arXiv subjects

Stephen G. Simpson

Publications and source records attributed to Stephen G. Simpson.

10 recordsLinked to original sources

arXiv2021

Pseudojump inversion in special r. b. $Π^0_1$ classes

The Jump Inversion Theorem says that for every real $A \ge_T 0'$ there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0'$. A known refinement of this theorem says that we can choose $B$ to be a member of any special $Π^0_1$ subclass of $\{0,1\}^ω$. We now consider the possibility of analogous refinements of two other well-known theorems: the Join Theorem -- for all reals $A$ and $Z$ such that $A \ge_T Z \oplus 0'$ and $Z >_T 0$, there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0' \equiv_T B \oplus Z$ -- and the Pseudojump Inversion Theorem -- for all reals $A \ge_T 0'$ and every $e \in ω$, there is a real $B$ such that $A \equiv_T B \oplus W_e^B \equiv_T B \oplus 0'$. We show that in these theorems, $B$ can be found in some special $Π^0_1$ subclasses of $\{0,1\}^ω$ but not in others.

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arXiv2021

Turing Degrees of Hyperjumps

The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z \oplus 0' \leq_\mathrm{T} A$ and $0 <_\mathrm{T} Z$, there exists $B$ such that $A \equiv_\mathrm{T} B' \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus 0'$. Consequently, any nonzero Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a Turing jump relative to some $B$. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 <_\mathrm{HYP} Z$, there exists $B$ such that $A \equiv_\mathrm{T} \mathcal{O}^B \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus \mathcal{O}$. As an analogous consequence, any nonhyperarithmetical Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a hyperjump relative to some $B$.

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arXiv2021

Very weak fragments of weak Kőnig's lemma

It is well-known that any finite $Π^{0}_{1}$-class of $2^{\mathbb N}$ has a computable member. Then, how can we understand this in the context of reverse mathematics? In this note, we consider several very weak fragments of Kőnig's lemma to answer this qeustion.

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arXiv2018

A symmetric $β$-model

We prove that there exists a countable $β$-model in which, for all reals $X$ and $Y$, $X$ is definable from $Y$ if and only $X$ is hyperarithmetical in $Y$. We also obtain some related results and pose some related questions.

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arXiv2017

Symbolic dynamics: entropy = dimension = complexity

Let $G$ be the group $\mathbb{Z}^d$ or the monoid $\mathbb{N}^d$ where $d$ is a positive integer. Let $X$ be a subshift over $G$, i.e., a closed and shift-invariant subset of $A^G$ where $A$ is a finite alphabet. We prove that the topological entropy of $X$ is equal to the Hausdorff dimension of $X$ and has a sharp characterization in terms of the Kolmogorov complexity of finite pieces of the orbits of $X$. In the version of this paper that has been published in Theory of Computing Systems, the proof of Lemma 4.3 contains a confusing typographical error. This version of the paper corrects that error.

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arXiv2015

Reverse mathematics, Young diagrams, and the ascending chain condition

Let $S$ be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over RCA$_0$ (or even over RCA$_0^*$) to the statement that $ω^ω$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

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arXiv2015

Comparing WO$(ω^ω)$ with $Σ^0_2$ induction

Let WO$(ω^ω)$ be the statement that the ordinal number $ω^ω$ is well ordered. WO$(ω^ω)$ has occurred several times in the reverse-mathematical literature. The purpose of this expository note is to discuss the place of WO$(ω^ω)$ within the standard hierarchy of subsystems of second-order arithmetic. We prove that WO$(ω^ω)$ is implied by I$Σ^0_2$ and independent of B$Σ^0_2$. We also prove that WO$(ω^ω)$ and B$Σ^0_2$ together do not imply I$Σ^0_2$.

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arXiv2014

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.

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arXiv2013

Propagation of partial randomness

Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-L"of random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree.

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arXiv2013

Harrington's results on arithmetical singletons

We exposit two previously unpublished theorems of Leo Harrington. The first theorem says that there exist arithmetical singletons which are arithmetically incomparable. The second theorem says that there exists a ranked point which is not an arithmetical singleton. Unlike Harrington's proofs of these theorems, our proofs do not use the finite- or infinite-injury priority method. Instead they use an oracle construction adapted from the standard proof of the Friedberg Jump Theorem.

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