近似近似推理：模糊集和Ershov层次
Approximating approximate reasoning: Fuzzy sets and the Ershov hierarchy
摘要：
可计算性理论家引入了多层次来衡量自然数集合的复杂性。Kleene Hierarchy根据定义公式的一阶复杂度对集合进行分类。Ershov层次根据近似所需的错误数量对$Δ^0_2$集进行分类。Biacino和Gerla将Kleene层次扩展到模糊集的领域，模糊集的隶属函数范围在完全格$L$中（例如，实区间$[0; 1]_\mathbb{R}$）。本文将Ershov层次与模糊集合论相结合，通过对模糊Ershov层次的介绍和研究，提出了模糊Ershov层次的概念。特别地，我们专注于模糊$n$ -c. e。形成这个层次结构的有限层次的集合。直觉上，模糊集是$n$ -c. e.如果它的隶属函数可以通过改变最多$n-1$次的单调性来近似。我们证明了模糊Ershov层次不崩溃，即，在类比与经典的情况下，每个模糊$n$ -c. e。集合可以表示为模糊c. e.但是，与经典的情况相反，模糊Ershov层次并没有用尽类的所有$Δ^0_2$模糊集。
英文摘要：
Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies $Δ^0_2$ sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., the real interval $[0; 1]_\mathbb{R}$). In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy $n$-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is $n$-c.e. if its membership function can be approximated by changing monotonicity at most $n-1$ times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy $n$-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all $Δ^0_2$ fuzzy sets.
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