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 The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

scholarly journals The Computational Strength of Extensions of Weak König’s Lemma

1998 ◽  
Vol 5 (41) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Natural Number ◽  
Arbitrary Function ◽  
The Other ◽  
Binary Trees ◽  
Subsequent Paper ◽  
Bounded Functions ◽  
The One ◽  
Weak König’S Lemma ◽  
König’S Lemma

The weak König's lemma WKL is of crucial significance in the study of fragments of mathematics which on the one hand are mathematically strong but on the other hand have a low proof-theoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL<br />is also `weak' in that the tree predicate is quantifier-free. Whereas in general the computational and proof-theoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are<br />given by formulas in a class Phi where we allow an arbitrary function quantifier prefix over bounded functions in front of a Pi^0_1-formula. This results in a schema Phi-WKL.<br />Another way of looking at WKL is via its equivalence to the principle<br /> For all x there exists y<=1 for all z A0(x; y; z) -> there exists f <= lambda x.1 for all x, z A0(x, fx, z);<br />where A0 is a quantifier-free formula (x, y, z are natural number variables). <br /> We generalize this to Phi-formulas as well and allow function quantifiers `there exists g <= s'<br />instead of `there exists y <= 1', where g <= s is defined pointwise. The resulting schema is called Phi-b-AC^0,1.<br />In the absence of functional parameters (so in particular in a second order context), the corresponding versions of Phi-WKL and Phi-b-AC^0,1 turn out to<br />be equivalent to WKL. This changes completely in the presence of functional<br />variables of type 2 where we get proper hierarchies of principles Phi_n-WKL and<br />Phi_n-b-AC^0,1. Variables of type 2 however are necessary for a direct representation<br />of analytical objects and - sometimes - for a faithful representation of<br />such objects at all as we will show in a subsequent paper. By a reduction of<br />Phi-WKL and Phi-b-AC^0,1 to a non-standard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in all finite types, we prove that Phi-WKL and Phi-b-AC^0,1 do<br />neither contribute to the provably recursive functionals of these fragments nor to their proof-theoretic strength. In a subsequent paper we will illustrate the greater mathematical strength of these principles (compared to WKL).

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 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 ${\cal S}$.

scholarly journals Weighted Interlace Polynomials

2009 ◽  
Vol 19 (1) ◽  
pp. 133-157 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Polynomial Time ◽  
Independent Set ◽  
Tutte Polynomial ◽  
Binary Trees ◽  
Computation Trees ◽  
Image Position

The interlace polynomials introduced by Arratia, Bollobás and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formulathat lacks the last term. It follows that interlace polynomial computations can be represented by binary trees rather than mixed binary–ternary trees. Binary computation trees provide a description ofq(G) that is analogous to the activities description of the Tutte polynomial. IfGis a tree or forest then these ‘algorithmic activities’ are associated with a certain kind of independent set inG. Three other novel properties are weighted pendant-twin reductions, which involve removing certain kinds of vertices from a graph and adjusting the weights of the remaining vertices in such a way that the interlace polynomials are unchanged. These reductions allow for smaller computation trees as they eliminate some branches. If a graph can be completely analysed using pendant-twin reductions, then its interlace polynomial can be calculated in polynomial time. An intuitively pleasing property is that graphs which can be constructed through graph substitutions have vertex-weighted interlace polynomials which can be obtained through algebraic substitutions.

Comparing incomparable kleene degrees

1985 ◽  
Vol 50 (1) ◽  
pp. 55-58
Author(s):  
Philip Welch
Keyword(s):  
Turing Degrees ◽  
Generic Extension ◽  
Simple Application ◽  
Whole Process ◽  
Simple Sets ◽  
Image Position

In the December 1982 issue of this Journal Weitkamp [W] posed some questions concerning the incomparability of certain “r.e.” sets for the notion of Kleene reducibility. He asked whether the incomparability of, for example, the Friedman set F (defined below) and the set WI0 (the set of reals coding wellfounded trees of admissible height) was equivalent to the existence of 0#, since forcing over L with a set of conditions could not achieve this. We answer this by showing that in a certain class generic extension of L they are comparable, but 0# does not exist. This is an application of Jensen's coding theorem (cf. [BJW]), using a modified construction due to René David [D]. Indeed the result here is a simple application of his result. Define F as follows:Harrington showed, in effect, that one could not add a cone of Turing degrees to this set by forcing with sets of conditions over L. The method used here does add a cone of Turing degrees to a much simpler set RI1 (defined below)—and indeed the whole process could be viewed as forcing over L to obtain the determinacy of certain rather simple sets. It is the determinacy of the game with payoff set RI1 that ensures the comparability of F and WI0 (amongst many others).We shall refrain from repeating all the basic definitions and lemmas since the reader can readily refer to [W]; we shall give the basic necessities.

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