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Peter Cholak

Publications and source records attributed to Peter Cholak.

At least 19 recordsLinked to original sources
arXiv2026

The finite big Ramsey degrees of Henson graphs are provable in $\mathrm{ACA}_0$

Let $\mathbb{H}_{n+1}$ denote a computable copy of the $(n+1)$-clique free universal homogeneous Henson graph, $G$ denote a finite subgraph of $\mathbb{H}_{n+1}$, and $k(G,n)$ denote the big Ramsey degree of $G$ in $\mathbb{H}_{n+1}$. We prove that for any computable coloring $\chi$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is a copy $\mathbb{H}'$ of $\mathbb{H}_{n+1}$ that is computable from $0^{(2\delta(G,n)-1)}$ in which $\chi$ takes no more than $k(G,n)$ colors, where $\delta(G,n)$ denotes the maximum number of levels of a diary for $G$ in $\mathbb{H}_{n+1}$ (this is a finite number). It follows that the statement, ``Henson graphs have finite big Ramsey degrees," is provable in ACA$_0'$. Combining this with a recent result of Cholak, Dobrinen, and McCoy \cite{CDM} yields the equivalence of the statement with ACA$_0'$ over RCA$_0$.

math.LO↗
arXiv2026

The Henson graphs: colorings and codings

By recent work of \citet{DobrinenICM} and \citet{Balko7} we know that every finite $G$ in the Henson graph $\mathbb{H}_{n+1}$ (the universal ultrahomogeneous $(n+1)$-clique free graph) has exact finite big Ramsey degree $k({G,n})$. That is, there is a positive integer $k({G,n})$ such that for each finite coloring $C$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is $\tilde{\mathbb{H}}$, a substructure of $\mathbb{H}_{n+1}$ and isomorphic to $\mathbb{H}_{n+1}$, such that in $\tilde{\mathbb{H}}$ at most $k({G,n})$ colors are used on the copies of $G$ in $\tilde{\mathbb{H}}$. Moreover, for exactness, for some coloring and all corresponding $\tilde{\mathbb{H}}$, all $k({G,n})$ colors are needed. The ultimate result here is that if $|G|\geq 2$, then there is a finite computable coloring $C$ such that, for all such $\tilde{\mathbb{H}}$, we have that $\tilde{\mathbb{H}}$ computes $\emptyset^{(|G|-1)}$ (and hence the halting set).

math.LO↗
arXiv2026

Ramsey Theory and Bounding in Arithmetic

We explore the relation between various versions of Ramsey theorem and bounding schemes in model ${N}$ of a fragment of arithmetic $F$. Our goal is to recast, in a different framework, and extend some results of Hirst \cite{Hirst-1987}, see Theorem 1. We will extract Weihrauch reductions from Hirst's and similar proofs. Our results, informally stated in the our terminology, all inside ${N}$, follow: First the following are equivalent: $B\Sigma_2$, the finite union of finite c.e.\ sets is finite, and Infinite Pigeonhole Principle, see Theorem 3. We also discuss the Weihrauch relations between these logically equivalent principles, see Section 4. The Infinite Pigeonhole Principle is Weihrauch reducible to $RT^2_2$, see Theorem 4. There are also another principle logically equivalent to $B\Sigma_2$ which is Weihrauch reducible to $SRT^2_2$, see Theorem 5. We show that there is a principle which is equivalent with $B\Sigma_3$, see Theorem 6, and Weihrauch reducible to $SRT^2_{<\infty}$, Theorem 7. We discuss some equivalencies with $B\Sigma_{n-1}$, see Subsection 6.1, and then end with a problem Weihrauch reducible to $RT^{n+1}_{2}$, Subsection 6.2. The reader should be aware since we working within ${N}$ many of the standard definitions need to be adjusted to work. Due to the expository nature of this short paper, these definitions are sprinkled throughout the paper. A quick read of the paper from start to finish will provide a better understanding of the ideas involved rather than a careful reading of theorems.

math.LO↗
arXiv2025

Algorithmic Information Bounds for Distances and Orthogonal Projections

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

cs.CC↗
arXiv2024

Low$_2$ computably enumerable sets have hyperhypersimple supersets

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 in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e.\ $A$ is low$_2$ then $A$ has an atomless hyperhypersimple superset. In fact, if $A$ is c.e.\ and low$_2$, then for any $\Sigma_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$.

math.LO↗
arXiv2024

Bounding the dimension of exceptional sets for orthogonal projections

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $\pi_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(\pi_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for $k=1$ and for $k=n-1$. Hence we completely resolve this question for $n=3$.

