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

Wesley Calvert

Journal of Symbolic Logic 70 (1):331 - 345 (2005)

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Computable structures and the hyperarithmetical hierarchy.C. J. Ash - 2000 - New York: Elsevier. Edited by J. Knight.details This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, (...) Download Export citation Bookmark 46 citations
Countable algebra and set existence axioms.Harvey M. Friedman - 1983 - Annals of Pure and Applied Logic 25 (2):141.details Download Export citation Bookmark 66 citations
Degree spectra and computable dimensions in algebraic structures.Denis R. Hirschfeldt, Bakhadyr Khoussainov, Richard A. Shore & Arkadii M. Slinko - 2002 - Annals of Pure and Applied Logic 115 (1-3):71-113.details Whenever 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 (...) Download Export citation Bookmark 53 citations
Labelling systems and R.E. structures.C. J. Ash - 1990 - Annals of Pure and Applied Logic 47 (2):99-119.details Download Export citation Bookmark 15 citations
Back and forth relations for reduced abelian p-groups.Ewan J. Barker - 1995 - Annals of Pure and Applied Logic 75 (3):223-249.details In order to apply known general theorems about the effective properties of recursive structures in a particular recursive structure, it is necessary to verify that certain decidability conditions are satisfied. This requires the determination of when certain relations, called back and forth relations, hold between finite strings of elements from the structure. Here we determine this for recursive reduced abelian p-groups, thus enabling us to apply these theorems. Download Export citation Bookmark 8 citations
The isomorphism problem for classes of computable fields.Wesley Calvert - 2004 - Archive for Mathematical Logic 43 (3):327-336.details 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 members is very complex become classifiable when we consider only computable members. We follow recent work (...) Download Export citation Bookmark 8 citations

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