arXiv subjects

Wesley Calvert

Publications and source records attributed to Wesley Calvert.

26 records · Page 2Linked to original sources

arXiv2008

Metric Structures and Probabilistic Computation

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role for structures described in continuous first-order logic. The main result of this paper is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. Later sections give examples of the application of this framework to various classes of structures, and to some problems of computational complexity theory.

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arXiv2008

Comparing Classes of Finite Structures

We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed.

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arXiv2008

Classification from a Computable Viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work, showing that certain classes of mathematical structures admit classification, while others do not. In the present paper, we describe some recent work on classification in computable structure theory.

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arXiv2008

Index Sets of Computable Structures

The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, $\mathbb{Q}$-vector spaces, Archimedean real closed ordered fields, reduced Abelian $p$-groups of length less than $ω^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $Π_{n}^{0}$, $d$-$Σ_{n}^{0}$, or $Σ_{n}^{0}$, for various $n$. In each case, the calculation involves finding an \textquotedblleft optimal\textquotedblright% \ sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable $Π_{n}$, $d$-$Σ_{n}$, or $Σ_{n}$) yields a bound on the complexity of the index set. When we show $m$% -completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.

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arXiv2008

Structures in Familiar Classes Which Have Scott Rank $ω_1^{CK}$

There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, which have Scott rank $ω_1^{CK}+1$. Makkai produced a structure of Scott rank $ω_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present paper, we show that there are further computable structures of Scott rank $ω_1^{CK}$ in the following classes: undirected graphs, fields of any characteristic, and linear orderings. The new examples share with the Harrison ordering, and the tree just mentioned, a strong approximability property.

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arXiv2005

Turing Degrees of Isomorphism Types of Algebraic Objects

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.

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arXiv2004

The isomorphism problem for classes of computable fields

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable models is very complex become classifiable when we consider only computable members. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non- classifiable. For some classes (undirected graphs, fields of fixed characteristic, and real closed fields) we show that the isomorphism problem is Σ^1_1 complete (the maximum possible), and for others it is of relatively low complexity. For instance, for algebraically closed fields, archimedean real closed fields, and vector spaces, we show that the isomorphism problem is Π^0_3 complete.

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arXiv2004

The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.

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