Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the \(\limsup \) of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the \(\limsup \) rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines (wITTMs). We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers.

中文翻译：

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无限时间图灵机的较弱变体

无限时间图灵机表示可计算性模型，通过将每个单元格的内容以极限步长定义为该单元格先前内容的序列的\（\ limsup \），将图灵机的操作扩展到有限序数时间。在本文中，我们研究了通过替换\（\ limsup \）获得的计算模型使用“最终恒定”规则的规则：在每个极限步骤中，当且仅当该单元格的内容在该极限步骤之前已经稳定并且然后等于该恒定值时，才定义每个单元格的值。我们称这些机器为无限时间图灵机（wITTM）。我们研究了wITTM的不同变体，添加了多个磁带，磁头或二维磁带。我们证明其中一些模型在计算强度方面彼此等效。我们证明了wITTM可以精确地决定自然数的算术关系。