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计算方案的克林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，并扩展了形式化实数计算的图灵"机器模型"。克林框架中的一个基本区分在于规范泛函与非规范泛函：前者计算相应的克林量词$\exists^n$，后者则不能。历史上研究重点集中于规范泛函，但近期基于著名定理涌现出新的非规范泛函研究，其中看似最弱的当属实数的不可数性。这些新非规范泛函与泰特扇泛函等经典范例存在本质差异：后者可由$\exists^2$计算，而前者可由$\exists^3$计算，但无法通过更弱的预言机实现。当然，$\exists^2$与$\exists^3$之间存在巨大鸿沟或深渊，我们识别出这些新非规范泛函的细微变体又可被$\exists^2$计算，即落在此深渊的不同侧。我们的例子基于主流数学概念，如来自实分析的拟连续性、贝尔类、有界变差与半连续性。