15 research outputs found
Recommended from our members
Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
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.Comment: 24 page
Degree spectra and computable dimensions in algebraic structures
AbstractWhenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings (with zero-divisors); integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories
Computable embeddings for pairs of linear orders
We study computable embeddings for pairs of structures, i.e. for classes
containing precisely two non-isomorphic structures. Surprisingly, even for some
pairs of simple linear orders, computable embeddings induce a non-trivial
degree structure. Our main result shows that is computably embeddable in iff divides .Comment: 20 page
Computable Categoricity of Trees of Finite Height
We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0n+1-categorical but not ∆0n-categorical
Computable Stone spaces
We investigate computable metrizability of Polish spaces up to homeomorphism.
In this paper we focus on Stone spaces. We use Stone duality to construct the
first known example of a computable topological Polish space not homeomorphic
to any computably metrized space. In fact, in our proof we construct a
right-c.e. metrized Stone space which is not homeomorphic to any computably
metrized space. Then we introduce a new notion of effective categoricity for
effectively compact spaces and prove that effectively categorical Stone spaces
are exactly the duals of computably categorical Boolean algebras. Finally, we
prove that, for a Stone space , the Banach space has a
computable presentation if, and only if, is homeomorphic to a computably
metrized space. This gives an unexpected positive partial answer to a question
recently posed by McNicholl.Comment: 16 page