## On decidable categoricity and almost prime models.(English)Zbl 1446.03069

Summary: The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. The first author [Proc. Steklov Inst. Math. 274, 105–115 (2011; Zbl 1294.03025); translation from Tr. Mat. Inst. Steklova 274, 119–129 (2011)] defined the degree of decidable categoricity of a decidable model $$\mathcal{M}$$ to be the least Turing degree, if it exists, which is capable of computing isomorphisms between arbitrary decidable copies of $$\mathcal{M}$$. If this degree is $$\mathbf{0}$$, we say that the structure $$\mathcal{M}$$ is decidably categorical. Goncharov established that every computably enumerable degree is the degree of categoricity of a prime model, and Bazhenov showed that there is a prime model with no degree of categoricity. Here we investigate the degrees of categoricity of various prime models with added constants, also called almost prime models. We relate the degree of decidable categoricity of an almost prime model $$\mathcal{M}$$ to the Turing degree of the set $$C(\mathcal{M})$$ of complete formulas. We also investigate uniform decidable categoricity, characterizing it by primality of $$\mathcal{M}$$ and Turing reducibility of $$C(\mathcal{M})$$ to the theory of $$\mathcal{M}$$.

### MSC:

 03C57 Computable structure theory, computable model theory 03C35 Categoricity and completeness of theories 03D45 Theory of numerations, effectively presented structures

Zbl 1294.03025
Full Text:

### References:

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