:90) (cid:74) σ i (cid:75) P ( X (cid:22) N ) M ( X (cid:22) N ) dM ( X ) (2)= k (cid:88) i =1 P ( (cid:74) σ i (cid:75) ) M ( (cid:74) σ i (cid:75) ) M ( (cid:74) σ i (cid:75) ) (3)= k (cid:88) i =1 P ( (cid:74) σ i (cid:75) ) = P ( A ) . (cid:3) Lemma 4.7 (V’yugin [V’y82]) . Let P be a left-c.e. semi-measure and suppose that for B ⊆ ω ,we have M ( B ) = 0 , where B = (cid:40) X ∈ B : dPdM ( X ) = 0 (cid:41) . Then P ( B ) = 0 implies that M ( B ) = 0 .Proof. By the hypothesis, 0 = P ( B \ B ) = (cid:90) B\B dPdM ( X ) dM ( X ) . Since dPdM ( X ) (cid:54) = 0 for every X ∈ B \ B , it follows that M ( B \ B ) = 0. Thus, M ( B ) = 0. (cid:3) It is well known that every martingale in the sense of algorithmic randomness (as given above) is a martingalein the classical sense, and thus the classical martingale convergence theorem is applicable. See Downey andHirschfeldt [DH10, Theorem 7.1.3] for a proof of an effective version of the martingale convergence theorem.
EGREES OF RANDOMIZED COMPUTABILITY 13
Lemma 4.8 (V’yugin [V’y82]) . Let µ be a computable measure, and let B ⊆
MLR µ be suchthat µ ( B ) = 0 . Then B is negligible.Proof. Since
B ⊆
MLR µ , by Theorem 3.8, for every X ∈ B , there is some c such that µ ( X (cid:22) n ) ≥ c · M ( X (cid:22) n )for every n . It follows that for all n , µ ( X (cid:22) n ) M ( X (cid:22) n ) = µ ( X (cid:22) n ) M ( X (cid:22) n ) ≥ µ ( X (cid:22) n ) M ( X (cid:22) n ) ≥ c. By Lemma 4.6, dµdM ( X ) (cid:54) = 0 for almost every X ∈ B , and so by Lemma 4.7 and the factthat µ ( B ) = 0, it follows that B is negligible. (cid:3) Lastly, we need one further classical result. Recall that
A ⊆ ω is a tailset if for all σ ∈ <ω and all Y ∈ ω with σY ∈ A we also have that τ Y ∈ A for every τ ∈ | σ | . That is, for atailset A , modifying a finite initial segment of an infinite binary sequence has no bearingon whether that sequence is an element of A or not. We will only use this result in thecontext of Cantor space; for a proof specific to that setting see Downey and Hirschfeldt [DH10,Theorem 1.2.4]. Theorem 4.9 (Kolmogorov’s 0-1 Law) . If A ⊆ ω is a measurable tailset, then λ ( A ) = 0 or λ ( A ) = 1 . We can now prove Theorem 4.5.
Proof of Theorem 4.5.
Suppose that r = a ∨ a and a ∧ a = for some a , a > . Let A , A ∈ I be collections of sequences as given by Lemma 4.3 where deg LV ( A i ) = a i and A i ⊆ (MLR) ≡ T for i = 0 ,
1. Note that for i = 0 ,
1, for each X ∈ A i there is some Y ∈ MLR ∩A i such that X ≡ T Y . Let us consider the subcollections of sequences A ∗ i = MLR ∩ A i for i = 0 , A i is non-negligible, it follows that λ (cid:32)(cid:91) e Φ − e ( A i ) (cid:33) > i = 0 ,
1. Since each X ∈ A i is Turing equivalent to some Y ∈ A ∗ i , it follows for i = 0 , (cid:91) e Φ − e ( A i ) = (cid:91) e Φ − e ( A ∗ i )and hence λ (cid:32)(cid:91) e Φ − e ( A ∗ i ) (cid:33) > . Then Proposition 3.11 and Lemma 4.8 imply that λ ( A ∗ i ) > i = 0 ,
1. But each A ∗ i is ameasurable tailset, so by Theorem 4.9 it follows that λ ( A ∗ i ) = 1 for i = 0 ,
1, which is impossibleas A ∗ and A ∗ are disjoint. (cid:3) Additional results about the LV -degrees. It is reasonable to ask whether the degree r ∨ c is the top degree in D LV . V’yugin gave a negative answer to this question by provingthat the complement of r ∨ c in D LV is non-negligible. We will give the details of his proof inSection 6, where it will provide the first instance of the technique of building semi-measuresthat we mentioned in the introduction. However, in this subsection, we provide a simplerproof of this result, and a number of new results about D LV .Given a ∈ D LV and A ⊆ ω such that deg LV (( A ) ≡ T ) = a , we say that A generates a orthat a is the LV-degree generated by A . We will use the following lemma repeatedly. Lemma 4.10.
Let A , B ⊆ ω be measurable sets. (i) If A \ B is negligible, then ( A ) ≡ T \ ( B ) ≡ T is also negligible. In particular, ( A ) ≡ T ≤ LV ( B ) ≡ T . (ii) If A ⊆ B , then ( A ) ≡ T ≤ LV ( B ) ≡ T .Proof. (i) First observe that ( A ) ≡ T \ ( B ) ≡ T ⊆ ( A \ B ) ≡ T . Indeed, given X ∈ ( A ) ≡ T \ ( B ) ≡ T ,there is some Y ≡ T X such that Y ∈ A and for all Z ∈ B , we have Z (cid:54)≡ T X . It followsthat Y / ∈ B , and hence X ∈ ( A \ B ) ≡ T .Now suppose that ( A ) ≡ T \ ( B ) ≡ T is non-negligible. By the above observation, ( A\B ) ≡ T is alsonon-negligible. For i, j ∈ ω define S i,j = { X ∈ ω : ( ∃ Y ∈ A \ B ) (Φ i ( Y ) = X ∧ Φ j ( X ) = Y ) } .Then we have ( A \ B ) ≡ T = (cid:91) ( i,j ) ∈ ω S i,j . Since (
A \ B ) ≡ T is non-negligible, there is some pair ( i, j ) ∈ ω such that S i,j is non-negligible.Then by Proposition 3.11, there is some Turing functional Ψ such that λ (Ψ − ( S i,j )) >
0. Bydefinition of S i,j , if Ψ( Z ) ∈ S i,j , then Φ j (Ψ( Z )) ∈ A \ B . Thus Ψ − ( S i,j ) ⊆ (Φ j ◦ Ψ) − ( A \ B ),and so λ ((Φ j ◦ Ψ) − ( A \ B )) >
0. Thus by Proposition 3.11,
A \ B is not negligible.(ii) If
A ⊆ B , then
A \ B = ∅ is trivially negligible. Thus by (i), ( A ) ≡ T ≤ LV ( B ) ≡ T . (cid:3) It is natural to ask how the LV-degree of the Martin-L¨of random Turing degrees comparesto the LV-degrees associated to other notions of algorithmic randomness. First we show thatthe LV-degree of the Schnorr random Turing degrees is also r . Theorem 4.11. deg LV ((SR) ≡ T ) = r .Proof. ( ⊇ ) MLR ⊆ SR, and thus by Lemma 4.10 (ii), (MLR) ≡ T ≤ LV (SR) ≡ T .( ⊆ ) We show that SR \ MLR is negligible, which by Lemma 4.10 (i) implies (SR) ≡ T ≤ LV (MLR) ≡ T . As shown by Nies, Stephan, and Terwijn [NST05], every X ∈ SR \ MLR has highdegree. But by Proposition 3.17, the collection of sequences of high degree is negligible. (cid:3)
Corollary 4.12.
Let R be any notion of algorithmic randomness such that MLR ⊆ R ⊆ SR .Then deg LV ((R) ≡ T ) = r . Proof.
By Lemma 4.10 (ii) and Theorem 4.11, we have r = deg LV ((MLR) ≡ T ) ≤ LV deg LV ((R) ≡ T ) ≤ LV deg LV ((SR) ≡ T ) = r . (cid:3) Thus, notions of randomness such as computable randomness, Kolmogorov-Loveland ran-domness, and the non-monotonic randomness notions studied in Bienvenu et al. [BHKM12] allare of LV-degree r . Similar results hold for notions of randomness stronger than Martin-L¨ofrandomness, as the following result shows. EGREES OF RANDOMIZED COMPUTABILITY 15
Theorem 4.13.
For every Z ∈ ω , deg LV ((MLR Z ) ≡ T ) = deg LV ((MLR) ≡ T ) .Proof. ( ⊇ ) MLR Z ⊆ MLR, and so by Lemma 4.10 (ii), (MLR Z ) ≡ T ≤ LV (MLR) ≡ T .( ⊆ ) We show that MLR \ MLR Z is negligible and apply Lemma 4.10 (i). Given any X ∈ MLR \ MLR Z , by the XYZ
Theorem of Miller and Yu [MY08], if X ≤ T Y ∈ MLR Z ,then X ∈ MLR Z . Thus no Y ∈ MLR Z computes any X ∈ MLR \ MLR Z . That is, no suffi-ciently random sequence computes an element of MLR \ MLR Z , and so by our heuristic ( P ),this latter collection is negligible. (cid:3) An immediate consequence of Theorem 4.13 is that for each n , the LV-degree of thecollection of n -random sequences is r . Another consequence is the following, the proof ofwhich is analogous to that of Corollary 4.12. Corollary 4.14.
