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. By this way the notion of computation is reduced to that of enumeration. Beside of 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 domain. Moreover, the domain can be computable 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：偏序性是计算中一种不可回避的自然现象，问题在于我们能否赋予偏序发生（即非终止情形）以更丰富的结构。本文考虑一类函数集合，其中除全函数外仅包含定义域为自然数初始段的有限函数。此类函数自然出现在计算过程中。我们证明，针对这些函数类可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——后者通常处理所有偏（可计算）函数。为此，我们推广了哥德尔数的概念，得到了更广泛的编号类。该方法的中心算法思想是在枚举列表中搜索，从而将计算概念归结为枚举。除为函数类发展可计算性理论外，本文还从编号论视角研究了新型编号（称为拟哥德尔编号）：它们具有完备性，且以这种方式编号的每个函数类都是所有偏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，获得了与偏可计算函数可计算编号情形类似的结果。这些函数类被证明是斯科特-叶尔绍夫意义上的有效给定代数域。拟哥德尔编号恰好是这些域的可容编号。进一步地，该域可通过可计算映射映满每个其他有效给定域，使得可计算域元素的每个可容编号都能通过此映射由拟哥德尔编号生成。