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.

翻译：基于S1-S9计算模式的Kleene可计算性理论构成了一个处理任意有限类型对象的计算模型，它将形式化实数计算的图灵"机器模型"进行了扩展。Kleene框架中的一个基本区分在于正规泛函与非正规泛函：前者能计算对应的Kleene量词$\exists^{n}$，而后者则不能。历史上研究焦点集中在正规泛函，但近期基于实数不可数性等著名定理，学者们开始研究新型非正规泛函。这些新型非正规泛函与Tait扇泛函等历史范例存在本质差异：后者可由$\exists^{2}$计算，而前者仅能由$\exists^{3}$计算。尽管$\exists^{2}$与$\exists^{3}$之间存在巨大鸿沟，但我们发现了某些紧密相关的非正规泛函恰落在这深渊的两侧。我们的示例基于拟连续性、贝尔类与半连续性等主流数学概念。