Let R be any notion of algorithmic randomness such that MLR ∅ (cid:48) ⊆ R ⊆ MLR .Then deg LV ((R) ≡ T ) = r . It follows that notions of randomness such as difference randomness, Demuth randomness,and weak 2-randomness are of LV-degree r .We now show that r ∨ c is not the top LV-degree by exhibiting a LV-degree that isincomparable with it. Let g be the LV-degree generated by the collection of 1-genericsequences. By Proposition 3.16 this collection is non-negligible. Proposition 4.15. (i) r and g are incomparable LV -degrees. (ii) r ∨ c and g are incomparable LV -degrees (iii) r and g ∨ c are incomparable LV -degrees.Proof. (i) As shown by Demuth and Kuˇcera [DK87], no 1-generic can compute a Martin-L¨ofrandom sequence. Thus the set of Turing degrees containing a Martin-L¨of random sequence isdisjoint from the set of Turing degrees containing a 1-generic sequence. Letting 1GEN denotethe collection of 1-generic sequences, it follows that (1GEN) ≡ T \ (MLR) ≡ T = (1GEN) ≡ T and(MLR) ≡ T \ (1GEN) ≡ T = (MLR) ≡ T , both of which are non-negligible collections of sequences.Thus (1GEN) ≡ T (cid:54)≤ LV (MLR) ≡ T and (MLR) ≡ T (cid:54)≤ LV (1GEN) ≡ T .Statements (ii) and (iii) follow from (i) and the fact that the collection of computablesequences is disjoint from the collection of 1-generic sequences and from the collection ofMartin-L¨of random sequences. (cid:3) Corollary 4.16.
Neither r ∨ c nor g ∨ c is the top LV -degree. Let h be the LV-degree of the collection of sequences of hyperimmune degree, which isnon-negligible by Proposition 3.16. Remark . As shown by Kurtz, a Turing degree is hyperimmune if and only if it contains aweakly 1-generic sequence, where a sequence is weakly 1-generic if for every dense c.e. S ⊆ <ω ,there is some σ ≺ X such that σ ∈ S . Here S ⊆ <ω is called dense if every element of 2 <ω has an extension in S . If we write the collection of weakly 1-generic sequences as W1GEN wehave h = deg LV ((W1GEN) ≡ T ).An additional characterization of h can be given in terms of the collection of Kurtz randomsequences. Proposition 4.18. h = deg LV ((KR) ≡ T ) .Proof. ( ⊆ ) Since every weakly 1-generic sequence is Kurtz random, by Lemma 4.10 (ii) wehave deg LV ((W1GEN) ≡ T ) ≤ LV deg LV ((KR) ≡ T ) . ( ⊇ ) We need to show that the collection of Kurtz random sequences that do not havehyperimmune degree is negligible. As shown by Yu in unpublished work (see Downey andHirschfeldt [DH10, Theorem 8.11.12]), every Kurtz random sequence of hyperimmune-freedegree is weakly 2-random. Since every 2-random sequence has hyperimmune degree, such asequence must be weakly 2-random and not 2-random. By Corollary 4.14, the collection ofweakly 2-random sequences that are not 2-random is negligible, from which the conclusionfollows. (cid:3) Since the collection of Kurtz random sequences includes every Martin-L¨of random sequenceand every 1-generic sequence, we obtain the following result.
Proposition 4.19. r < LV h and g < LV h .Proof. Since MLR ⊆ KR and 1GEN ⊆ KR, by Lemma 4.10 (ii) we have r ≤ LV h and g ≤ LV h .Moreover, 1GEN ⊆ KR \ MLR, so this latter collection is non-negligible, which implies r < LV h .Similarly, MLR ⊆ KR \ g < LV h . (cid:3) h < LV h ∨ c , as the collection of computable sequences is disjoint from the collection ofsequences of hyperimmune degree. Moreover, we have the following results. Proposition 4.20. h ∨ c is the top degree in D LV .Proof. By Proposition 3.17 (iv) the collection of noncomputable sequences of hyperimmune-freedegree is negligible, from which the result immediately follows. (cid:3)
The following corollary, pointed out to the authors by Frank Stephan, allows identifying h also as the LV-degree of immunity notions. Definition 4.21. (i)
Let IM denote the collection of immune sequences, where a sequence is immune if ithas no infinite computably enumerable subsets. (ii) Let BI denote the collection of biimmune sequences, where a sequence is biimune if itand its complement are immune. (iii) Let
BHI denote the collection of bihyperimmune sequences, where a sequence is bi-hyperimmune if it and its complement are hyperimmune.Then set i = deg LV ((IM) ≡ T ) , b = deg LV ((BI) ≡ T ) , and bh = deg LV ((BHI) ≡ T ) . Corollary 4.22.
We have i = b = h = bh = c c .Proof. Let COMP denote the computable and HI denote the hyperimmune sequences. Then(2 ω \ COMP) ≡ T = (IM) ≡ T ⊇ (BI) ≡ T ⊇ (BHI) ≡ T = (HI) ≡ T . Here the first equality is by Dekker and Myhill [DM58] (see, for example, Odifreddi [Odi89,item 1 on page 498]), the first inequality is by definition, and the final equality is byKurtz [Kur83, Corollary 2.1]. Using the definition of hyperimmunity given in terms ofstrong c.e. arrays (see, for example, Odifreddi [Odi89, Definition III.3.7]), it is easy to see that
EGREES OF RANDOMIZED COMPUTABILITY 17 every hyperimmune set is immune, and by applying this to both a set and its complement, wesee that every bihyperimmune set is biimmune, giving the second inequality.Therefore, by Lemma 4.10, we have h = bh ≤ LV b ≤ LV i = c c = h , where the last equality is by Proposition 4.20. (cid:3) We can also conclude that there is no intermediate LV-degree between h and . Corollary 4.23.
There is no LV-degree e such that h < LV e < LV .Proof. By Proposition 4.4, c is an atom of D LV , and by Corollary 4.22, c c = h . It is a generalfact that in Boolean algebras the complement of an atom is a co-atom, that is, an element k such that there is no k (cid:48) such that k < k (cid:48) < (see, for instance Blythe [Bly05, item (3) onpage 79]). (cid:3) Let d denote the LV-degree of the collection of sequences of DNC degree, which is non-negligible by Proposition 3.16. Given that every Martin-L¨of random sequence has DNC degree,we have r ≤ LV d ; Bienvenu and Patey [BP17] showed the strictness of the relation. Theorem 4.24 (Bienvenu, Patey [BP17]) . r < LV d . Corollary 4.25. r ∨ c and d are incomparable. We can also easily derive the following result.
Proposition 4.26. d and g are incomparable.Proof. No 1-generic sequence has DNC degree, and so the result follows from the samereasoning used in the proof of Proposition 4.15 (i). (cid:3)
In Section 6, our new application of V’yugin’s technique for building semi-measures impliesthat the collection of non-computable sequences that do not have DNC degree is non-negligible,which in turn implies that d ∨ c is not the top LV-degree. However, we can alternatively derivethis latter fact as follows. Proposition 4.27. d < LV h .Proof. By Proposition 4.20, d ≤ LV h ∨ c , which implies that the collection of sequences ofDNC degree that are neither computable nor of hyperimmune degree is negligible. But clearlyno sequence of DNC degree is computable, and thus we have d ≤ LV h . Since every 1-genericsequence has hyperimmune degree and does not have DNC degree, we have h (cid:2) LV d , andthus d < LV h . (cid:3) The following results about joins in D LV are immediate. Corollary 4.28. (i) c < LV r ∨ c < LV d ∨ c < LV . (ii) c < LV g ∨ c < LV d ∨ g ∨ c . (iii) d ∨ c , g ∨ c , and d ∨ g are pairwise incomparable LV-degrees. The results of this section are summarized in Figure 1.
Open questions.
We conclude with the following open questions.
Question 4.29. Is d ∨ g = h ? In particular, is d ∨ g ∨ c = ? Given that r is a D LV -atom, it is also reasonable to ask whether the same holds for g . Question 4.30. Is g a D LV -atom? = h ∨ cd ∨ g ∨ c hd ∨ c d ∨ g g ∨ c r ∨ c dc r g0 Figure 1.
Standard arrows represent strict separations in the LV-degrees.Dotted arrows represent the following open questions: (a) Is g a D LV -atom?(b) Is d ∨ g = h , and thus is d ∨ g ∨ c = ?For the definitions of the notions appearing in the following open question, see a standardreference such as Downey and Hirschfeldt [DH10]. Question 4.31.
What are the LV -degrees of the collections of sequences that are Turingequivalent to some sequence of Hausdorff dimension , of packing dimension , of Hausdorffdimension < , of packing dimension < ? Given some α ∈ (0 , , what are the LV -degrees ofthe collections of sequences that are Turing equivalent to some sequence of Hausdorff dimension α or of packing dimension α ? How to build a Semi-measure
In this section, we outline a template for building left-c.e. semi-measures that was devel-oped [V’y82] and applied [V’y08, V’y09, V’y12] by V’yugin and which has several applicationsin the study of D LV as well as the study of Π classes. The main idea of V’yugin’s constructionis that a semi-measure on 2 <ω can be seen as a network flow on a directed graph G such that EGREES OF RANDOMIZED COMPUTABILITY 19 (i) the nodes of G , V G , are the elements of 2 <ω , and(ii) the edges of G , E G , are pairs ( σ, τ ) of nodes σ, τ ∈ <ω such that σ ≺ τ .For σ, τ ∈ <ω with σ (cid:22) τ we will say that σ is above τ and that τ is below σ ; that is, inthis article the binary tree 2 <ω grows downward. Note that, while this goes against the usualconvention in computability theory, it has the intuitive advantage that measure will flow fromthe root ε downwards, as liquids naturally do.Given σ, τ ∈ <ω with σ ≺ τ , the length of ( σ, τ ), written as | ( σ, τ ) | , is defined to be | τ | − | σ | .If | ( σ, τ ) | = 1 then we always have ( σ, τ ) ∈ E G ; such edges of G will be referred to as normaledges and the set of normal edges will be denoted by N G . If | ( σ, τ ) | > σ, τ ) may ormay not be in E G ; if it is, we call ( σ, τ ) an extra edge of G . The set of extra edges will bedenoted by X G . We will omit the subscripts if G is clear from context.Directed graphs G that satisfy V G = 2 <ω as described above will be called 2 <ω -digraphs . Inthe sequel, we will restrict our attention to computable 2 <ω -digraphs. Definition 5.1.
