连续数据的量化编码与复杂性理论
Quantitative Coding and Complexity Theory of Continuous Data
摘要：
解决计算问题需要固定输入和输出的编码：将图形编码为邻接矩阵，将字符编码为整数，将整数编码为位串，反之亦然。对于这样的离散数据，实际的编码通常是直接的和/或复杂的-理论上是不重要的（比如说，直到多项式时间）;但是对于连续数据，实数已经自然地提出了各种编码（所谓的代表）具有非常不同的属性，范围从可计算的“不合理的”二进制扩展，通过定性多项式，甚至线性复杂性-理论上“合理的”签署，数字扩展但是，如何区分微积分和数值中常见的其他空间（如Sobolev）的不合适的编码？关于定性的可计算性，Kreitz和Weihrauch（1985）已经将可容许性确定为无限二元序列在康托空间上的表示是“合理的”的关键标准。[doi：10.1007/11780342_48]。将拓扑空间上的可计算性细化为度量空间上的复杂性，发展了多项式/线性可容许性理论，作为定性可容许性的两个定量细化。我们还将数量容许性重新表述为表示及其集值逆的数量连续性，后者采用了[doi：10.4115/jla.2013.5.7]多函数的“序列”连续性的新概念。通过建立紧超度量空间之间的集值函数的定量连续选择定理，我们可以将上述定量主定理从函数推广到集值函数，也就是搜索问题。更高类型的复杂性是通过将Cantor（和Baire）的基空间编码推广到其他（紧凑）ULRA度量空间来捕获的。
英文摘要：
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.
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