math.CA↗
arXiv2023

Tighter Bounds on the Expressivity of Transformer Encoders

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. Bhattamishra and others have shown that transformer encoders are at least as expressive as a certain kind of counter machine, while Merrill and Sabharwal have shown that fixed-precision transformer encoders recognize only languages in uniform $TC^0$. We connect and strengthen these results by identifying a variant of first-order logic with counting quantifiers that is simultaneously an upper bound for fixed-precision transformer encoders and a lower bound for transformer encoders. This brings us much closer than before to an exact characterization of the languages that transformer encoders recognize.

cs.LG↗
arXiv2022

Overcoming a Theoretical Limitation of Self-Attention

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 transformer's classification decisions become less and less confident (that is, with cross-entropy approaching 1 bit per string) as input strings get longer and longer. We examine this limitation using two languages: PARITY, the language of bit strings with an odd number of 1s, and FIRST, the language of bit strings starting with a 1. We demonstrate three ways of overcoming the limitation suggested by Hahn's lemma. First, we settle an open question by constructing a transformer that recognizes PARITY with perfect accuracy, and similarly for FIRST. Second, we use layer normalization to bring the cross-entropy of both models arbitrarily close to zero. Third, when transformers need to focus on a single position, as for FIRST, we find that they can fail to generalize to longer strings; we offer a simple remedy to this problem that also improves length generalization in machine translation.

cs.LG↗
arXiv2020

Realizing Computably Enumerable Degrees in Separating Classes

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 $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.

math.LO↗
arXiv2019

Thin set theorems and cone avoidance

The thin set theorem $\mathsf{RT}^n_{<\infty,\ell}$ asserts the existence, for every $k$-coloring of the subsets of natural numbers of size $n$, of an infinite set of natural numbers, all of whose subsets of size $n$ use at most $\ell$ colors. Whenever $\ell = 1$, the statement corresponds to Ramsey's theorem. From a computational viewpoint, the thin set theorem admits a threshold phenomenon, in that whenever the number of colors $\ell$ is sufficiently large with respect to the size $n$ of the tuples, then the thin set theorem admits strong cone avoidance. Let $d_0, d_1, \dots$ be the sequence of Catalan numbers. For $n \geq 1$, $\mathsf{RT}^n_{<\infty, \ell}$ admits strong cone avoidance if and only if $\ell \geq d_n$ and cone avoidance if and only if $\ell \geq d_{n-1}$. We say that a set $A$ is $\mathsf{RT}^n_{<\infty, \ell}$-encodable if there is an instance of $\mathsf{RT}^n_{<\infty, \ell}$ such that every solution computes $A$. The $\mathsf{RT}^n_{<\infty, \ell}$-encodable sets are precisely the hyperarithmetic sets if and only if $\ell < 2^{n-1}$, the arithmetic sets if and only if $2^{n-1} \leq \ell < d_n$, and the computable sets if and only if $d_n \leq \ell$.

math.LO↗
arXiv2017

Effective Prime Uniqueness

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 the theorem that there are two atomic decidable models which are not computably isomorphic. The implications of this work in reverse mathematics is that "effectively prime implies effectively atomic" holds in topped models. But due to an observation of David Belanger, "effectively prime implies effectively atomic" fails for some Scott sets. The reserve mathematical strength of "Prime Uniqueness" remains open.

math.LO↗
arXiv2016

On Splits of Computably enumerable sets

Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly.

math.LO↗
arXiv2016

Density-1-bounding and quasiminimality in the generic degrees

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: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees. We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding. We also consider a historic difficulty in constructing a potential counterexample: By previous results \cite{I1} any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal. Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them. Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp \cite{HJKS}, this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.

math.LO↗
arXiv2016

Any FIP real computes a 1-generic

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 extended to establish the same result with 2IP in place of FIP.

math.LO↗
arXiv2015

Computably Enumerable Sets that are Automorphic to Low Sets

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 that every set with semilow complement is $Δ^0_3$ automorphic to a low set. While it remains open whether every set with semilow complement is effectively automorphic to a low set, we show that there are sets without semilow complement that are effectively automorphic to low sets. We also consider other lowness notions such as having a semilow$_{1.5}$ complement, having the the outer splitting property, and having a semilow$_2$ complement. We show that in every non low \ce degree, there are sets with semilow$_{1.5}$ complements without semilow complements as well as sets with semilow$_2$ complements and the outer splitting property that do not have semilow$_{1.5}$ complements. We also address the question of which sets are automorphic to low$_2$ sets.

math.LO↗
arXiv2014

$\mathcal{D}$-maximal sets

Soare proved that the maximal sets form an orbit in $\mathcal{E}$. We consider here $\mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $\mathcal{D}$-maximal sets are well understood, e.g., hemimaximal sets, but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the $\mathcal{D}$-maximal sets. Although these invariants help us to better understand the $\mathcal{D}$-maximal sets, we use them to show that several classes of $\mathcal{D}$-maximal sets break into infinitely many orbits.

math.LO↗