scholarly journals ON THE INTERPLAY BETWEEN EFFECTIVE NOTIONS OF RANDOMNESS AND GENERICITY

2019 ◽  
Vol 84 (1) ◽  
pp. 393-407
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Random Sequence ◽  
Complete Classification ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Random Oracles ◽  
Image Position

AbstractIn this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

scholarly journals Computing from projections of random points

2019 ◽  
Vol 20 (01) ◽  
pp. 1950014
Author(s):  
Noam Greenberg ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Cost Function ◽  
Random Sequence ◽  
Turing Degrees ◽  
Orthogonal Projections ◽  
Open Set ◽  
Proper Subclass ◽  
The Family ◽  
Function Characterization ◽  
The Cost ◽  
The Ideal

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call [Formula: see text]-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the [Formula: see text]-trivial sets. We characterize [Formula: see text]-bases as the sets computable from both halves of Chaitin’s [Formula: see text], and as the sets that obey the cost function [Formula: see text]. Generalizing these results yields a dense hierarchy of subideals in the [Formula: see text]-trivial degrees: For [Formula: see text], let [Formula: see text] be the collection of sets that are below any [Formula: see text] out of [Formula: see text] columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be [Formula: see text]. Furthermore, the corresponding cost function characterization reveals that [Formula: see text] is independent of the particular representation of the rational [Formula: see text], and that [Formula: see text] is properly contained in [Formula: see text] for rational numbers [Formula: see text]. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the [Formula: see text]-trivial sets; we can calculate from the family which [Formula: see text] it characterizes. We finish by studying the union of [Formula: see text] for [Formula: see text]; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic 81(3) (2016) 1028–1046], who showed that it is a proper subclass of the [Formula: see text]-trivial sets. We prove that all such sets are robustly computable from [Formula: see text], and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a [Formula: see text] subideal of the [Formula: see text]-trivial sets.

scholarly journals THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

scholarly journals A Minimal Pair of Turing Degrees

2014 ◽  
Vol 6 (1) ◽  
Author(s):  
Patrizio Cintioli
Keyword(s):  
Minimal Pair ◽  
Turing Degrees

λ-Mappings Between Representation Rings of Lie Algebras

1983 ◽  
Vol 35 (5) ◽  
pp. 898-960 ◽  
Author(s):  
R. V. Moody ◽  
A. Pianzola
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Complete Classification ◽  
Representation Rings ◽  
Mathematical Formalization ◽  
Intuitive Idea ◽  
Image Position

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

scholarly journals Classification of universal formality maps for quantizations of Lie bialgebras

2020 ◽  
Vol 156 (10) ◽  
pp. 2111-2148
Author(s):  
Sergei Merkulov ◽  
Thomas Willwacher
Keyword(s):  
Complete Classification ◽  
Poisson Manifolds ◽  
Minimal Resolution ◽  
Lie Bialgebras ◽  
Induced Representation ◽  
Gauge Equivalence ◽  
Image Position ◽  
Drinfeld Associator ◽  
Deformation Complex

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$. We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

scholarly journals New Dichotomies for Borel Equivalence Relations

1997 ◽  
Vol 3 (3) ◽  
pp. 329-346 ◽  
Author(s):  
Greg Hjorth ◽  
Alexander S. Kechris
Keyword(s):  
Quotient Space ◽  
Complete Classification ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Recent Developments ◽  
Borel Map ◽  
Image Position ◽  
Completely Metrizable

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).Intuitively, E ≤BF might be interpreted in any one of the following ways:(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).(ii) One can classify E by using as invariants F-equivalence classes.(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.

