Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider function classes which besides the total functions only contain finite functions whose domain of definition is an initial segment of the natural numbers. Such functions appear naturally in computation. We show that a rich computability theory can be developed for these functions classes which embraces the central results of classical computability theory, in which all partial (computable) functions are considered. To do so, the concept of a G\"odel number is generalised, resulting in a broader class of numberings. The central algorithmic idea in this approach is to search in enumerated lists. In this way, function computability is reduced to set listability. Besides the development of a computability theory for the functions classes, the new numberings -- called quasi-G\"odel numberings -- are studied from a numbering-theoretic perspective: they are complete, and each of the function classes numbered in this way is a retract of the G\"odel numbered set of all partial computable functions. Moreover, the Rogers semi-lattice of all computable numberings of the considered function classes is studied and results as in the case of the computable numberings of the partial computable functions are obtained. The function classes are shown to be effectively given algebraic domains in the sense of Scott-Ershov. The quasi-G\"odel numberings are exactly the admissible numberings of the computable elements of the domain. Moreover, the domain can be computably mapped onto every other effectively given one so that every admissible numbering of the computable domain elements is generated by a quasi-G\"odel numbering via this mapping.

翻译：部分性是计算中不可避免的自然现象。因此，问题在于我们能否为出现部分性（即非终止性）的领域赋予更多结构。本文考虑这样一类函数：除全函数外，仅包含定义域为自然数初始段的有限函数。这类函数在计算中自然出现。我们证明，针对这些函数类可以发展出丰富的可计算性理论，该理论涵盖了经典可计算性理论的核心结论——在经典理论中所有部分（可计算）函数均被考虑。为此，我们推广了哥德尔数的概念，从而得到更广泛的编号类。该方法的核心算法思想是枚举列表搜索，由此将函数可计算性归约为集合可列性。除了为这些函数类建立可计算性理论外，我们还从编号论视角研究了新型编号（称为拟哥德尔编号）：它们是完备的，且每个以此方式编号的函数类都是所有部分可计算函数的哥德尔编号集的收缩核。此外，我们研究了所考虑函数类的所有可计算编号构成的罗杰斯半格，并获得了与部分可计算函数的可计算编号情形相同的结果。本文证明这些函数类构成斯科特-叶尔绍夫意义上的有效给定代数域。拟哥德尔编号恰是该域中可计算元素的容许编号。进一步，该域可通过计算方式映满每个其他有效给定域，使得该可计算域元素的每个容许编号都能通过此映射由某个拟哥德尔编号生成。