Peter Cholak
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View article: LOW$_2$ COMPUTABLY ENUMERABLE SETS HAVE HYPERHYPERSIMPLE SUPERSETS
LOW$_2$ COMPUTABLY ENUMERABLE SETS HAVE HYPERHYPERSIMPLE SUPERSETS Open
A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal {L}^*(A)$ , of a low $_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal {L}^*(A)\cong {\mathcal E}^*$ . In spite of cl…
View article: Algorithmic Information Bounds for Distances and Orthogonal Projections
Algorithmic Information Bounds for Distances and Orthogonal Projections Open
We develop quantitative algorithmic information bounds for orthogonal projections and distances in the plane. Under mild independence conditions, the distance $|x-y|$ and a projection coordinate $p_e x$ each retain at least half the algori…
View article: Low$_2$ computably enumerable sets have hyperhypersimple supersets
Low$_2$ computably enumerable sets have hyperhypersimple supersets Open
A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal{L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal{L}^*(A)\cong {\mathcal E}^*$. In spite of claims …
View article: Bounding the dimension of exceptional sets for orthogonal projections
Bounding the dimension of exceptional sets for orthogonal projections Open
It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(π_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ a…
View article: Tighter Bounds on the Expressivity of Transformer Encoders
Tighter Bounds on the Expressivity of Transformer Encoders Open
Characterizing neural networks in terms of better-understood formal systems has the potential to yield new insights into the power and limitations of these networks. Doing so for transformers remains an active area of research. Bhattamishr…
View article: A definable relation between c.e. sets and ideals
A definable relation between c.e. sets and ideals Open
The Pi-0-1 classes have become important structures in computability theory. Related to the study of properties of individual classes is the study of the lattice of all Pi-0-1 classes, denoted E_Pi. We define a substructure of E_Pi, G = [N…
View article: Overcoming a Theoretical Limitation of Self-Attention
Overcoming a Theoretical Limitation of Self-Attention Open
Although transformers are remarkably effective for many tasks, there are some surprisingly easy-looking regular languages that they struggle with. Hahn shows that for languages where acceptance depends on a single input symbol, a transform…
View article: Overcoming a Theoretical Limitation of Self-Attention
Overcoming a Theoretical Limitation of Self-Attention Open
Although transformers are remarkably effective for many tasks, there are some surprisingly easy-looking regular languages that they struggle with. Hahn shows that for languages where acceptance depends on a single input symbol, a transform…
View article: Extending Properly n-REA Sets
Extending Properly n-REA Sets Open
In [5] Soare and Stob prove that if $A$ is an r.e. set which isn't computable then there is a set of the form $A \oplus W^A_e$ which isn't of r.e. Turing degree. If we define a properly $n+1$-REA set to be an $n+1$-REA set which isn't Turi…
View article: Extending Properly n-REA Sets.
Extending Properly n-REA Sets. Open
In [5] Soare and Stob prove that if $A$ is an r.e. set which isn't computable then there is a set of the form $A \oplus W^A_e$ which isn't of r.e. Turing degree. If we define a properly $n+1$-REA set to be an $n+1$-REA set which isn't Turi…
View article: Realizing Computably Enumerable Degrees in Separating Classes
Realizing Computably Enumerable Degrees in Separating Classes Open
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}…
View article: Milliken's tree theorem and its applications: a computability-theoretic perspective
Milliken's tree theorem and its applications: a computability-theoretic perspective Open
Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences. Motivated by a question of Dobrinen, we initiat…
View article: Some results concerning the SRT 2 2 vs. COH problem
Some results concerning the SRT 2 2 vs. COH problem Open
The $\\mathsf{SRT}^2_2$ vs.\\ $\\mathsf{COH}$ problem is a central problem in\ncomputable combinatorics and reverse mathematics, asking whether every Turing\nideal that satisfies the principle $\\mathsf{SRT}^2_2$ also satisfies the\nprinci…
View article: Some results concerning the $\mathsf{SRT}^2_2$ vs. $\mathsf{COH}$ problem
Some results concerning the $\mathsf{SRT}^2_2$ vs. $\mathsf{COH}$ problem Open
The $\mathsf{SRT}^2_2$ vs.\ $\mathsf{COH}$ problem is a central problem in computable combinatorics and reverse mathematics, asking whether every Turing ideal that satisfies the principle $\mathsf{SRT}^2_2$ also satisfies the principle $\m…
View article: Effective prime uniqueness
Effective prime uniqueness Open
Assuming the obvious definitions (see paper) we show the a decidable model that is effectively prime is also effectively atomic. This implies that two effectively prime (decidable) models are computably isomorphic. This is in contrast to t…
View article: Computably enumerable sets that are automorphic to low sets
Computably enumerable sets that are automorphic to low sets Open
We work with the structure consisting of all computably enumerable (c.e.) sets ordered by set inclusion. The question we will partially address is which c.e.\ sets are autormorphic to low (or low$_2$ sets. Using work of Miller, we can see …
View article: The Rado Path Decomposition Theorem
The Rado Path Decomposition Theorem Open
We discuss a theorem of Rado: Every r-coloring of the pairs of natural numbers has a path decomposition.
View article: Any FIP real computes a 1-generic
Any FIP real computes a 1-generic Open
We construct a computable sequence of computable reals such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is extended to establish th…
View article: Density-1-bounding and quasiminimality in the generic degrees
Density-1-bounding and quasiminimality in the generic degrees Open
We consider the question "Is every nonzero generic degree a density-1-bounding generic degree?" By previous results \cite{I2} either resolution of this question would answer an open question concerning the structure of the generic degrees:…
View article: Generics for Mathias forcing over general Turing ideals
Generics for Mathias forcing over general Turing ideals Open
In Mathias forcing, conditions are pairs $(D,S)$ of sets of natural numbers, in which $D$ is finite, $S$ is infinite, and $\max D < \min S$. The Turing degrees and computational characteristics of generics for this forcing in the special (…
View article: Any FIP real computes a 1-generic
Any FIP real computes a 1-generic Open
We construct a computable sequence of computable reals $\langle X_i\rangle$ such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is exte…