Complexity of \(\Sigma ^0_n\)-classifications for definable subsets

IF 0.3 4区 数学 Q1 Arts and Humanities
Svetlana Aleksandrova, Nikolay Bazhenov, Maxim Zubkov
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引用次数: 0

Abstract

For a non-zero natural number n, we work with finitary \(\Sigma ^0_n\)-formulas \(\psi (x)\) without parameters. We consider computable structures \({\mathcal {S}}\) such that the domain of \({\mathcal {S}}\) has infinitely many \(\Sigma ^0_n\)-definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of \(\Sigma ^0_n\)-formulas is a \(\Sigma ^0_n\)-classification for \({\mathcal {S}}\) if the list enumerates all \(\Sigma ^0_n\)-definable subsets of \({\mathcal {S}}\) without repetitions. We show that an arbitrary computable \({\mathcal {S}}\) always has a \({{\mathbf {0}}}^{(n)}\)-computable \(\Sigma ^0_n\)-classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no \({{\mathbf {0}}}^{(n-1)}\)-computable \(\Sigma ^0_n\)-classifications.

可定义子集的\(\Sigma ^0_n\) -分类的复杂性
对于非零自然数n,我们使用不带参数的有限元\(\Sigma^0_n\)-公式\(\psi(x)\)。我们考虑可计算结构({\mathcal{S}}),使得({\ mathcal{S}})的域具有无限多个(\ Sigma ^0_n)可定义子集。继Goncharov和Kogabaev之后,我们说一个\(\ Sigma ^0_n\)-公式的无限列表是\({\mathcal{S}})的\(\西格玛^0_n)-分类,如果该列表枚举了\({\ mathcal{S}})所有\(\∑^0_n\n)-可定义的子集而不重复。我们证明了一个任意可计算的\({\mathcal{S}})总是具有\({\mathbf{0}}}^{(n)}\)-可计算\(\ Sigma ^0_n\)-分类。另一方面,我们证明了这个界是尖锐的:我们建立了一个不具有\({\mathbf{0}})^{(n-1)}\)-可计算\(\ Sigma ^0_n\)-分类的可计算结构。
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来源期刊
Archive for Mathematical Logic
Archive for Mathematical Logic MATHEMATICS-LOGIC
CiteScore
0.80
自引率
0.00%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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