COMPUTABILITY IN UNCOUNTABLE BINARY TREES

2019 ◽  
Vol 84 (3) ◽  
pp. 1049-1098
Author(s):  
REESE JOHNSTON
Keyword(s):  
Binary Trees ◽  
Turing Degrees ◽  
Image Position ◽  
König’S Lemma

AbstractComputability, while usually performed within the context of ω, may be extended to larger ordinals by means of α-recursion. In this article, we concentrate on the particular case of ω1-recursion, and study the differences in the behavior of ${\rm{\Pi }}_1^0$-classes between this case and the standard one.Of particular interest are the ${\rm{\Pi }}_1^0$-classes corresponding to computable trees of countable width. Classically, it is well-known that the analog to König’s Lemma—“every tree of countable width and uncountable height has an uncountable branch”—fails; we demonstrate that not only does it fail effectively, but also that the failure is as drastic as possible. This is proven by showing that the ω1-Turing degrees of even isolated paths in computable trees of countable width are cofinal in the ${\rm{\Delta }}_1^1\,{\omega _1}$-Turing degrees.Finally, we consider questions of nonisolated paths, and demonstrate that the degrees realizable as isolated paths and the degrees realizable as nonisolated ones are very distinct; in particular, we show that there exists a computable tree of countable width so that every branch can only be ω1-Turing equivalent to branches of trees with ${\aleph _2}$-many branches.

An extension of the nondiamond theorem in classical and α-recursion theory

1984 ◽  
Vol 49 (2) ◽  
pp. 586-607 ◽  
Author(s):  
Klaus Ambos-Spies
Keyword(s):  
Boolean Algebra ◽  
Recursion Theory ◽  
Upper Bounds ◽  
Minimal Pair ◽  
Splitting Theorem ◽  
Recursively Enumerable ◽  
Interesting Corollary ◽  
Image Position

Lachlan's nondiamond theorem [7, Theorem 5] asserts that there is no embedding of the four-element Boolean algebra (diamond) in the recursively enumerable degrees which preserves infima, suprema, and least and greatest elements. Lachlan observed that, essentially by relativization, the theorem can be extended toUsing the Sacks splitting theorem he concluded that there exists a pair of r.e. degrees which does not have an infimum, thus showing that the r.e. degrees do not form a lattice.We will first prove the following extension of (1):where an r.e. degree a is non-b-cappable if . From (2) we obtain more information about pairs of r.e. degrees without infima: For every nonzero low r.e. degree there exists an incomparable one such that the two degrees do not have an infimum and there is an r.e. degree which is not half of a pair of incomparable r.e. degrees which has an infimum in the low r.e. degrees. Probably the most interesting corollary of (2) is that the join of any cappable r.e. degree (i.e. half of a minimal pair) and any low r.e. degree is incomplete. Consequently there is an incomplete noncappable degree above every incomplete r.e. degree. Cooper's result [3] that ascending sequences of uniformly r.e. degrees can have minimal upper bounds in the set R of r.e. degrees is another corollary of (2).

A jump class of noncappable degrees

1989 ◽  
Vol 54 (2) ◽  
pp. 324-353 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Pair ◽  
Recursively Enumerable ◽  
Image Position

Friedberg [3] showed that every degree of unsolvability above 0′ is the jump of some degree, and Sacks [9] showed that the degrees above 0′ which are recursively enumerable (r.e.) in 0′ are the jumps of the r.e. degrees.In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree c > 0′ which is r.e. in 0′ but which is not the jump of half a minimal pair of r.e. degrees.This extends Yates' result [13] proving the existence of noncappable degrees (that is, r.e. degrees a < 0′ for which there is no corresponding r.e. b > 0 with a ∩ b = 0).It also throws more light on the class PS of promptly simple degrees. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that PS coincides with the class NC of noncappable degrees, and with the class LC of all low-cuppable degrees, and (using earlier work of Maass, Shore and Stob [5]) that PS splits every class Hn or Ln, n ≥ 0, in the high-low hierarchy of r.e. degrees.If c > 0′, with c r.e. in 0′, letand call c−1 the jump class for c. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low a ∈ LC, where of course [a, 0′] (= {br.e. ∣a ≤ b ≤ 0′}) ⊆ LC = PS. But by Robinson [7] [a, 0′] intersects with every jump class.

Sign in / Sign up

Export Citation Format

Share Document