Given a <ω -digraph G , a network on G is a function q : E G → Q ∩ [0 , satisfying, for each σ ∈ <ω , (cid:88) ( σ,τ ) ∈E G q ( σ, τ ) ≤ . The idea here is that for a node σ , q ( σ, τ ) gives the proportion of the flow arriving in σ thatcontinues to flow into τ .In the remainder of the article, we will always have q ( σ, τ ) > σ, τ ) ∈ X . In fact, if | ( σ, τ ) | >
1, we will silently identify the two properties q ( σ, τ ) = 0 and( σ, τ ) / ∈ E since both cases equally have no effect on the outcome of the construction. Notehowever that for normal edges ( σ, τ ) ∈ N the case q ( σ, τ ) = 0 can and will occur frequently. Definition 5.2.
The amount of flow into a node τ , denoted R ( τ ) , is defined inductively by R ( ε ) = 1 ,R ( τ ) = (cid:88) ( σ,τ ) ∈E G q ( σ, τ ) R ( σ ) . Hereafter we will refer to R as the in-flow function associated to q . Observe further that if q is computable, then so is R . Remark . σ ≺ τ does not necessarily imply that R ( σ ) ≥ R ( τ ). In particular, not all of theflow that we observe below σ must have flowed through σ itself, as there could be an extraedge that bypasses σ and diverts flow to an extension of σ .To correct for this lack of monotonicity of R , we define the q -flow associated with a network q .Given σ ∈ <ω , let T σ be the collection of finite prefix-free sets of strings τ such that σ (cid:22) τ . Definition 5.4.
Let q be a network on a <ω -digraph G , and let R be the in-flow functionassociated to q . Then the q -flow P is defined by P ( σ ) = sup D ∈ T σ (cid:88) τ ∈ D R ( τ ) .P ( σ ) is thus the maximal amount of flow that can be observed passing through a set ofextensions of the node σ . The motivation for looking at prefix-free sets D of nodes is to avoidcounting the same quantity of flow more than once. Note that since { σ } ∈ T σ , we always have P ( σ ) ≥ R ( σ ), but equality need not hold due to the reason discussed in Remark 5.3. We have the following important fact.
Lemma 5.5.
Let q be a computable <ω -digraph. Then the q -flow P is a left-c.e. semi-measure.Proof. Clearly, P ( ε ) = 1. Let s = sup D ∈ T σ (cid:80) τ ∈ D R ( τ ) and s = sup D ∈ T σ (cid:80) τ ∈ D R ( τ ).Given δ >
0, there are D ∈ T σ and D ∈ T σ such that (cid:88) τ ∈ D i R ( τ ) ≥ s i − δ/ i = 0 ,
1. Then D ∪ D ∈ T σ , and hencesup D ∈ T σ (cid:88) τ ∈ D R ( τ ) ≥ (cid:88) τ ∈ D ∪ D R ( τ ) ≥ s + s − δ, for every δ >
0. Thus P ( σ ) ≥ P ( σ
0) + P ( σ P ( σ ) is left-c.e. uniformly in σ , as G , q ,and R are all computable. (cid:3) Definition 5.6.
A network q is elementary if q ( σ, τ ) = 1 / for almost every ( σ, τ ) ∈ N . By the definition of a network q , it follows that the set of extra edges X is finite if q is elementary. Since by definition networks q only take rational values, every elementarynetwork q is computable. Given a computable network q , we can write q as a limit of elementarynetworks ( q n ) n ∈ ω by requiring that(i) q n ( σ, τ ) = q ( σ, τ ) if | τ | ≤ n ;(ii) q n ( σ, τ ) = 1 / σ, τ ) ∈ N and | τ | > n ;(iii) q n ( σ, τ ) = 0 if ( σ, τ ) ∈ X and | τ | > n ;Note that these conditions imply that q n − and q n agree on every edge ( σ, τ ) except possiblyon edges ( σ, τ ) satisfying | τ | = n . We refer to such a sequence of elementary networks as the sequence of elementary restrictions of q . Moreover, we will refer to each q n as the level n elementary restriction of q .5.1. The General Template.
The semi-measure P that we construct will be one induced bya network flow q as described in the previous paragraphs. Here, q will be constructed throughan infinite procedure which works in stages. At each stage n , an elementary network q n together with its extra edge set X n will be built. In the end we will then let q = lim n q n and X = (cid:83) n X n . We first make some general intuitive remarks about the overall procedure, andthen go on to describe in formal detail the individual stages.The general construction template depends upon three parameters:(1) A computable function t : ω → ω , called the task function , such that the values t (0) , t (1) , t (2) , t (3) , . . . follow the pattern 0 , , , , , , , , , , , , , , . . . In particular, for each i , the set { n : t ( n ) = i } is infinite and t ( n ) (cid:54) = t ( n + 1) for every n .Every node will be assigned a task; in particular, each σ ∈ <ω will be assigned thetask t ( | σ | ). For a given task i , the i -nodes are the nodes σ ∈ <ω with t ( | σ | ) = i .(2) A computable predicate B ( q (cid:48) , σ, τ ) which is defined for elementary networks q (cid:48) on a2 <ω -digraph G and strings σ , τ such that both are i -nodes for the same i ∈ ω .(3) A computable, strictly increasing function c : ω → ω . EGREES OF RANDOMIZED COMPUTABILITY 21
The predicate B will be determined by the requirements we are attempting to satisfy, while thefunction c will be specifically used to provide the initial values for countdowns to expirationfor certain nodes that are “active” in the construction, in a technical sense to be explainedshortly.We take action towards fulfilling the task i if we add an extra edge connecting two i -nodes;we will refer to such an edge as an i -edge (or as an edge that is assigned to task i ). Thatis, an edge ( σ, τ ) ∈ E G is an i -edge if t ( | σ | ) = t ( | τ | ) = i . Let X [ i ] be the set of extra edgesassigned to task i . Note that we never assign normal edges to any task i , since t ( n ) (cid:54) = t ( n + 1)for every n .In the course of the construction, if j < i , we would ideally want to first perform all actionsnecessary for task j before beginning to work on task i . That is, for every extra edge ( σ, τ )between a pair of i -nodes σ and τ and for any extra edge ( σ (cid:48) , τ (cid:48) ) assigned to some task j with j < i , we would like to have | τ (cid:48) | < | σ | . But, in fact, during the construction we will notbe able to always ensure this property. After having added ( σ, τ ) for task t ( | σ | ) = t ( | τ | ) = i itmay turn out later in the construction that further edges for task j need to be added. Addingthem will then invalidate our previous actions for task i . The edge ( σ, τ ) stays in the graph,but we will consider it a failure, as it does not help us achieve the desired goal for task i . Whilethe presence of ( σ, τ ) also causes no harm, we will, at some later stage, have to completelyrestart the construction for task i . The construction can therefore be thought of as a type offinite injury argument.For a given task i , we will need to talk about the minimal length of an i -node to which anextra edge can be attached. We thus define the following auxiliary function w : Let q (cid:48) be an elementary network on G , with the associated set of extra edges X (cid:48) through which some ofthe flow passes. Then for each i ∈ ω , we define w ( i, q (cid:48) ) = min { n : t ( n ) = i ∧ ( ∀ j < i )( ∀ ( σ, τ ) ∈ X (cid:48) [ j ]) | τ | < n } . That is, w ( i, q (cid:48) ) is the least n such that (i) t ( n ) = i and (ii) every edge in G assigned to task j for some j < i ends in a node of length less than n .For an arbitrary (that is, not necessarily elementary) computable network q (cid:48) , w ( i, q (cid:48) ) maybe undefined in general. But in fact, for q ’s built using the template described here, w ( i, q (cid:48) )will always be well-defined by the above equation, and w ( i, q ) = lim n w ( i, q n ), where q n is thelevel n elementary restriction of q . When taking the limit in this last equation, the lengths n where w ( i, q n − ) (cid:54) = w ( i, q n ) correspond to the “failures” described in the previous paragraph.Another component of our construction is that at each stage, a number of nodes may be setas active, serving as candidates to which an extra edge may be attached. Before activation,all flow into a node σ will be equally divided to flow into σ ’s direct successor nodes σ σ activate a node we reduce the flow from σ into σ σ
1, resulting in a certain amount of flow into σ being temporarily unused. We say thatwe have delayed part of the flow. In a later step we may then attach an extra edge to σ anddirect the delayed, leftover flow through this new edge.More formally, for an elementary network q (cid:48) , we have a function d (cid:48) , called a flow-delayfunction , which satisfies d (cid:48) ( σ ) = 1 − q (cid:48) ( σ, σ − q (cid:48) ( σ, σ σ ∈ <ω . This is precisely the proportion of flow into σ that is prevented fromflowing into σ σ
1. The active nodes consist of those nodes σ such that 0 < d (cid:48) ( σ ) <
1; theconstruction will be such that if we block all of the flow through a node σ by setting d (cid:48) ( σ ) = 1,then it, and all of its extensions, will never be activated from that point on. Moreover, for j < i , to enforce the requirement that all j -edges end before any i -edges begin, whenever we attachan extra edge to a j -node τ , all active i -nodes whose length is less than | τ | become unusableas, by the conditions in the construction, we will never attach edges to such nodes. Intuitively,we can then think of the flow that was delayed at such i -nodes as wasted.Next, given a node σ to which we would like to attach an extra edge, there is a func-tion β ( σ, q (cid:48) , n ) that selects (somewhat arbitrarily) a candidate τ (cid:31) σ of length n for connectingan edge between σ and τ in an elementary network q (cid:48) in such a way as to satisfy the predicate B ,if such a τ exists. Specifically, β ( σ, q (cid:48) , n ) = min { τ ∈ n : τ (cid:31) σ ∧ t ( | σ | ) = t ( | τ | ) ∧ B ( q (cid:48) , σ, τ ) } , where the minimum refers to the length-lexicographic ordering of strings τ .After these informal remarks, we describe in detail how to construct the network q , withits set of extra edges X and its flow-delay function d : As mentioned at the start of thissubsection, we first build a sequence ( q n , X n , d n ) n ∈ ω , where each q n is an elementary networkwith associated set of extra edges X n . For each n ∈ ω we will let d n denote the flow-delayfunction associated with q n . In the end we will set q = lim n q n , X = (cid:83) n X n and d = lim n d n .The definition of the sequence ( q n , X n , d n ) n ∈ ω proceeds in stages as follows: For n = 0, q ( σ, τ ) = (cid:40) / τ = σ τ = σ . Clearly d ( σ ) = 0 for all σ ∈ <ω and X = ∅ .Suppose we have defined ( q n − , X n − , d n − ), where for all ( σ, τ ) ∈ X n − , | τ | < n . We willfirst define X n , d n , and then q n . The goal of this stage of the construction is to attach anextra edge connecting a t ( n )-node whose length is strictly less than n − t ( n )-node oflength n . We consider two cases. Case 1: w ( t ( n ) , q n − ) = n . This means that the extra edges in X n − assigned to sometask j < t ( n ) terminate in nodes of length ≤ n −
