We consider notions of space by Winter [21, 22]. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P|$_{+}=$|PSPACE|$_{+}$| and P|$_{++}=$|PSPACE|$_{++}$| holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in Section 7.2.2 of Carl (2019, Ordinal Computability: An Introduction to Infinitary Machines), which appear here as Lemma |$6$| up to Corollary |$9$|⁠.

中文翻译：

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无限时间图灵机的时空复杂性

我们考虑到冬天的空间概念[21岁 22]。我们回答关于这些概念的几个公开问题，其中包括低空间复杂度是否意味着低时间复杂度（不是）以及等式P = PSPACE，P | $ _ {+ {=} = $ |。PSPACE | $ _ {+} $ | 和P | $ _ {++} = $ | PSPACE | $ _ {++} $ | 持有ITTM（三个都是错误的）。我们还显示了ITTM的空间复杂度类别之间的各种分离结果。这大大扩展了我们先前在Carl的7.2.2节（2019年，有序可计算性：无穷机器的简介）中对该主题的观察力，在此处显示为引理| $ 6 $ | 直至推论| $ 9 $ |⁠。