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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
##### Keywords:
stability; recursive structures; recursive categoricity
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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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