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Linda Brown Westrick

Publications and source records attributed to Linda Brown Westrick.

13 recordsLinked to original sources
arXiv2021

Effectiveness for the Dual Ramsey Theorem

We analyze the Dual Ramsey Theorem for $k$ partitions and $\ell$ colors ($\mathsf{DRT}^k_\ell$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}_0$, the Dual Ramsey Theorem stated for Baire colorings is equivalent to the statement for clopen colorings and to a purely combinatorial theorem $\mathsf{cDRT}^k_\ell$. When the theorem is stated for Borel colorings and $k\geq 3$, the resulting principles are essentially relativizations of $\mathsf{cDRT}^k_\ell$. For each $α$, there is a computable Borel code for a $Δ^0_α$ coloring such that any partition homogeneous for it computes $\emptyset^{(α)}$ or $\emptyset^{(α-1)}$ depending on whether $α$ is infinite or finite. For $k=2$, we present partial results giving bounds on the effective content of the principle. A weaker version for $Δ^0_n$ reduced colorings is equivalent to $\mathsf{D}^n_2$ over $\mathsf{RCA}_0+\mathsf{I}Σ^0_{n-1}$ and in the sense of strong Weihrauch reductions.

math.LO↗
arXiv2020

The determined property of Baire in reverse math

We define the notion of a determined Borel code in reverse math, and consider the principle $DPB$, which states that every determined Borel set has the property of Baire. We show that this principle is strictly weaker than $ATR$. Any $ω$-model of $DPB$ must be closed under hyperarithmetic reduction, but $DPB$ is not a theory of hyperarithmetic analysis. We show that whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $DPB$, then for every $Z \in M$, there is a $G \in M$ such that $G$ is $Δ^1_1$-generic relative to $Z$.

math.LO↗
arXiv2019

An effective analysis of the Denjoy rank

We analyze the descriptive complexity of several $Π^1_1$ ranks from classical analysis which are associated to Denjoy integration. We show that $VBG, VBG_\ast, ACG$ and $ACG_\ast$ are $Π^1_1$-complete, answering a question of Walsh in case of $ACG_\ast$. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most $α$ steps of the transfinite process of Denjoy totalization: if $|\cdot|$ is the $Π^1_1$-rank naturally associated to $VBG, VBG_\ast$ or $ACG_\ast$, and if $α<ω_1^{ck}$, then $\{F \in C(I): |F| \leq α\}$ is $Σ^0_{2α}$-complete. These finer results are an application of the author's previous work on the limsup rank on well-founded trees. Finally, $\{(f,F) \in M(I)\times C(I) : F\in ACG_\ast \text{ and } F'=f \text{ a.e.}\}$ and $\{f \in M(I) : f \text{ is Denjoy integrable}\}$ are $Π^1_1$-complete, answering more questions of Walsh.

math.LO↗
arXiv2019

Three topological reducibilities for discontinuous functions

We define a family of three related reducibilities, $\leq_T$, $\leq_{tt}$ and $\leq_m$, for arbitrary functions $f,g:X\rightarrow\mathbb R$, where $X$ is a compact separable metric space. The $\equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $α$-jump functions $j_α:2^ω\rightarrow \mathbb R$ are $\leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $\leq_{tt}$ and $\leq_m$, finding an exact match to the $α$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.

