Abstract

The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. Goncharov [26] 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 \(\mathcal {M} \).

中文翻译：

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可判定的分类和几乎素数模型

摘要

可计算和可确定结构的同构的复杂性在可计算模型理论中起着重要作用。贡恰洛夫[26]将可判定模型\（\ mathcal {M} \）的可判定分类程度定义为最小图灵度（如果存在），该图灵能够计算\（\ mathcal { M} \）。如果该度为\（\ mathbf {0} \），则可以说结构\（\ mathcal {M} \）是绝对分类的。贡恰洛夫确定，每个可计算的度数都是素数模型的分类度，而巴珍诺夫则表明存在一个没有分类度的素数模型。在这里，我们研究带有附加常数的各种素数模型（也称为几乎素数模型）的分类度。我们将几乎素数模型\（\ mathcal {M} \）的可判定分类程度与完整公式集\（C（\ mathcal {M}）\）的Turing度联系起来。我们还调查统一可判定categoricity，由素性表征它\（\ mathcal {M} \）和图灵还原\（C（\ mathcal {M}）\），以理论\（\ mathcal {M} \）。