arXiv subjects

Matthew Harrison-Trainor

Publications and source records attributed to Matthew Harrison-Trainor.

49 records · Page 3Linked to original sources

arXiv2017

A first-order theory of Ulm type

The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an abelian $p$-group and a pure set (and so that every abelian $p$-group is bi-interpretable with a model of $T_p$) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of "Ulm type": Any two models, low for $ω_1^{CK}$, and with the same computable infinitary theory, are isomorphic. It also gives a new example of an elementary first-order theory whose isomorphism problem is $\mathbfΣ^1_1$-complete but not Borel complete.

math.LO↗

arXiv2017

There is no classification of the decidably presentable structures

A computable structure $\mathcal{A}$ is decidable if, given a formula $φ(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $φ$ holds of $\bar{a}$. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is $Σ^1_1$-complete. This result holds even if we restrict out attention to groups, graphs, or fields. We also show that the index sets of the computable structures with $n$-decidable presentations is $Σ^1_1$-complete for any $n$.

math.LO↗

arXiv2016

Left-orderable Computable Groups

We answer a question of Downey and Kurtz on left-orderable groups by showing that there is a computable left-orderable group which is not classically isomorphic to a computable group with a computable left-order.

math.LO↗

arXiv2016

On computable field embeddings and difference closed fields

We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields.

math.LO↗

arXiv2016

Borel Functors and Infinitary Interpretations

We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: Every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show the complexities are maintained in the sense that if the functor is $\mathbfΔ^0_α$, then the interpretation that induces it is $Δ^{\mathtt{in}}_α$ up to $\mathbfΔ^0_α$ equivalence.

math.LO↗

arXiv2016

Some new computable structures of high rank

We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $ω_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank $ω_1^{CK}+1$ have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.

math.LO↗

arXiv2015

Scott ranks of models of a theory

The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mathcal{L}_{ω_1 ω}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $\boldsymbolΣ^1_1$ classes of ordinals. Our investigation of Scott spectra leads to the resolution (in $ZFC$) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each $α< ω_1$, that there is a $Π^{\mathtt{in}}_2$ theory with no models of Scott rank less than $α$. We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that $δ^1_2$ is the least ordinal $α$ such that if the models of a computable theory $T$ have Scott rank bounded below $ω_1$, then their Scott ranks are bounded below $α$.

math.LO↗

arXiv2015

Independence in computable algebra

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.

math.LO↗

arXiv2015

Computable functors and effective interpretability

Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.

math.LO↗

arXiv2015

Degrees of categoricity on a cone

We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is $\bf{0^{(α)}}$ for some $α$. To prove this, we extend Montalbán's $η$-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinal $α$ and a cone in the Turing degrees such that the exact complexity of computing an isomorphism between the given structure and another copy $\mathcal{B}$ in the cone is a c.e. degree in $Δ^0_α(\mathcal{B})$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions.

math.LO↗

arXiv2014

Degree Spectra of Relations on a Cone

Let $\mathcal{A}$ be a mathematical structure with an additional relation $R$. We are interested in the degree spectrum of $R$, either among computable copies of $\mathcal{A}$ when $(\mathcal{A},R)$ is a "natural" structure, or (to make this rigorous) among copies of $(\mathcal{A},R)$ computable in a large degree \textbf{d}. We introduce the partial order of degree spectra \textit{on a cone} and begin the study of these objects. Using a result of Harizanov---that, assuming an effectiveness condition on $\mathcal{A}$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e.\ degrees---we see that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e.\ degrees. We show that this does not generalize to d.c.e.\ degrees by giving an example of two incomparable degree spectra on a cone. We also give a partial answer to a question of Ash and Knight: they asked whether (subject to some effectiveness conditions) a relation which is not intrinsically $Δ^0_α$ must have a degree spectrum which contains all of the $α$-CEA degrees. We give a positive answer to this question for $α= 2$ by showing that any degree spectrum on a cone which strictly contains the $Δ^0_2$ degrees must contain all of the 2-CEA degrees. We also investigate the particular case of degree spectra on the structure $(ω,<)$. This work represents the beginning of an investigation of the degree spectra of "natural" structures, and we leave many open questions to be answered.

math.LO↗

arXiv2013

Differential-algebraic jet spaces preserve internality to the constants

This paper concerns the model theory of jet spaces (i.e., higher-order tangent spaces) in differentially closed fields. Suppose p is the generic type of the jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constant field called "preserving internality to the constants". This strengthening is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. A counterexample is constructed showing that only this generic analogue holds in differential-algebraic geometry.

math.LO↗

arXiv2011

Nonstandard methods for bounds in differential polynomial rings

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91, 1984] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz.

math.AC↗