We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability theory. We begin with a treatment of partial functions, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps. We relate this and other ideas from synthetic domain theory to other approaches to partiality in type theory. We show that the notion of dominance is difficult to apply in our setting: the set of $\Sigma_0^1$ propositions investigated by Rosolini form a dominance precisely if a weak, but nevertheless unprovable, choice principle holds. To get around this problem, we suggest an alternative notion of partial function we call disciplined maps. In the presence of countable choice, this notion coincides with Rosolini's. Using a general notion of partial function, we take the first steps in constructive computability theory. We do this both with computability as structure, where we have direct access to programs; and with computability as property, where we must work in a program-invariant way. We demonstrate the difference between these two approaches by showing how these approaches relate to facts about computability theory arising from topos-theoretic and type-theoretic concerns. Finally, we tie the two threads together: assuming countable choice and that all total functions $\mathbb{N}\to\mathbb{N}$ are computable (both of which hold in the effective topos), the Rosolini partial functions, the disciplined maps, and the computable partial functions all coincide. We observe, however, that the class of all partial functions includes non-computable partial functions.

中文翻译：

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单价类型理论中的偏函数和递归

我们从构造性的单价类型理论中研究偏函数和可计算性理论。重点是将可计算性置于更大的数学环境中，而不是完全发展可计算性理论。我们从处理部分函数开始，使用支配性的概念，在合成域理论中使用支配性的概念来讨论部分映射的类。我们将这个和其他来自合成领域理论的想法与其他类型理论中的偏向性方法联系起来。我们表明支配的概念很难在我们的设置中应用：罗索里尼研究的 $\Sigma_0^1$ 命题集合恰好在弱但仍然无法证明的选择原则成立的情况下形成支配。为了解决这个问题，我们提出了一个替代的部分函数概念，我们称之为规范映射。在可数选择的存在下，这个概念与罗索里尼的概念相吻合。使用偏函数的一般概念，我们迈出了建设性可计算性理论的第一步。我们以可计算性为结构来做到这一点，我们可以直接访问程序；并且将可计算性作为属性，我们必须以程序不变的方式工作。我们通过展示这些方法如何与由拓扑理论和类型理论问题引起的可计算性理论相关的事实来证明这两种方法之间的差异。最后，我们将两个线程联系在一起：假设可数选择并且所有全函数 $\mathbb{N}\to\mathbb{N}$ 都是可计算的（两者都在有效拓扑中），Rosolini 偏函数，严格的映射和可计算的偏函数都重合。然而，我们观察到，