Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say); but concerning continuous data, already real numbers naturally suggest various encodings (so-called REPRESENTATIONS) with very different properties, ranging from the computably 'unreasonable' binary expansion via qualitatively to polynomially and even linearly complexity-theoretically 'reasonable' signed-digit expansion. But how to distinguish between un/suitable encodings of other spaces common in Calculus and Numerics, such as Sobolev? With respect to qualitative computability, Kreitz and Weihrauch (1985) had identified ADMISSIBILITY as crucial criterion for a representation over the Cantor space of infinite binary sequences to be 'reasonable'; cmp. [doi:10.1007/11780342_48]. Refining computability over topological to complexity over metric spaces, we develop the theory of POLYNOMIAL/LINEAR ADMISSIBILITY as two quantitative refinements of qualitative admissibility. We also rephrase quantitative admissibility as quantitative continuity of both the representation and of its set-valued inverse, the latter adopting from [doi:10.4115/jla.2013.5.7] a new notion of 'sequential' continuity for multifunctions. By establishing a quantitative continuous selection theorem for multifunctions between compact ultrametric spaces, we can extend our above quantitative MAIN THEOREM from functions to multifunctions aka search problems. Higher-type complexity is captured by generalizing Cantor's (and Baire's) ground space for encodings to other (compact) ULRAmetric spaces.
翻译:指定计算问题需要固定输入和输出的编码: 将图形编码为匹配矩阵, 字符为整数, 整数为整数, 整数为比字符, 反之亦然。 对于这种离散的数据, 实际编码通常是直截了当的和(或)复杂的理论性不必要( 例如, ) ; 但是关于连续数据, 已经真实的数字自然地表明, 各种编码( 所谓的表示) 具有非常不同的属性, 从可比较的“ 不合理的” 二进制扩展为多元矩阵, 甚至是线性复杂性- 理论性“ 合理” 签名数字扩展。 但对于 Calcululs 和 Nummerical( 如 Sobolev ) 中常见的其他空间的不适宜编码, 实际编码通常显示, Kreitz 和 Weihracheruch 1985 将“ DMISIL ” 确定一个关键标准, 从“ 直径” 到“ 直径” 。 [ dodo: 10/ 10: 2013- broal- oral- oralal- distrueal relizeal listal lizeal real real reliversal matium 。