1, and this is the least n for which this holds.This further implies that there is no active and usable t ( n )-node of length less than n to whichwe can attach an extra edge. We thus take the following steps:(i) Set X n = X n − .(ii) Set d n ( σ ) = (cid:40) / c ( n ) if | σ | = nd n − ( σ ) otherwise.Setting d n ( σ ) = 1 / c ( n ) for each σ of length n has the effect of activating these nodes, inanticipation of attaching a t ( n )-edge to them later in the construction. We call this the initialization of the nodes.Recall that c provides the initial value for a countdown mechanism that we will use duringthe construction; once we implement this template for a specific application, we will have tochoose c carefully to ensure that a positive amount of flow stays in the network in the limit. Case 2: w ( t ( n ) , q n − ) < n . Our hope in this case is that we can attach some extra edges from t ( n )-nodes of length ≥ w ( t ( n ) , q n − ) to t ( n )-nodes of length n . Thus we search for σ ∈ <ω such that the following four conditions hold:( a ) w ( t ( n ) , q n − ) ≤ | σ | < n ;( b ) 0 < d n − ( σ ) < EGREES OF RANDOMIZED COMPUTABILITY 23 ( c ) β ( σ, q n − , n ) is defined; and( d ) σ ≺ ρ implies that ( σ, ρ ) / ∈ X n − .Condition ( a ) guarantees that the start of the new edge occurs beyond the end of any currentlypresent j -edge for j < t ( n ). Condition ( b ) guarantees that σ has been activated previouslyand is still active (henceforth, we will refer to a node σ such that 0 < d n ( σ ) < activeat stage n ). Condition ( c ) guarantees that σ is assigned to task t ( n ) and that there is alength n node that can serve as the endpoint of a new t ( n )-edge we want to attach at σ (thatis, the predicate B is satisfied). Finally, condition ( d ) guarantees that no extra edge has beenpreviously attached starting at σ .Let C n be the set of σ ∈ <ω such that conditions ( a )–( d ) are satisfied. Then we have twosubcases to consider. Subcase 2.1: C n (cid:54) = ∅ . For every σ ∈ C n and every τ (cid:31) σ with | τ | = n we let d n ( τ ) = (cid:26) τ = β ( σ, q n − , n ) ,d n − ( σ ) / (1 − d n − ( σ )) else . For all other τ we let d n ( τ ) = d n − ( τ ).By condition ( b ) above, setting d n ( τ ) = 0 deactivates τ , meaning that we will not add anyfurther t ( n )-edges to any extensions of τ , with one possible exception: It may be that at alater stage, a new j -edge for j < t ( n ) is added, which would lead to Case 1 above occurringagain for task t ( n ). This would in turn lead to all extensions of τ getting freshly initialized fortask t ( n ).Note that when initializing a node σ , we assign a delay of the form 1 /k , where c ( | σ | ) = k for some k ∈ ω . Moreover, the mapping d (cid:55)→ d/ (1 − d ), as in the second line above, maps sucha number to 1 / ( k − / ( k − t ( n )-nodes. The reciprocalof these assigned delay values are positive integers, and we can interpret them as a countercounting down by 1 along a path every time a t ( n )-edge branches off it; see Figure 2.Even on the same path different tasks are initialized separately at different nodes ofappropriate length. The countdown happens separately for all tasks, as a new delay valueassigned to an i -node depends on the delay value of an i -node of shorter length, and not onthe delay values of j -nodes with i (cid:54) = j . It therefore makes sense to talk about the i -counter for task i along a given path, and we will use this expression in the informal explanations inthe sequel.As we continue to add edges for task t ( n ) that branch off a path, the t ( n )-counter along thatpath may eventually reach 1 (or, more formally, the delay value may increase until all flow isblocked at a value of 1). Once this happens, by construction, we stop attaching t ( n )-edges onany extension of the current initial segment of that path.Next, we set X n = X n − ∪ { ( σ, β ( σ, q n − , n )) : σ ∈ C n } and q n ( σ, τ ) = (1 − d n ( σ )) if τ = σ τ = σ ,d n ( σ ) if ( σ, τ ) ∈ X n , j > t ( n ), w ( j, q k ) > n for all k ≥ n . In particular, we are now prevented fromattaching an edge to any j -nodes that were active at the beginning of this stage; we will thussay that these nodes have become unusable. ε t ( n ) t ( n ) w ( t ( n ) , · ) d ( · ) = c ( n ) d ( · ) = c ( n ) − d ( · ) = c ( n ) − d ( · ) = 0 t ( n ) t ( n ) t ( n ) Figure 2.
An edge for task t ( n ) is added. The root of the edge was initializedwith counter value c ( n ). The node at the end of the new edge has delay value 0,that is, it is not active. All other extensions of the root keep a positive delayvalue, and therefore stay active. The counter on these nodes, which is thereciprocal of the value of d on the respective node, has been reduced by 1.Note how other, completely independent t ( n )-edges can occur off to the side. Subcase 2.2: C n = ∅ . Then we set d n = d n − , X n = X n − , and q n = q n − . No new nodes areinitialized, nor do any active nodes become unusable.To finalize the outline of the construction template we lastly set q = lim n q n , d = lim n d n ,and X = (cid:83) n X n . It is not difficult to check that q and d are computable functions and that X is a computable set. It then follows from Lemma 5.5 that the resulting q -flow P , as inDefinition 5.4, is a left-c.e. semi-measure.5.2. Verification of the General Template.
We now work to establish the desired proper-ties of the constructed objects q , d , X , R , P , and so on. For the sake of notational simplicity,during this verification part, we will again use the letters q n , d n , and X n , for n ∈ ω , to referto the finite approximations of q , d , and X that we built in the previous subsection. Inparticular note that, for q , these finite approximations q n , n ∈ ω , coincide with the sequenceof elementary restrictions discussed on page 20.The following can be verified by carefully following the construction. EGREES OF RANDOMIZED COMPUTABILITY 25
Fact 5.7.
There do not exist strings σ ≺ σ ≺ σ ≺ σ such that ( σ , σ ) ∈ X and ( σ , σ ) ∈ X . Before we implement this template, we show that a number of features of the constructioncan be established independently of the concrete implementation.
Lemma 5.8 (Stability Lemma) . For every i ∈ ω , X [ i ] is finite and w ( i, q ) < ∞ .Proof. First, observe that if X [ j ] is finite for every j < i , then w ( i, q ) < ∞ . It is thereforesufficient to prove the first part of the statement.So suppose that i is minimal such that X [ i ] is infinite. Then by the previous observationwe have w ( i, q ) < ∞ . For σ with | σ | ≥ w ( i, q ), define m σ to be the maximal m > w ( i, q ) suchthat there is an edge ( σ (cid:22) m, τ ) ∈ X [ i ] where τ is incomparable with σ . If no such m exists,set m σ = w ( i, q ).Then define a function u via u ( σ ) = (cid:40) /d ( σ (cid:22) m σ ) if d ( σ (cid:22) m σ ) > , | σ | ≥ w ( i, q ) , c ( w ( i, q )) if | σ | < w ( i, q ) . Note that these two cases are exhaustive; to see this assume that | σ | ≥ w ( i, q ). If m σ = w ( i, q ),then by construction d ( σ (cid:22) m σ ) = 1 / c ( m σ ) >
0. The only other possibility is that m σ is themaximal m > w ( i, q ) such that there is an edge ( σ (cid:22) m, τ ) ∈ X [ i ] where τ is incomparablewith σ . But then d ( σ (cid:22) m σ ) > σ (cid:22) m σ , τ ) would not have beenadded according to the conditions in the construction.We claim that u ( σ ) is an upper bound on the number of possible i -edges branching offbelow length max( w ( i, q ) , | σ | ) from any path going through σ .First, consider σ meeting the conditions of the first line of the definition, and such that anedge ( σ (cid:22) m σ , τ ) as in the definition of m σ exists. Since by the choice of m σ the edge ( σ (cid:22) m σ , τ )is the last edge branching off above σ , and by the discussion of the i -counter mechanism above,we know that then at most d ( σ (cid:22) m σ ) − i -edges can branch off below σ from any pathextending σ , and the claim in this case follows.Secondly, consider σ meeting the conditions of the first line of the definition, but where anedge of the form ( σ (cid:22) m σ , τ ) as in the definition of m σ does not exist. For those σ we have thata parent ρ of σ with | ρ | = w ( i, q ) has been initialized, but that there is no extra i -edge thatbranches off between ρ and σ . Again by the discussion of the i -counter mechanism, we knowthat then at most c ( w ( i, q )) − i -edges can branch off below σ from any path extending σ .Since u ( σ ) = 1 /d ( σ (cid:22) m σ ) = 1 /d ( σ (cid:22) w ( i, q )) = c ( w ( i, q )) , the claim in this case follows.Lastly, consider σ satisfying | σ | < w ( i, q ). Let τ (cid:31) σ be of length w ( i, q ). Then τ isinitialized with d ( τ ) = 1 / c ( | τ | ). By the definition of u , u ( σ ) = u ( τ ), and we can argue as inthe previous paragraph to conclude that at most c ( w ( i, q )) − i -edges can branch off belowlength w ( i, q ) from any path extending σ .It should now be clear that u is constant on all strings σ with | σ | ≤ w ( i, q ); and that for arbi-trary strings σ and τ with σ (cid:22) τ we have u ( σ ) ≥ u ( τ ). We then define the function (cid:98) u : 2 ω → ω by letting, for every A ∈ ω , (cid:98) u ( A ) = min { n : u ( A (cid:22) n ) = u ( A (cid:22) (cid:96) ) for all (cid:96) ≥ n } . ε t ( n ) t ( n ) t ( n ) t ( n ) t ( n ) t ( n ) w ( t ( n ) , . ) A Figure 3.
