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Strong \(\Delta ^ 0_ 2\) categoricity. (English) Zbl 0596.03043
Algebra Logic 24, 471-476 (1985); and Algebra Logika 24, No. 6, 718-727 (1985).
The notion of strong \(\Delta^ 0_ 2\) stability for recursive structures is first introduced and discussed. A recursive structure \({\mathfrak A}\) is strongly \(\Delta^ 0_ 2\) stable if there is a total \(\Delta^ 0_ 2\) function f on \(A\times {\mathbb{N}}\) such that for every recursive structure \({\mathfrak B}\), every possible isomorphism from \({\mathfrak A}\) to \({\mathfrak B}\) is f(a,n) for some n. This notion lies between that of recursive stability, previously studied by Goncharov and that of \(\Delta^ 0_ 2\)-stability, studied by Ash. The analogous notion of strong \(\Delta^ 0_ 2\)-categoricity is then also defined.
Several useful-looking examples are produced to show that various combinations of properties can occur, while several questions are posed which remain unanswered. The main question asked is whether there is a natural syntactical characterization (under reasonable assumptions) of the strongly \(\Delta^ 0_ 2\)-categorical recursive structures, along the lines of those obtained by the authors respectively for recursive categoricity and \(\Delta^ 0_ 2\)-categoricity.
The corresponding question is also asked for strong \(\Delta^ 0_ 2\)- stability, although the possibility may be thought to be precluded in this case by the result proved here that a 1-recursive structure is strongly \(\Delta^ 0_ 2\)-stable if and only if it is recursively stable. By contrast, an example shows that a 2-recursive structure may be strongly \(\Delta^ 0_ 2\)-categorical but not recursively categorical.
MSC:
03D45 Theory of numerations, effectively presented structures
03D25 Recursively (computably) enumerable sets and degrees
03C57 Computable structure theory, computable model theory
03D55 Hierarchies of computability and definability
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References:
[1] C. J. Ash, ”Stability of recursive structures in hyperarithmetical degrees,” Preprint, Monash Univ. (1984). · Zbl 0631.03017
[2] C. J. Ash, ”Categoricity in hyperarithmetical degrees,” Preprint, Monash Univ. (1984). · Zbl 0617.03016
[3] S. S. Goncharov, ”On the number of nonautoequivalent constructivizations,” Algebra Logika,16, No. 3, 257–282 (1977). · Zbl 0407.03040
[4] S. S. Goncharov, ”Self-stability and computable families of constructivizations,” Algebra Logika,14, No. 6, 647–680 (1975). · Zbl 0382.03033
[5] S. S. Goncharov, ”Limit equivalent constructivizations,” Tr. Inst. Mat. Sib. Otd. Akad. Nauk SSSR, 2, Nauka, Novosibirsk (1982). · Zbl 0543.03017
[6] S. S. Goncharov, ”The problem of the number of nonautoequivalent constructivizations,” Dokl. Akad. Nauk SSSR,251, No. 2, 271–274 (1980). · Zbl 0476.03045
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