scholarly journals ON REALS WITH -BOUNDED COMPLEXITY AND COMPRESSIVE POWER

2016 ◽  
Vol 81 (3) ◽  
pp. 833-855
Author(s):  
IAN HERBERT
Keyword(s):  
Kolmogorov Complexity ◽  
Constant Factor ◽  
Binary String ◽  
Turing Degrees ◽  
Image Position ◽  
Bounded Complexity

AbstractThe (prefix-free) Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some ‘standard’ lowness notions for reals: A isK-trivial if its initial segments have the lowest possible complexity and A is low forKif using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold only up to all${\rm{\Delta }}_2^0$orders, and call the new notions${\rm{\Delta }}_2^0$-bounded K-trivialand${\rm{\Delta }}_2^0$-bounded low for K. Several of the ‘nice’ properties ofK-triviality are lost with this weakening. For instance, the new weaker definitions both give uncountable set of reals. In this paper we show that the weaker definitions are no longer equivalent, and that the${\rm{\Delta }}_2^0$-boundedK-trivials are cofinal in the Turing degrees. We then compare them to other previously studied weakenings, namelyinfinitely-often K-trivialityandweak lowness for K(in each, the defining inequality must hold up to a constant, but only for infinitely many inputs). We show that${\rm{\Delta }}_2^0$-boundedK-trivial implies infinitely-oftenK-trivial, but no implication holds between${\rm{\Delta }}_2^0$-bounded low forKand weakly low forK.

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 Complex tilings

2008 ◽  
Vol 73 (2) ◽  
pp. 593-613 ◽  
Author(s):  
Bruno Durand ◽  
Leonid A. Levin ◽  
Alexander Shen
Keyword(s):  
Kolmogorov Complexity ◽  
Turing Degrees ◽  
Degrees Of Unsolvability

AbstractWe study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings—that we also develop in terms of Turing degrees of unsolvability.

scholarly journals Bootstrap Percolation in High Dimensions

2010 ◽  
Vol 19 (5-6) ◽  
pp. 643-692 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
BÉLA BOLLOBÁS ◽  
ROBERT MORRIS
Keyword(s):  
Positive Root ◽  
Constant Factor ◽  
The Other ◽  
Bootstrap Percolation ◽  
Main Question ◽  
Critical Probability ◽  
High Dimensions ◽  
Critical Window ◽  
Fixed Constant ◽  
Image Position

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d ≫ log n.The bootstrap process has been extensively studied on [n]d when d is a fixed constant and 2 ⩽ r ⩽ d, and in these cases pc([n]d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2]d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n]d, d) asymptotically if d ≥ (log log n)2+ϵ, and gave much sharper bounds for the hypercube.Here we prove the following result. Let λ be the smallest positive root of the equation so λ ≈ 1.166. Then if d is sufficiently large, and moreover as d → ∞, for every function n = n(d) with d ≫ log n.

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.

scholarly journals Degrees of monotone complexity

2006 ◽  
Vol 71 (4) ◽  
pp. 1327-1341 ◽  
Author(s):  
William C. Calhoun
Keyword(s):  
Kolmogorov Complexity ◽  
Minimal Pair ◽  
Partial Ordering ◽  
Binary String ◽  
Linear Ordering ◽  
Relative Complexity ◽  
Computably Enumerable

AbstractLevin and Schnorr (independently) introduced the monotone complexity, Km (α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α | n) ≤ Km(β | n)+ c for all n. The monotone degree of α is the set of all β such that α Km β and β Km α. We show the monotone degrees contain an antichain of size , a countable dense linear ordering (of degrees of cardinality ), and a minimal pair.Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalban. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We defineareal to be (Km,K)-trivial if for some constant c, Km(α | n) ≤ K(n) + c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

scholarly journals THE COMPLEMENTS OF LOWER CONES OF DEGREES AND THE DEGREE SPECTRA OF STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 997-1006 ◽  
Author(s):  
URI ANDREWS ◽  
MINGZHONG CAI ◽  
ISKANDER SH. KALIMULLIN ◽  
STEFFEN LEMPP ◽  
JOSEPH S. MILLER ◽  
...  
Keyword(s):  
Turing Degrees ◽  
Degree Spectra ◽  
Degree Spectrum ◽  
Image Position ◽  
Countable Structure

AbstractWe study Turing degrees a for which there is a countable structure ${\cal A}$ whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ > 0‴.

scholarly journals On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

2006 ◽  
Vol 71 (1) ◽  
pp. 119-136 ◽  
Author(s):  
Stephen Binns ◽  
Bjørn Kjos-Hanssen ◽  
Manuel Lerman ◽  
Reed Solomon
Keyword(s):  
Set Theory ◽  
Computable Function ◽  
Turing Degrees ◽  
Fair Coin ◽  
Almost Everywhere ◽  
Regularity Properties ◽  
Turing Reducibility ◽  
Image Position ◽  
Almost All ◽  
Generic Extensions

Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article, we examine one of their conjectures concerning these notions.Throughout this article, ≤T denotes Turing reducibility and μ denotes the Lebesgue (or “fair coin”) probability measure on 2ω given byA property holds almost everywhere or for almost all X ∈ 2ω if it holds on a set of measure 1. For f, g ∈ ωω, f dominatesg if ∃m∀n < m(f(n) > g(n)).(Dobrinen, Simpson). A set A ∈ 2ωis almost everywhere (a.e.) dominating if for almost all X ∈ 2ω and all functions g ≤TX, there is a function f ≤TA such that f dominates g. A is uniformly almost everywhere (u.a.e.) dominating if there is a function f ≤TA such that for almost all X ∈ 2ω and all functions g ≤TX, f dominates g.There are several trivial but useful observations to make about these definitions. First, although these properties are stated for sets, they are also properties of Turing degrees. That is, a set is (u.)a.e. dominating if and only if every other set of the same degree is (u.)a.e. dominating. Second, both properties are closed upwards in the Turing degrees. Third, u.a.e. domination implies a.e. domination. Finally, if A is u.a.e. dominating, then there is a function f ≤TA which dominates every computable function.

scholarly journals COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

2016 ◽  
Vol 81 (3) ◽  
pp. 1028-1046 ◽  
Author(s):  
DENIS R. HIRSCHFELDT ◽  
CARL G. JOCKUSCH ◽  
RUTGER KUYPER ◽  
PAUL E. SCHUPP
Keyword(s):  
Main Tool ◽  
The Other ◽  
Compactness Theorem ◽  
Random Sets ◽  
Symmetric Difference ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Random Set ◽  
Natural Embedding ◽  
Image Position

AbstractA coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

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 WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

2018 ◽  
Vol 83 (1) ◽  
pp. 392-394
Author(s):  
LINDA BROWN WESTRICK
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Turing Degrees ◽  
Image Position ◽  
Scott Set

AbstractLet ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

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