A sequence of extra edges branching off a given path A . Note howthe length of the start point of every edge has to coincide with the length ofthe endpoint of the previous edge branching off the path.We claim that the function (cid:98) u is continuous. This is because ( a ) u is non-increasing over longerand longer initial segments of a path A , ( b ) u only takes integer, positive values, and ( c ) adecrease in u cannot happen arbitrarily late along A . This last point ( c ) follows from the twofacts that (i) at every node at most one edge starts (by construction) and that (ii) for an i -edgebranching off A at A (cid:22) (cid:96) we must have that (cid:96) is either w ( i, q ) or the length of the endpoint of theprevious i -edge branching off A ; otherwise A (cid:22) (cid:96) would not be active; see Figure 3. Therefore,for a long enough initial segment A (cid:22) k of A , u ( A (cid:22) k ) has stabilized; meaning that A (cid:22) k alreadydetermines (cid:98) u ( A ).Because 2 ω is compact, (cid:98) u is bounded by some N ∈ ω , meaning in particular that u ( σ ) = u ( σ (cid:22) N ) for all σ with | σ | ≥ N . But then no new i -edge ( σ, τ ) can be attachedto any such σ , as that would imply u ( τ ) < u ( σ ), contradicting the choice of N . Thus X [ i ] cannot be infinite. (cid:3) Definition 5.9.
For a finite sequence σ ∈ <ω we call an infinite sequence X ∈ ω an i -continuation of σ if i = t ( | σ | ) , σ ≺ X , and B ( q n − , σ, X (cid:22) n ) holds for almost all n with t ( n ) = i . Definition 5.10.
A sequence X is called i -discarded if d ( X (cid:22) n ) = 1 for some n where t ( n ) = i . EGREES OF RANDOMIZED COMPUTABILITY 27
Note that a sequence X ∈ ω becomes i -discarded if there exists an initial segment X (cid:22) k such that the counter for task i has reached the final value 1 on X (cid:22) k . By the conditions statedin Case 2 of the construction, below such an X (cid:22) k no further extra edges for task i will branchoff of X , hence the name “discarded.” Lemma 5.11 (Edge Existence Lemma) . Assume that for X ∈ ω and for all k ∈ ω suchthat t ( k ) = i it holds that X (cid:22) k has an i -continuation and that X is not i -discarded. Then X contains an i -edge ( σ, τ ) , that is, σ ≺ τ ≺ X .Proof. Assume that X is not i -discarded. Let m be maximal with t ( m ) = i and d ( X (cid:22) m ) > m is defined: First, an m as described exists, sinceby Stability Lemma 5.8 we have that w ( i, q ) is finite, and, by construction, d ( X (cid:22) w ( i, q )) isset to a value strictly between 0 and 1. Secondly, by construction, any positive value d ( X (cid:22) (cid:96) )for some (cid:96) > w ( i, q ) with t ( (cid:96) ) = i must be the result of a chain of i -edges branching off X ,as illustrated in Figure 3. Again by Stability Lemma 5.8, X [ i ] is finite, and therefore anysuch chain can only have finite length, therefore only finitely many (cid:96) with t ( (cid:96) ) = i canhave d ( X (cid:22) (cid:96) ) >
0. As a result a maximal m as described must exist.Since, by assumption, X (cid:22) m has an i -continuation and d ( X (cid:22) m ) <
1, the conditions ofSubcase 2.1 of the construction are met. Therefore, eventually an i -edge of the form ( X (cid:22) m, τ )is attached at X (cid:22) m . By construction d ( τ ) = 0 and d ( ρ ) (cid:54) = 0 for all ρ such that X (cid:22) m ≺ ρ , τ (cid:54) = ρ , and | ρ | = | τ | . By the choice of m we must therefore have τ ≺ X . (cid:3) Lemma 5.12 (Continuity Lemma) . The semi-measure P has no atoms.Proof. Note that by definition of the function w , there are no extra edges ( σ, τ ) ∈ X suchthat | σ | < w ( i, q ) ≤ | τ | for any i . That is, for any i , all flow that flows from nodes of lengthless than w ( i, q ) to nodes extending them flows through normal edges. Let σ be a node oflength w ( i, q ) −
1. By construction q ( σ, σ
0) = q ( σ, σ ≤ / , and hence, for b ∈ { , } , P ( σ (cid:95) b ) = R ( σ (cid:95) b ) = q ( σ, σ (cid:95) b ) · R ( σ ) ≤ / · P ( σ ) . Since there are infinitely many numbers of the form w ( i, q ), i ∈ ω , we have lim n →∞ P ( X (cid:22) n ) = 0for every X ∈ ω . (cid:3) The Roadmap.
Everything discussed thus far in this section forms the common partof the construction. In particular, we do not need to re-prove Fact 5.7, Stability Lemma 5.8,Edge Existence Lemma 5.11, and Continuity Lemma 5.12 for each application of the template.However, when applying the template to obtain different results, some parts of the constructionneed to be adapted to the statement that should be proved. There will still be a commonstructure with the following components.
Predicate B : The predicate B determines when edges are added to the graph, andtherefore the information that will be coded into the semi-measure constructed. Cut-off Lemma:
Here we show that if any positive flow occurs beyond a node τ , thenat least some part of that flow must have passed through normal edges. Continuation Existence Lemma:
To be able to apply the Edge Existence Lemma 5.11to all of the sequences in the support of the semi-measure we construct, we need toprove that the hypotheses of the lemma are satisfied by these sequences. That is, weneed to prove that every sequence X in the support is an i -continuation for all of itsown initial segments X (cid:22) n with t ( n ) = i . Measure Lemma:
This shows that the support of the constructed semi-measure P has positive P -measure. Note that, together with Continuity Lemma 5.12 and usingProposition 3.14, this implies that the support of P does not exclusively containcomputable elements. Verifying the desired properties:
Finally we need to verify that the semi-measurewe constructed has the desired properties needed for the statement that was to beshown. 6.
Implementing the Template
A first example.
We begin by giving V’yugin’s proof of Theorem 1.1.
Theorem 1.1 (V’yugin [V’y12]) . For any δ > , there is a probabilistic algorithm thatproduces with probability at least − δ a non-computable sequence that does not compute anyMartin-L¨of random sequence. To prove this, we will show the following more general statement.
