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首页> 外文期刊>Journal of Symbolic Logic >Effectivity and effective continuity of multifunctions
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Effectivity and effective continuity of multifunctions

机译：多功能的有效性和有效连续性

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摘要

If one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed into better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) into (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest. In this paper the setting is extended to the case of multifunctions. Various ways of coding (indexing) sets are discussed and their relationship is investigated. Moreover, effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Mostly, in addition to being effective the multifunctions need also possess certain witnessing functions. Important special cases are discussed where such witnessing functions always exist.
机译：如果要使用实数或数据流之类的无限对象进行计算，则连续性是必要的要求：输入的越来越好（有限）近似值将转换成输出的越来越好（有限）近似值。如果以构造方式生成对象，则可以通过生成过程的有限描述来表示它们。通过有效地将用于生成输入的这些描述（分别是其代码）转换为用于生成输出的描述（的代码），可以得到另一种可计算操作类型。这种操作也称为有效。这两类行动的关系一直是人们非常感兴趣的问题。在本文中，该设置扩展到了多功能的情况。讨论了各种编码（索引）集的方法，并研究了它们之间的关系。此外，还介绍了几种多功能连续性概念的有效版本。对于这些概念中的每一个，都展示了用于集合的索引系统，以便对该索引系统有效的多功能正是针对所考虑的连续性概念有效连续的多功能。通常，除了有效之外，多功能功能还需要具有某些见证功能。讨论了重要的特殊情况，这些情况总是存在。

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