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On the existence of hidden machines in computational time hierarchies
arXiv - CS - Logic in Computer Science Pub Date : 2020-09-02 , DOI: arxiv-2009.01145
Felipe S. Abrah\~ao, Klaus Wehmuth, Artur Ziviani

Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total, cannot be proved to be total. In this article we show that this implies the existence of an infinite hierarchy of time complexity classes whose representative members are hidden from (or unknown by) the respective formal axiomatic systems. Although these classes contain total computable functions, there are some of these functions for which the formal axiomatic system cannot recognize as belonging to a time complexity class. This leads to incompleteness results regarding formalizations of computational complexity.

中文翻译:

关于计算时间层次中隐藏机器的存在

挑战可计算函数中总体的标准概念,人们认为,给定任何足够表达的形式公理系统,存在尽管可计算且“直观地”理解为总体的总体函数,但不能证明是总体的。在本文中,我们表明这意味着存在一个无限层次的时间复杂度类,其代表成员隐藏于(或未知于)相应的正式公理系统。尽管这些类包含全部可计算函数,但其​​中一些函数的形式公理系统无法识别为属于时间复杂度类。这导致关于计算复杂性形式化的不完整性结果。
更新日期:2020-09-03
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