Theorem 6.1 (V’yugin [V’y12]) . For each δ ∈ (0 , , there is a left-c.e. semi-measure P suchthat (i) P has no atoms; (ii) P (2 ω ) = P (Supp( P )) > − δ ; and (iii) for each X ∈ Supp( P ) and each Turing functional Φ , if Φ( X ) is defined, then Φ( X ) (cid:54)∈ MLR . We obtain the desired probabilistic algorithm from Theorem 6.1 by applying Theorem 3.4(ii):Since P is a left-c.e. semi-measure, there is some Turing functional Ψ such that P = λ Ψ . Thefunctional Ψ equipped with a random oracle provides the probabilistic algorithmic satisfyingthe conditions of Theorem 1.1.One additional consequence of Theorem 6.1 is that r ∨ c is not the top degree of D LV , whichwe already showed via an alternative method in Section 4. Indeed, since Supp( P ) contains noatoms and every atom of a left-c.e. semi-measure is computable, it follows that P (Supp( P ) \ { X : X computable } ) > . This implies that Supp( P ) \ { X : X computable } is non-negligible. But, by construction, theLevin-V’yugin degree generated by Supp( P ) \ { X : X computable } is disjoint from r ∨ c .V’yugin originally proved this result in [V’y76] without use of the machinery laid out in theprevious section, but in a later article [V’y12] he gave the proof discussed here.To prove Theorem 6.1, we first need to specify the predicate B and the function c , as inthe template outlined above. For an elementary network q (cid:48) and nodes σ , τ with t ( | σ | ) = t ( | τ | ), B ( q (cid:48) , σ, τ ) is defined to hold if and only if(a) σ (cid:22) τ ,(b) d (cid:48) ( τ (cid:22) k ) < k such that 1 ≤ k ≤ | τ | , where d (cid:48) is the flow-delay function of q (cid:48) ,and(c) (cid:12)(cid:12) Φ τj, | τ | (cid:12)(cid:12) > (cid:104) σ ) , s (cid:105) , where t ( | σ | ) = (cid:104) j, s (cid:105) . Here σ ) denotes the position of σ inthe canonical lexicographic ordering of 2 <ω and (cid:104)· , ·(cid:105) denotes a pairing function thatsatisfies (cid:104) m, n (cid:105) ≥ m + n for all m, n ∈ ω .The idea of this choice of B is that for each i ∈ ω such that i = (cid:104) j, s (cid:105) for j, s ∈ ω , we attachan i -edge between i -nodes σ and τ only if (cid:12)(cid:12) Φ τj, | τ | (cid:12)(cid:12) > (cid:104) σ ) , s (cid:105) , that is, Φ τj, | τ | is sufficiently long. EGREES OF RANDOMIZED COMPUTABILITY 29
Moreover, we will ensure that for each X ∈ ω , either there is some n such that the flow outof X (cid:22) n is completely blocked, or, for each Turing functional Φ j such that Φ j ( X ) is defined,Φ j ( X ) / ∈ MLR. This latter condition will be accomplished by enumerating β ( σ, q n − , n ) intoa Martin-L¨of test for each i -edge σ such that t ( | σ | ) = i = (cid:104) j, s (cid:105) for some s ∈ ω .As for the choice of c , given δ >
0, we let c ( n ) = ( n + n ) , where n is such that (cid:88) n ∈ ω ( n + n ) − < δ. This will be used to prove Measure Lemma 6.2 below.Now let P be the semi-measure produced by the template outlined in Section 5.1 whenused with this specific choice of B and c . We establish that P has the desired properties. Lemma 6.2 (Measure Lemma) . P (Supp( P )) > − δ . For X ∈ Supp( P ) we already have that P ( X (cid:22) n ) > n ; that is, at any finite level n ,not all measure has dissipated. We will show that for all n , the amount of flow that flows intobut not out of strings of length n is bounded from above by ( n + n ) − with n a constant.This implies that the total dissipation is (cid:80) n ( n + n ) − < δ , thus establishing the result.In the construction, when an i -counter runs out along a path, the delay value is set to 1 atsome node σ that is an initial segment of that path to remove the path from the support ofthe constructed semi-measure. As this means that all flow arriving in σ is blocked at σ , theamount of measure lost this way could be very large. This is why we start the countdownwith larger and larger numbers in the construction, as this ensures that there are more andmore chances to add edges, which preserves more and more measure.On the other hand we do need that after finitely many attempts to add an edge we giveup and block all flow along that path completely, as otherwise a single task might causeinfinitely many of the failures described on page 21, which might prevent the constructionfrom ever successfully handling the remaining tasks. Furthermore, if a currently investigatedfunctional Φ j stops producing output somewhere, then we only lose the measure currentlydelayed there; all the remaining measure keeps flowing through normal edges. The measurelost this way is another quantity that we need to control.The trade-offs needed to reconcile these necessities make the construction quite complexand are the reason why establishing a lower bound for the remaining measure requires thefollowing involved argument. Proof of Lemma 6.2.
By the definition of R and d , (cid:88) | σ | = n +1 R ( σ ) = (cid:88) | τ | = n q ( τ, τ R ( τ ) + (cid:88) | τ | = n q ( τ, τ R ( τ ) + (cid:88) ( ρ,ξ ) ∈X , | ξ | = n +1 q ( ρ, ξ ) R ( ρ )= (cid:88) | τ | = n (1 − d ( τ )) R ( τ ) + (cid:88) ( ρ,ξ ) ∈X , | ξ | = n +1 q ( ρ, ξ ) R ( ρ ) . (4)We set S n = (cid:88) | σ | = n R ( σ ) − (cid:88) ( ρ,ξ ) ∈X , | ξ | = n q ( ρ, ξ ) R ( ρ ) , (5)so that it follows from (4) and (5) that S n +1 = (cid:88) | τ | = n (1 − d ( τ )) R ( τ ) . (6) That is, S n +1 is the amount of flow into nodes of length n + 1 that comes directly from nodesof length n (and not through extra edges whose end nodes have length n + 1).We claim that S n +1 ≥ S n − ( n + n ) − for all n . For fixed n , we consider the possible valuesof w ( t ( n ) , q n − ). First, we consider Subcase 2.2 of the construction, where w ( t ( n ) , q n − ) < n but we added no extra edge ( σ, τ ) where | τ | = n . In this case, for each ρ such that | ρ | = n , d ( ρ ) = d n ( ρ ) = d n − ( ρ ) = 0. It then follows from (5) and (6) that S n +1 = S n .Next, suppose that we are in Subcase 2.1 of the construction, where w ( t ( n ) , q n − ) < n andwe added at least one extra edge ( σ, τ ) with | τ | = n . For σ, τ ∈ <ω , letFan( σ, τ ) = { ρ : | ρ | = | τ | ∧ σ ≺ ρ ∧ ρ (cid:54) = τ } . In Figures 2 and 3 the fans of extra edges were represented by dotted cones.
Sublemma 6.3.
For every ( σ, τ ) ∈ X , (cid:88) ρ ∈ Fan( σ,τ ) R ( ρ ) ≤ (1 − d ( σ )) R ( σ ) . (7) Proof.
The term on the left-hand side of the inequality is the total amount of flow that flowsinto all nodes in Fan( σ, τ ), while the term on the right-hand side is the total flow into σ (thenode at the base of the fan) minus the flow that is diverted into the extra edge ( σ, τ ). Theonly case where this inequality can fail to hold is if there is some flow through an extra edge( ζ, ξ ) ∈ X such that ζ ≺ σ ≺ ξ (cid:22) ρ for some ρ ∈ Fan( σ, τ ). However, since ( σ, τ ) ∈ X , theexistence of such an extra edge ( ζ, ξ ) contradicts Fact 5.7. Thus the inequality must hold. ♦ The sum (cid:80) | ρ | = n d ( ρ ) R ( ρ ) can be understood as the total amount of measure that is delayedat level n . Indeed, since R ( ρ ) is the absolute amount of flow into ρ and d ( ρ ) is the relativefraction of flow blocked at ρ , we have that d ( ρ ) R ( ρ ) is the absolute quantity of flow blockedat ρ .Since we are in Subcase 2.1 (and therefore a non-trivial delay value at a node ρ cannotbe caused by activation of ρ but must be caused by an extra edge ending in a node τ with ρ ∈ Fan( σ, τ )), we have: (cid:88) | ρ | = n d ( ρ ) R ( ρ ) = (cid:88) ( σ,τ ) ∈X , | τ | = n (cid:88) ρ ∈ Fan( σ,τ ) d ( ρ ) R ( ρ )By definition of d on ρ ∈ Fan( σ, τ ):= (cid:88) ( σ,τ ) ∈X , | τ | = n d ( σ )1 − d ( σ ) (cid:88) ρ ∈ Fan( σ,τ ) R ( ρ )By Sublemma 6.3: ≤ (cid:88) ( σ,τ ) ∈X , | τ | = n d ( σ ) R ( σ )= (cid:88) ( σ,τ ) ∈X , | τ | = n q ( σ, τ ) R ( σ ) . EGREES OF RANDOMIZED COMPUTABILITY 31
Then S n +1 = (cid:88) | ρ | = n (1 − d ( ρ )) R ( ρ ) = (cid:88) | σ | = n R ( σ ) − (cid:88) | ρ | = n d ( ρ ) R ( ρ ) ≥ (cid:88) | σ | = n R ( σ ) − (cid:88) ( σ,τ ) ∈X , | τ | = n q ( σ, τ ) R ( σ ) = S n . (8)Lastly, in Case 1 of the construction, we have w ( t ( n ) , q n − ) = n , and hence (cid:88) | ρ | = n d ( ρ ) R ( ρ ) ≤ / c ( n ) = ( n + n ) − . Consequently, S n +1 = (cid:88) | ρ | = n (1 − d ( ρ )) R ( ρ ) = (cid:88) | σ | = n R ( σ ) − (cid:88) | ρ | = n d ( ρ ) R ( ρ ) ≥ (cid:88) | σ | = n R ( σ ) − ( n + n ) − ≥ (cid:88) | σ | = n R ( σ ) − (cid:88) ( σ,τ ) ∈X , | τ | = n q ( σ, τ ) R ( σ ) − ( n + n ) − = S n − ( n + n ) − . (9)Now since S n +1 ≥ S n − ( n + n ) − for every n and S = 1, we have S n ≥ − ∞ (cid:88) i =1 ( i + n ) − > − δ. Lastly, by the definition of the support of a semi-measure, we have P (Supp( P )) = inf n (cid:88) | ρ | = n P ( ρ ) ≥ inf n (cid:88) | ρ | = n R ( ρ ) ≥ inf n S n > − δ. (cid:3) Lemma 6.4 (Cut-off Lemma) . For τ ∈ <ω , P ( τ ) = 0 if and only if there is some σ ≺ ρ (cid:22) τ such that ρ ∈ { σ , σ } and q ( σ, ρ ) = 0 .Proof. Assume that, for all 0 ≤ i < | τ | , q (cid:0) τ (cid:22) i, τ (cid:22) ( i + 1) (cid:1) > R we have R ( τ ) ≥ | τ |− (cid:89) i =0 q (cid:0) τ (cid:22) i, τ (cid:22) ( i + 1) (cid:1) > , which together with P ( τ ) ≥ R ( τ ) implies P ( τ ) > n < | τ | such that q (cid:0) τ (cid:22) n, τ (cid:22) ( n + 1) (cid:1) = 0,but P ( τ ) (cid:54) = 0. Then there must be some extra edge ( σ, ρ ) such that σ (cid:22) τ (cid:22) n and τ (cid:22) ( n + 1) (cid:22) ρ .We have that q (cid:0) τ (cid:22) n, τ (cid:22) ( n + 1) (cid:1) = 0 implies d ( τ (cid:22) n ) = 1. But, by condition (b) in the definitionof B above, ( σ, ρ ) can only be added if d ( ρ (cid:22) k ) < k such that 1 ≤ k ≤ | ρ | , contradictingthe fact that d ( ρ (cid:22) n ) = d ( τ (cid:22) n ) = 1. (cid:3) Lemma 6.5 (Continuation Existence Lemma) . For every Turing functional Φ j , every X ∈ Supp( P ) such that Φ j ( X ) is defined, and every i = (cid:104) j, s (cid:105) for s ∈ ω , X is an i -continuationof X (cid:22) m for every m ∈ ω such that t ( m ) = i . Proof.