math.LO↗
arXiv2018

The reverse mathematics of Hindman's theorem for sums of exactly two elements

Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation of restricted versions of HT from the computability-theoretic and reverse-mathematical perspectives has been a productive line of research recently. In particular, HT$^{\leqslant n}_k$ is the restriction of HT to sums of at most $n$ many elements, with at most $k$ colors allowed, and HT$^{=n}_k$ is the restriction of HT to sums of \emph{exactly} $n$ many elements and $k$ colors. Even HT$^{\leqslant 2}_2$ appears to be a strong principle, and may even imply HT itself over RCA$_0$. In contrast, HT$^{=2}_2$ is known to be strictly weaker than HT over RCA$_0$, since HT$^{=2}_2$ follows immediately from Ramsey's Theorem for $2$-colorings of pairs. In fact, it was open for several years whether HT$^{=2}_2$ is computably true. We show that HT$^{=2}_2$ and similar results with addition replaced by subtraction and other operations are not provable in RCA$_0$, or even WKL$_0$. In fact, we show that there is a computable instance of HT$^{=2}_2$ such that all solutions can compute a function that is diagonally noncomputable relative to $\emptyset'$. It follows that there is a computable instance of HT$^{=2}_2$ with no $Σ^0_2$ solution, which is the best possible result with respect to the arithmetical hierarchy. Furthermore, a careful analysis of the proof of the result above about solutions DNC relative to $\emptyset'$ shows that HT$^{=2}_2$ implies RRT$^{=2}_2$, the Rainbow Ramsey Theorem for $2$-colorings of pairs, over RCA$_0$. The most interesting aspect of our construction of computable colorings as above is the use of an effective version of the Lovász Local Lemma due to Rumyantsev and Shen.

math.LO↗
arXiv2017

Weakly 2-randoms and 1-generics in Scott sets

Let $S$ be a Scott set, or even an $ω$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if $A_1,\dots, A_n \in S$ are non-computable, there is $X \in S$ such that each $A_i$ is Turing incomparable with $X$, answering a question of Kučera and Slaman. More generally, any $\forall\exists$ sentence in the language of partial orders that holds in $\mathcal D$ also holds in $\mathcal D_S$, where $\mathcal D_S$ is the partial order of Turing degrees of elements of $S$.

math.LO↗
arXiv2017

Dimension 1 sequences are close to randoms

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly $s$-random sequence. Further, for any $s<t$, every sequence of effective dimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of its bits to produce a sequence of effective dimension $t$, and this bound is optimal.

math.LO↗
arXiv2017

Finding bases of uncountable free abelian groups is usually difficult

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.

math.LO↗
arXiv2016

Seas of squares with sizes from a $Π^0_1$ set

For each $Π^0_1$ $S\subseteq \mathbb{N}$, let the $S$-square shift be the two-dimensional subshift on the alphabet $\{0,1\}$ whose elements consist of squares of 1s of various sizes on a background of 0s, where the side length of each square is in $S$. Similarly, let the distinct-square shift consist of seas of squares such that no two finite squares have the same size. Extending the self-similar Turing machine tiling construction of Durand, Romashchenko and Shen, we show that if $X$ is an $S$-square shift or any effectively closed subshift of the distinct square shift, then $X$ is sofic.

math.DS↗
arXiv2016

Ramsey's theorem for singletons and strong computable reducibility

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for $\ell$-colorings, $\mathsf{SRT}^2_\ell$. Our proof actually establishes the following considerably stronger fact: given $k > \ell$, there is a coloring $c : ω\to k$ such that for every stable coloring $d : [ω]^2 \to \ell$ (computable from $c$ or not), there is an infinite homogeneous set $H$ for $d$ that computes no infinite homogeneous set for $c$. This also answers a separate question of Dzhafarov, as it follows that the cohesive principle, $\mathsf{COH}$, is not strongly computably reducible to the stable Ramsey's theorem for all colorings, $\mathsf{SRT}^2_{<\infty}$. The latter is the strongest partial result to date in the direction of giving a negative answer to the longstanding open question of whether $\mathsf{COH}$ is implied by the stable Ramsey's theorem in $ω$-models of $\mathsf{RCA}_0$.

math.LO↗
arXiv2016

Effectiveness of Hindman's theorem for bounded sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F \subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ elements of $X$ have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in $\mathsf{RCA}_0$. We also show that there is a computable instance of $\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in $\mathsf{RCA}_0$.

math.LO↗
arXiv2013

A lightface analysis of the differentiability rank

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than omega_1 which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal alpha>0, the set of Turing indices of C[0,1] functions that are differentiable with rank at most alpha is Pi_{2 alpha + 1}-complete. This result is expressed in the notation of Ash and Knight.

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