Abstract

Let \(\simeq\) be a binary relation between sets of integers, and \(\leq_{R}\) be a Post reducibility, i.e. a reflexive and transitive relation between sets of integers such that if \(A\leq_{R}B\) then the computational complexity of recognition of elements of \(A\) is easier than (or equal to) the recognition of elements of \(B\). Suppose that for a class \({\mathcal{A}}\) of arithmetical sets, which have an effective enumeration \(\{\Omega_{e}\}_{e\in\omega}\), there are \(R\)-complete sets, i.e. such sets \(D\) that for any \(A\in{\mathcal{A}}\), \(A\leq_{R}D\). Earlier we considered completeness criteria for such reducibilities roughly of the following type: For any \(A\in{\mathcal{A}}\), \(A\) is \(R\)-complete if and only if there is a function \(f\), defined on \(\omega\) such that \(f\leq_{R}D\) and \(\Omega_{f(i)}\not\simeq\Omega_{i}\) for all \(i\in\omega\). This means that for any set \(A\in{\mathcal{A}}\), if it is non-complete, then any function \(f\leq_{R}A\) has a fixed-point \(e\): \(\Omega_{f(e)}\simeq\Omega_{e}\). In this paper we introduce a notion of fixed-point selection function for sequences of such sets and study their complexity characteristics.

中文翻译：

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定点选择函数

摘要

令\(\simeq\)是整数集之间的二元关系，而\(\leq_{R}\)是后可约性，即整数集之间的自反和传递关系，如果\(A\leq_{ R}B\)那么识别\(A\)的元素的计算复杂度比识别\(B\)的元素更容易（或等于）。假设对于具有有效枚举\(\{\Omega_{e}\}_{e\in\omega}\)的算术集的类\({\mathcal{A} }\)，有\ （R \） -complete集，即，使得套\（d \）对于任何\（A \在{\ mathcal {A}} \） ，\（A \ leq_ {R} d \）. 早些时候，我们考虑了以下类型的可约性的完整性标准：对于任何\(A\in{\mathcal{A}}\)，\(A\)是\(R\) -complete 当且仅当存在一个函数\(f\)，在\(\omega\)上定义使得\(f\leq_{R}D\)和\(\Omega_{f(i)}\not\simeq\Omega_{i}\ )对于所有\(i\in\omega\)。这意味着对于任何集合\(A\in{\mathcal{A}}\)，如果它是不完整的，那么任何函数\(f\leq_{R}A\)都有一个不动点\(e \) : \(\Omega_{f(e)}\simeq\Omega_{e}\). 在本文中，我们为此类集合的序列引入了定点选择函数的概念并研究了它们的复杂性特征。