不可计算谱集
Noncomputable Spectral Sets
摘要：
可以枚举所有计算机程序。特别是，对于每个部分可计算函数，都有一个最短的程序来计算该函数。f-MIN是最短程序的索引集。在1972年，Meyer证明了f-MIN是图灵等价于0“的，即带有停机集oracle的停机集。本文推广了最短程序的概念，并使用可计算性理论中的各种度量来描述所得到的“谱集”的复杂性。“我们证明了在某些哥德尔数下，谱集正好是规范集0 '，0''，0''，..直到图灵等价。这可能是不正确的一般情况下，但我们表明，谱集总是包含一些有用的信息。我们表明，免疫力，或“薄”是一个有用的特性，用于区分光谱集。在最后一章中，我们构造了一个集合，它既不包含任何无穷算术集，也不与任何无穷算术集不相交，但它是0-优化的，并且包含一个自然谱集。因此，一个病态的集合变得更友好一点。最后，一些有趣的开放性问题留给有灵感的读者。
英文摘要：
It is possible to enumerate all computer programs. In particular, for every partial computable function, there is a shortest program which computes that function. f-MIN is the set of indices for shortest programs. In 1972, Meyer showed that f-MIN is Turing equivalent to 0'', the halting set with halting set oracle. This paper generalizes the notion of shortest programs, and we use various measures from computability theory to describe the complexity of the resulting "spectral sets." We show that under certain Godel numberings, the spectral sets are exactly the canonical sets 0', 0'', 0''', ... up to Turing equivalence. This is probably not true in general, however we show that spectral sets always contain some useful information. We show that immunity, or "thinness" is a useful characteristic for distinguishing between spectral sets. In the final chapter, we construct a set which neither contains nor is disjoint from any infinite arithmetic set, yet it is 0-majorized and contains a natural spectral set. Thus a pathological set becomes a bit more friendly. Finally, a number of interesting open problems are left for the inspired reader.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
