Kleene可计算性理论中的非正规深渊
The non-normal abyss in Kleene's computability theory
摘要：
Kleene的可计算性理论基于他的S1-S9计算方案，构成了一个模型，用于计算任何有限型的对象，并扩展了图灵的“机器模型”，该模型将计算形式化为实数。Kleene框架中的一个基本区别是正规和非正规泛函之间的区别，前者计算相关的Kleene量词$\exists^{n}$，而后者不计算。历史上，重点是正规泛函，但最近新的非正规泛函已经研究，基于众所周知的定理，如实数的不可数性。这些新的非正规泛函从根本上不同于历史上的例子，如泰特的风扇功能：后者是可计算的$\exists^{2}$，而前者只能计算在$\exists^{3}$。虽然$\exists^{2}$和$\exists^{3}$之间有一个巨大的鸿沟，但我们确定了某些密切相关的非正规泛函，它们落在这个深渊的不同两侧。我们的例子是基于主流的数学概念，如准连续性，Baire类和半连续性。
英文摘要：
Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $\exists^{n}$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied, based on well-known theorems like the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $\exists^{2}$ while the former are only computable in $\exists^{3}$. While there is a great divide separating $\exists^{2}$ and $\exists^{3}$, we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.
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