Fix j, m, s ∈ ω , and let i = (cid:104) j, s (cid:105) . Recall that X is an i -continuation of σ ∈ <ω with t ( | σ | ) = i if σ ≺ X and B ( q n − , σ, X (cid:22) n ) holds for almost all n such that t ( n ) = i . Thus,to show that X is an i -continuation of X (cid:22) m , it suffices to show that, for almost every n ,(b) d ( X (cid:22) k ) < k such that 1 ≤ k ≤ n , and(c) (cid:12)(cid:12) Φ X (cid:22) nj,n (cid:12)(cid:12) > (cid:104) X (cid:22) m ) , s (cid:105) .Since X ∈ Supp( P ), P ( X (cid:22) n ) > n , and it follows from the Cut-off Lemma 6.4 that d ( X (cid:22) n ) < n , and so (b) holds. Moreover, as Φ j ( X ) is defined, for each N ∈ ω , | Φ X (cid:22) nj,n | ≥ N for all sufficiently large n ; thus, (c) holds. (cid:3) Lemma 6.6.
For any X ∈ Supp( P ) and any Turing functional Φ j such that Φ j ( X ) is defined, Φ j ( X ) / ∈ MLR . Proof.
For s ∈ ω , let U s = (cid:91) n : t ( n )= (cid:104) j,s (cid:105) (cid:91) σ ∈C n (cid:74) Φ β ( σ,q n − ,n ) j,n (cid:75) where C n is the set of the same name that was defined during the construction.Fix s ∈ ω . Since X ∈ Supp( P ) and Φ j ( X ) is defined, by Continuation ExistenceLemma 6.5, X is an i -continuation of X (cid:22) m for i = (cid:104) j, s (cid:105) and every m ∈ ω such that t ( m ) = i .Since X ∈ Supp( P ), X cannot be i -discarded. Then, by Edge Existence Lemma 5.11, thereare n, m ∈ ω with m < n such that there is an extra i -edge ( X (cid:22) m, β ( X (cid:22) m, q n − , n )) such that β ( X (cid:22) m, q n − , n ) = X (cid:22) n . It follows that (cid:74) Φ X (cid:22) nj (cid:75) is enumerated into U s .Lastly, since (cid:12)(cid:12) Φ β ( σ,q n − ,n ) j,n (cid:12)(cid:12) > (cid:104) σ ) , s (cid:105) for each n ∈ ω and σ ∈ C n , λ ( U s ) ≤ (cid:88) n : t ( n )= (cid:104) j,s (cid:105) (cid:88) σ ∈C n −(cid:104) σ ) ,s (cid:105) ≤ − s . Hence, ( U s ) s ∈ ω is a Martin-L¨of test covering Φ j ( X ), and thus Φ j ( X ) / ∈ MLR. (cid:3)
This completes the proof of Theorem 6.1, as Continuity Lemma 5.12 establishes theTheorem’s condition (i), Measure Lemma 6.2 establishes condition (ii), and Lemma 6.6establishes condition (iii).In light of the second paragraph of the proof of Lemma 6.6 we can now formulate an intuitiveunderstanding of Edge Existence Lemma 5.11: It states that every path (that meets theconditions in the statement of the lemma) will eventually either be removed from the supportof the semi-measure during its construction, or, if not, will be treated using the predicate B to make sure all paths that remain in the support have the desired properties. In either case,the construction succeeds.6.2. A new application of the technique.
We now turn to the proof of Theorem 1.2, anextension of V’yugin’s Theorem 1.1.
Theorem 1.2.
For any δ > , there is a probabilistic algorithm that produces with probabilityat least − δ a non-computable sequence that does not compute any sequence of DNC degree. To prove Theorem 1.2, we prove a strengthening of Theorem 6.1 in terms of a familyof weak notions of randomness; just as Theorem 1.1 follows from Theorem 6.1, so too willTheorem 1.2 follow from this strengthening. The following notion was explicitly defined byHiguchi et al. [HHSY14] and was further studied by Simpson and Stephan [SS15].
EGREES OF RANDOMIZED COMPUTABILITY 33
Definition 6.7.
Let f : 2 <ω → ω be a total computable function. (i) An f -Martin-L¨of test is a sequence of uniformly c.e. sets of strings ( U i ) i ∈ ω such that (cid:88) σ ∈ U i − f ( σ ) ≤ − i for every i ∈ ω . (ii) A sequence X ∈ ω is f -random if X / ∈ (cid:84) i ∈ ω (cid:74) U i (cid:75) for every f -Martin-L¨of test ( U i ) i ∈ ω . We will focus our attention on notions of f -randomness for sequences X and functions f where f is unbounded along X , that is, lim n →∞ f ( X (cid:22) n ) = ∞ . We can now state our generalizationof Theorem 6.1. Theorem 6.8.
There is a left-c.e. semi-measure P such that (i) P has no atoms; (ii) P (Supp( P )) > ; and (iii) for each X ∈ Supp( P ) and each Turing functional Φ , if Φ( X ) is defined, then Φ( X ) isnot f -random for any computable function f that is unbounded along Φ( X ) . We call a function f : 2 <ω → ω monotone if for any σ, τ ∈ <ω with σ (cid:22) τ we havethat f ( σ ) ≤ f ( τ ). For any f : 2 <ω → ω , we define f ∗ : 2 <ω → ω by letting, for each σ ∈ <ω , f ∗ ( σ ) = max { f ( τ ) : τ (cid:22) σ } . Clearly, f ∗ is monotone and we have f ( σ ) ≤ f ∗ ( σ ) for all σ ∈ <ω . If f is furthermorecomputable and unbounded along some X ∈ ω , then the same holds for f ∗ . The proof ofTheorem 6.8 below will only ensure that (iii) holds for monotone f . The following argumentestablishes that this is sufficient. Lemma 6.9.
Let f : 2 <ω → ω be a total computable function. Then X ∈ ω is f -random ifand only if X is f ∗ -random.Proof. ( ⇐ ) Suppose that X ∈ ω is not f -random. Then there is some f -Martin-L¨of test( U i ) i ∈ ω such that X ∈ (cid:84) i ∈ ω (cid:74) U i (cid:75) . We claim that ( U i ) i ∈ ω is an f ∗ -Martin-L¨of test. Indeed,since f ( σ ) ≤ f ∗ ( σ ) for all σ ∈ <ω , (cid:88) σ ∈ U i − f ∗ ( σ ) ≤ (cid:88) σ ∈ U i − f ( σ ) ≤ − i . It thus follows that X is not f ∗ -random.( ⇒ ) Now suppose that X is not f ∗ -random. Then there is some f ∗ -Martin-L¨of test ( U i ) i ∈ ω such that X ∈ (cid:84) i ∈ ω (cid:74) U i (cid:75) . We modify ( U i ) i ∈ ω to produce an f -Martin-L¨of test covering X asfollows. First note that for every σ ∈ <ω , if f ( σ ) (cid:54) = f ∗ ( σ ), then there is some τ ≺ σ such that f ( τ ) = f ∗ ( τ ) = f ∗ ( σ ). In this case, let us set (cid:98) σ = τ . In the case that f ( σ ) = f ∗ ( σ ), set (cid:98) σ = σ ;in either case, we have (cid:98) σ (cid:22) σ . Then for each i ∈ ω and σ ∈ <ω , we define (cid:98) U i so that (cid:98) σ ∈ (cid:98) U i if and only if σ ∈ U i . It follows that ( (cid:98) U i ) i ∈ ω is an f -Martin-L¨of test, since (cid:88) (cid:98) σ ∈ (cid:98) U i − f ( (cid:98) σ ) = (cid:88) σ ∈ U i − f ∗ ( σ ) ≤ − i . Next, since for every σ ∈ <ω we have (cid:98) σ (cid:22) σ , it follows that (cid:74) U i (cid:75) ⊆ (cid:74) (cid:98) U i (cid:75) for every i ∈ ω , andhence X ∈ (cid:84) i ∈ ω (cid:74) U i (cid:75) ⊆ (cid:84) i ∈ ω (cid:74) (cid:98) U i (cid:75) . We thus conclude that X is not f -random. (cid:3) The general strategy of the proof of Theorem 6.8 is much like that of the proof of Theorem 6.1,but with several modifications. First, since we want that elements of Supp( P ) cannotcompute any f -random sequences for any monotone unbounded computable f : 2 <ω → ω ,our construction will have to involve all total computable functions. Of course, there is noeffective enumeration of such functions, so we have to work with an enumeration of all partialcomputable functions ( ϕ e ) e ∈ ω (where each ϕ e is viewed as a map from 2 <ω to ω ). Moreover,we can assume that all functions ϕ e are monotone simply by replacing each ϕ e with thecorresponding monotone ϕ ∗ e .Second, we have to modify the definition of the predicate B from the proof of Theorem 6.1as follows: For an elementary network q (cid:48) and nodes σ , τ with t ( | σ | ) = t ( | τ | ), B ( q (cid:48) , σ, τ ) isdefined to hold if and only if(a) σ (cid:22) τ ,(b) d (cid:48) ( τ (cid:22) k ) < k such that 1 ≤ k ≤ | τ | , where d (cid:48) is the flow-delay function of q (cid:48) ,and(c ∗ ) there is some ρ σ,τ (cid:22) Φ τj, | τ | such that ϕ e, | τ | ( ρ σ,τ ) ↓ > (cid:104) σ ) , s (cid:105) , where t ( | σ | ) = (cid:104) j, s, e (cid:105) . Observe that for non-total ϕ e condition (c ∗ ) may never become true and as a result wemay never attach a t ( | σ | )-edge to σ . This is not a problem, as condition (iii) in Theorem 6.8only makes a promise about total functions, so no action is required in this case. Tasks of theform (cid:104) j, · , e (cid:105) can therefore be safely ignored when verifying that the construction yields thedesired semi-measure.As in the previous subsection, that Continuity Lemma 5.12 holds is an inherent feature