, Volume 41, Issue 6, pp 351373
First online:
Computable Structure and NonStructure Theorems
 S. S. GoncharovAffiliated withInstitute of Mathematics SB RAS, Akademika Koptyuga Prospekt
 , J. F. KnightAffiliated withNotre Dame University
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In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shore's question. We consider structures Д of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify Д with its atomic diagram D(Д), which, via the Gödel numbering, may be treated as a subset of ω. In particular, Д is computable if D(Д) is computable. If K is some class, then K^{c} denotes the set of computable members of K. A computable characterization for K should separate the computable members of K from other structures, that is, those that either are not in K or are not computable. A computable classification (structure theorem) should describe each member of K^{c} up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable nonstructure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the “correct” answer for vector spaces over Q, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.
 Title
 Computable Structure and NonStructure Theorems
 Journal

Algebra and Logic
Volume 41, Issue 6 , pp 351373
 Cover Date
 200211
 DOI
 10.1023/A:1021758312697
 Print ISSN
 00025232
 Online ISSN
 15738302
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 computable characterization
 computable classification
 structure theorem
 nonstructure theorem
 Authors

 S. S. Goncharov ^{(1)}
 J. F. Knight ^{(2)}
 Author Affiliations

 1. Institute of Mathematics SB RAS, Akademika Koptyuga Prospekt, Novosibirsk, 630090, Russia
 2. Notre Dame University, Notre Dame, USA