Computable Presentability of Countable Linear Orders

The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Earlier, in 1989, R....

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Published inJournal of mathematical sciences (New York, N.Y.) Vol. 256; no. 2; pp. 199 - 233
Main Author Frolov, A. N.
Format Journal Article
LanguageEnglish
Published New York Springer US 02.07.2021
Springer
Springer Nature B.V
Subjects
Mathematics
Mathematics and Statistics
Number theory
low degrees
computable structures
computable presentability
countable linear orders
03D45
03C57
Online AccessGet full text
ISSN1072-3374
1573-8795
DOI10.1007/s10958-021-05426-y

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Summary:The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Earlier, in 1989, R. Downey and M. Moses showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations the existence of a computable presentation is sufficient. This question (namely, research program) was stated by R. Downey in 1998: Describe the order property P such that, for any low linear order L , P ( L ) implies the existence of a computable presentation of L . In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship.
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ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-021-05426-y