arXiv subjects

Johanna N. Y. Franklin

Publications and source records attributed to Johanna N. Y. Franklin.

12 recordsLinked to original sources

arXiv2026

Structural vs. computational complexity

We consider highness in the context of computable structure theory and, particularly, the Scott rank of a structure. We define highness for Scott rank and highness for computably defined Scott rank $α$ and characterize them in terms of the ability to compute $Δ^0_β$ sets for appropriate $β$. We close with a discussion of the index sets of structures with a given Scott rank or computably defined Scott rank and a few words about highness for noncomputable Scott ranks.

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arXiv2026

Algorithmic randomness in harmonic analysis

Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-π,π]$ converges are precisely the Martin-Löf random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-Löf random reals.

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arXiv2025

Failure Modes for Structural Highness Notions

In a previous paper, entitled "Structural Highness Notions," we defined several classes of degrees that are high in senses related to computable structure theory. Each class of degrees is characterized by a structural feature (e.g., an isomorphism) that it can compute if such a feature exists. In this paper, we examine each of these classes and characterize them based on what they do if no such object exists. We describe, in particular, reticent, loquacious, and collegiate senses of being high. These, respectively, reflect the case where a computation from the degree can give output only if the desired feature exists, the case where it will give output of some kind whether or not the feature exists, and the case where the degree will either compute the feature or the best available approximation to it.

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arXiv2025

Computable classifications of continuous, transducer, and regular functions

We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipschitz functions is $Σ^0_2$-complete. (Every regular function is pointwise linear-time Lipschitz.) We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular. As one of many consequences, our $Σ^0_2$-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space $C[0,1]$ of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.

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arXiv2022

Free structures and limiting density

Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the non-abelian free groups. We ask Gromov's question more generally, for structures in an arbitrary algebraic variety (in the sense of universal algebra), with presentations of a specific form. We focus on elementary properties. We give examples illustrating different behaviors of the limiting density. Based on the examples, we identify sufficient conditions for the elementary first-order theory of the free structure to match that of the typical structure; i.e., a sentence is true in the free structure iff it has limiting density 1.

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arXiv2021

Structural Highness Notions

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

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arXiv2020

Key developments in algorithmic randomness

The goal of this introductory survey is to present the major developments of algorithmic randomness with an eye toward its historical development. While two highly comprehensive books and one thorough survey article have been written on the subject, our goal is to provide an introduction to algorithmic randomness that will be both useful for newcomers who want to develop a sense of the field quickly and interesting for researchers already in the field who would like to see these results presented in chronological order.

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arXiv2019

Degrees of and lowness for isometric isomorphism

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces.

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arXiv2019

Lowness for isomorphism, countable ideals, and computable traceability

We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune-free degrees, lowness for isomorphism is entirely independent of computable traceability.

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arXiv2019

Taking the path computably travelled

We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $Π^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.

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arXiv2012

Randomness and Non-ergodic Systems

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.

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arXiv2011

Anti-complex sets and reducibilities with tiny use

In contrast with the notion of complexity, a set $A$ is called anti-complex if the Kolmogorov complexity of the initial segments of $A$ chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e.\ traceability or being weak truth-table reducible to a Schnorr trivial set. A set $A$ is anti-complex if and only if it is reducible to another set $B$ with \emph{tiny use}, whereby we mean that the use function for reducing $A$ to $B$ can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

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