Kleene's computability theory based on the 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, the weakest among which seems to be 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 computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.

翻译：基于S1-S9计算方案的克莱尼可计算性理论构成了一种对任意有限类型对象进行计算的模型，并扩展了形式化实数计算的图灵机模型。克莱尼框架中的一个基本区分在于正规泛函与非正规泛函：前者可计算关联的克莱尼量词∃ⁿ，而后者则不能。历史上，研究焦点集中于正规泛函，但近年来基于著名定理的新非正规泛函得到研究，其中最弱的似乎与实数的不可数性相关。这些新非正规泛函与泰特扇形泛函等历史范例存在根本差异：后者可从∃²可计算，而前者在∃³中可计算，但无法通过更弱的神谕计算。当然，∃²与∃³之间存在着巨大的鸿沟或深渊，我们识别出这些新非正规泛函的细微变体，它们仍可在∃²中计算，即落于该深渊的不同侧。我们的示例基于主流数学概念，如来自实分析中的拟连续性、贝尔类、有界变差与半连续性。