ofthe construction technique, independently of the specific choice of the predicate B and thecountdown function c . Measure Lemma 6.2 also still holds since its truth does not depend onthe specific choice of B while c is unchanged. As for Cut-off Lemma 6.4, an inspection of itsproof shows that it only relies on condition (b) of the predicate B which we haven’t changedfrom the last subsection; so this lemma still holds as well. The Continuation Existence Lemma,however, needs to be modified. Lemma 6.10 (Modified Continuation Existence Lemma) . Suppose that we have a Turingfunctional Φ j , some X ∈ Supp( P ) ∩ dom(Φ j ) , and a monotone total computable function ϕ e such that ϕ e (Φ j ( X ) (cid:22) n ) is unbounded in n . Then for every i of the form (cid:104) j, s, e (cid:105) for some s ∈ ω , X is an i -continuation of X (cid:22) m for every m ∈ ω such that t ( m ) = i .Proof. That condition (b) in the predicate B eventually holds is shown as in the proof ofLemma 6.5.For condition (c), since X ∈ dom(Φ j ) and ϕ e (Φ j ( X ) (cid:22) n ) is unbounded in n , there are n and n such that ϕ e,n (Φ X (cid:22) n j ) ↓ > (cid:104) X (cid:22) m ) , s (cid:105) . Then for any n ≥ max { n , n } , Φ X (cid:22) nj has theinitial segment ρ X (cid:22) m, X (cid:22) n = Φ X (cid:22) n j such that ϕ e,n ( ρ X (cid:22) m, X (cid:22) n ) = ϕ e,n ( ρ X (cid:22) m, X (cid:22) n ) > (cid:104) X (cid:22) m ) , s (cid:105) . Therefore, condition (c) holds for any large enough n with t ( n ) = i . (cid:3) Lastly, we prove the following. We cannot simply let ρ σ,τ = Φ τj, | τ | as there is no guarantee that running ϕ e with input Φ τj, | τ | terminates within | τ | steps. EGREES OF RANDOMIZED COMPUTABILITY 35
Lemma 6.11.
Let f : 2 <ω → ω be a monotone total computable function. For any X ∈ Supp( P ) and any Turing functional Φ such that X ∈ dom(Φ) , if f (Φ( X ) (cid:22) n ) is unbounded in n , then Φ( X ) is not f -random.Proof. Let e be the index of f as a partial computable function and let j be the indexof Φ. Then we define an f -Martin-L¨of test ( U s ) s ∈ ω , where U s consists of all strings of theform Φ β ( σ, q n − , n ) j,n where n ∈ ω , σ ∈ C n , and t ( n ) = (cid:104) j, s, e (cid:105) .For each X ∈ Supp( P ) ∩ dom(Φ j ) such that f (Φ j ( X ) (cid:22) n ) is unbounded in n , by theModified Continuation Existence Lemma 6.10, X is an i -continuation of X (cid:22) m for every i such that i = (cid:104) j, s, e (cid:105) for some s ∈ ω and every m ∈ ω such that t ( m ) = i . Further-more, X is not i -discarded, and hence by Edge Existence Lemma 5.11 there is an i -edge (cid:0) X (cid:22) m, β ( X (cid:22) m, q n − , n ) (cid:1) such that β ( X (cid:22) m, q n − , n ) = X (cid:22) n for some n, m ∈ ω such that m < n .Since t ( n ) = (cid:104) j, s, e (cid:105) , it follows that Φ X (cid:22) nj,n is enumerated into U s .Choose any η ∈ U s . Then, by definition of U s , there is some τ such that Φ τj,n = η andsome σ ∈ C | τ | such that ( σ, τ ) ∈ X . By condition (c ∗ ) of the predicate B , the fact that thisextra edge was added to the graph implies that there is a witnessing initial segment ρ σ,τ (cid:22) Φ τj,n such that ϕ e ( η ) = ϕ e (Φ τj,n ) ≥ ϕ e ( ρ σ,τ ) = ϕ e, | τ | ( ρ σ,τ ) > (cid:104) σ ) , s (cid:105) ;here the first inequality follows from the monotonicity of ϕ e . As this line of reasoning appliesto every η ∈ U s , we obtain (cid:88) η ∈ U s − ϕ e ( η ) ≤ (cid:88) n : t ( n )= (cid:104) j,s,e (cid:105) (cid:88) σ ∈C n −(cid:104) σ ) ,s (cid:105) ≤ − s . Hence, ( U s ) s ∈ ω is an f -Martin-L¨of test covering Φ j ( X ), and so Φ j ( X ) is not f -random. (cid:3) This completes the proof of Theorem 6.8. We can recast this result in terms of autocomplexityas well as in terms of having DNC degree. Recall that the Kolmogorov complexity of astring σ ∈ <ω is defined by K ( σ ) = min {| τ | : U ( τ ) ↓ = σ } , where U is a universal, prefix-freeTuring machine. Moreover, a function f : ω → ω is called an order if f is unbounded andnondecreasing. Definition 6.12. X ∈ ω is autocomplex if there is an X -computable order f such that K ( X (cid:22) n ) ≥ f ( n ) for every n ∈ ω . The following two propositions provide alternative characterizations of f -randomness. Proposition 6.13 (Higuchi, Hudelson, Simpson, Yokoyama [HHSY14]) . X ∈ ω is autocom-plex if and only if there is some computable function f : 2 <ω → ω such that f is unboundedalong X and X is f -random. Proposition 6.14 (Kjos-Hanssen, Merkle, Stephan [KHMS11]) . X ∈ ω is autocomplex ifand only if X has DNC degree. We can now recast Theorem 6.8 as follows.
Corollary 6.15.
There is a left-c.e. semi-measure P such that (i) P has no atoms; (ii) P (Supp( P )) > ; and (iii) for each X ∈ Supp( P ) and each Turing functional Φ , if Φ( X ) is defined, then Φ( X ) isnot autocomplex, or equivalently, Φ( X ) does not have DNC degree. By the same argument as the one that immediately follows the statement of Theorem 6.1,Corollary 6.15 yields an alternative proof of the fact that d ∨ c is not the top LV-degree.7. Applications to Π classes As we have seen, V’yugin’s construction as laid out in Sections 5 and 6 yields significantresults in the study of the LV-degrees. As noted in the introduction, the construction also hassome interesting consequences for the study of Π classes, that is, effectively closed subsetsof 2 ω . In particular, for each of the semi-measures P defined via V’yugin’s construction,for σ ∈ <ω , the condition P ( σ ) = 0 is computable, as P ( σ ) = 0 if and only if q ( σ ) is set to 0at stage | σ | in the construction of P . This implies that in each case, the support of P is aΠ class. We thus establish the two corollaries stated in the introduction. Corollary 1.3.
For every δ > , there is a Turing functional Φ such that (i) the domain of Φ has Lebesgue measure − δ , (ii) the range of Φ is a Π class, and (iii) no sequence in the range of Φ computes a Martin-L¨of random sequence. Corollary 1.4.
For every δ > , there is a Turing functional Φ such that (i) the domain of Φ has Lebesgue measure − δ , (ii) the range of Φ is a Π class, and (iii) no sequence in the range of Φ computes a sequence of DNC degree. Acknowledgments
The authors would like to thank Laurent Bienvenu, Frank Stephan, Mushfeq Khan, andPaul Shafer for helpful discussions. In particular, Bienvenu contributed to the reconstructionof the proof of Theorem 4.5, Khan informed us of the proof of the negligibility of the collectionof noncomputable sequences of hyperimmune-free degree in Proposition 3.17, and Stephanpointed out the relation with immunity notions discussed in Corollary 4.22.H¨olzl was supported by a Feodor Lynen postdoctoral research fellowship by the Alexandervon Humboldt Foundation and by the Ministry of Education of Singapore through grantR146-000-184-112 (MOE2013-T2-1-062). Porter was supported by the National Science Foun-dation through grant OISE-1159158 as part of the International Research Fellowship Programand by the John Templeton Foundation as part of the project “Structure and Randomness inthe Theory of Computation,” and by the National Security Agency Mathematical SciencesProgram grant H98230-I6-I-D310 as part of the Young Investigator’s Program.Both authors received travel support from the Bayerisch-Franz¨osisches Hochschulzentrum/Centre de Coop´eration Universitaire Franco-Bavarois. In addition, H¨olzl received travelsupport from the above Templeton grant for a collaborative visit to work with Porter at theUniversity of Florida.The opinions expressed in this publication are those of the authors and do not necessarilyreflect the views of the John Templeton Foundation.
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Institut 1, Fakult¨at f¨ur Informatik, Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany
E-mail address : [email protected] URL : http://hoelzl.fr Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311,USA
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