Wesley Calvert

• Southern Illinois University Carbondale

We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable $\Pi^0_1$ classes, in a sense made precise here. This family of concept classes is sufficient to cover all standa...

The decision tree model, aka the query model, perhaps due to its simplicity and fundamental nature has been extensively studied over decades. Yet there remain some fascinating open questions about it. The purpose of this paper is to revisit three such ...

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets. We prove as-sorted theorems about the new reducibilities and especially about functions which have nonincreasing computable appr...

A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Ma...

The generalized constraint language (GCL), introduced by Zadeh, serves as a basis for computing with words (CW). It provides an agenda to express the imprecise and fuzzy information embedded in natural language and allows reasoning with perceptions. Despite its fundamental role, the definition of GCL has remained informal since its introduction by...

This opinionated essay discusses the role of intellectual and instrumental values in scientific fields, and the way these values evolve due to internal and external forces. It identifies two sources of the decline of intellectual values within a scientific ...

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets, and we prove assorted theorems about the new reducibilities and about functions which have nonincreasing computable approximatio...

We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree.

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 probab...

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d...

We examine the relation of BSS-reducibility on subsets of the real numbers. The question was asked recently (and anonymously) whether it is possible for the halting problem H in BSS-computation to be BSS-reducible to a countable set. Intuitively, it seems that a countable set ought not to contain enough information to decide membership in a reasona...

Reconfiguration means changing the set of processes executing a distributed system. We explain several methods for reconfiguring a system implemented using the state-machine approach, including some new ones. We discuss the relation between these methods ...

Using the model of real computability developed by Blum, Cucker, Shub, and Smale, we investigate the diculty of determining the answers to several basic topological questions about manifolds. We state definitions of real-computable manifold and of real-computable paths in such manifolds, and show that, while BSS machines cannot in general decide su...

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev characterized the Abelian p-groups with computable copies. A computable structure A is said to be $\Delta^0_\alpha...

Computable structures of Scott rank w1CK{\omega_1^{CK}} are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of Lw1 w{\mathcal{L}_{\omega_1 \omega}}, this sentence may not be computable. We give examples, in several familiar classes of structures, of computable str...

The main result of this paper, as previously presented to arxiv, was incorrect. See the full text for details and for reference to the remaining results. Comment: This submission has been withdrawn by the author [arXiv admin]

An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic. The present p...

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev characterized the Abelian p-groups with computable copies. A computable structure A is said to be $\Delta^0_\alpha...

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 o...

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 the...

The index set of a computable structure \(\mathcal{A}\) is the set of indices for computable copies of \(\mathcal{A}\). We determine complexity of the index sets of various mathematically interesting structures including different finite structures, ℚ-vector spaces, Archimedean real-closed ordered fields, reduced Abelian p-groups of length less tha...

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 $\omega_1^{CK}+1$. Makkai produced a structure of Scott rank $\omega_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present pape...

We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an $F$-parameterizable structure, defined by Morozov and based on Mal'tsev's notio...

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory o...

We describe the use of innitary logics computable over the real num- bers (i.e. in the sense of Blum{Shub{Smale, with full-precision arithmetic) as a constraint query language for spatial databases. We give a charac- terization of the sets denable in various syntactic classes corresponding to the classical hyperarithmetical hierarchy.

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class K!CK 1 of structures of Scott rank !CK 1 , the class K!CK 1 +1 of structures of Scott rank ! CK 1 +1, and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete 1 1, I(K!CK 1 ) is m-com...

There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, that have Scott rank !CK 1 + 1. Makkai (13) produced a structure of Scott rank !CK 1 , which can be made computable (12), and simplified so that it is just a tree (4). In the present paper, w...

We investigate effective categoricity of computable equivalence structures A. We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that...

Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai’s example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass...

The Turing degree spectrum of a countable structure 𝒜 is the set of all Turing degrees of isomorphic copies of 𝒜. The Turing degree of the isomorphism type of 𝒜, is the least Turing degree in its degree spectrum. We show that there are structures with isomorphism types of arbitrary Turing degrees in each of the following classes: countable fields,...

We compare classes of structures using the notion of a computable embedding, which is a partial order on the classes of structures. Our attention is mainly, but not exclusively, focused on classes of finite structures. Also, a number of problems are formulated.

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...

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 s...

We compare three notions of effectiveness on uncountable struc-tures. The first notion is that of a R-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an F -parameterizable structure, defined by Morozov and based on Mal'tsev's notion of a...

We present a foundation for the effective theory of random vari-ables and stochastic processes. To accomplish this, we show that certain ele-ments of the Itô stochastic calculus have a natural expression in an extension of continuous first-order (CFO) logic and are effectively true in the sense of randomized computation. In particular, the Itô stoc...

Thesis (Ph. D.)--University of Notre Dame, 2005. Thesis directed by Julia F. Knight for the Department of Mathematics. "March 2005." Includes bibliographical references (leaves 80-84).