The position of index sets of identifiable sets in the arithmetical hierarchy

Q4 Mathematics 信息与控制 Pub Date : 1986-01-01 DOI:10.1016/S0019-9958(86)80034-1
Ulrike Brandt
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引用次数: 9

Abstract

We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in Σ3Π3. An identifiable set is presented with index set in Σ2Π3 but neither in Σ2 nor in Π2. Furthermore we show that there is no nonempty identifiable set with index set in Σ1. In Π1 it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.

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可识别集的索引集在算术层次结构中的位置
我们证明了每一个可以被归纳推理机识别的部分递归函数集合都包含在一个索引集在Σ3∩Π3中的可识别函数集合中。一个可识别的集合在Σ2∩Π3中以索引集表示,但在Σ2和Π2中都没有。此外,我们还证明了Σ1中的索引集不存在非空的可识别集。在Π1中可以找到这个集合之王。在文章的最后,我们证明了所有部分递归函数的辨识问题和停止问题具有相同程度的不可解性。
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来源期刊
信息与控制
信息与控制 Mathematics-Control and Optimization
CiteScore
1.50
自引率
0.00%
发文量
4623
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