arXiv subjects

Jason Block

Publications and source records attributed to Jason Block.

3 recordsLinked to original sources

arXiv2026

Measuring the Complexity of Countable Presburger Models

We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger arithmetic. Many of our results will be achieved by showing how given a linear order $\mathcal{L}$, we can construct a Presburger group $P_\mathcal{L}$ that maintains much of the structure of $\mathcal{L}$.

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arXiv2025

Computability for tree presentations of continuum-size structures

We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the focus to first-order properties, using computable structure theory (on countable structures) as a guide. We present three basic questions to be asked when a structure is presented as the set of paths through a computable tree, as in our definition, and also propose the concept of tree-decidability as an analogue to the notion of decidability for a countable structure. As examples, we prove decidability results for certain additive and multiplicative groups of $p$-adic integers, products of these (such as the profinite completion of $\mathbb Z$), and the field of real numbers.

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arXiv2024

Elementarity of Subgroups and Complexity of Theories for Profinite Groups

Although $S_\infty$ (the group of all permutations of $\mathbb{N}$) is size continuum, both it and its closed subgroups can be presented as the set of paths through a countable tree. The subgroups of $S_\infty$ that can be presented this way with finite branching trees are exactly the profinite ones. We use these tree presentations to find upper bounds on the complexity of the existential theories of profinite subgroups of $S_\infty$, as well as to prove sharpness for these bounds. These complexity results enable us to distinguish a simple subclass of profinite groups, those with \emph{orbit independence}, for which we find an upper bound on the complexity of the entire first order theory. Additionally, given a profinite subgroup $G$ of $S_\infty$ and a Turing ideal $I$ we define $G_I$ to be the set of elements in $G$ whose Turing degree lies in $I$. We examine to what extent and under what conditions $G_I$ will be an elementary subgroup of $G$. In particular, we construct a profinite group whose subgroup of computable elements is not elementary even for existential formulas.

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