[论文解读] On function spaces and polynomial-time computability.

Akitoshi Kawamura, Arno Pauly|arXiv (Cornell University)|Jan 13, 2014

Computability, Logic, AI Algorithms参考文献 16被引用 4

一句话总结

本文提出了一种连续实值函数空间的自然表示方法，确保函数求值的多项式时间可计算性，将类型二有效拓扑扩展至多项式时间复杂度。研究证明，表示法 Œ!ı� 是使逐点求值多项式时间可计算的最小表示法，为实函数的复杂性理论分析提供了基础工具。

ABSTRACT

In Computable Analysis, elements of uncountable spaces, such as the real line R, are represented by functions on strings and fed to Turing machines as oracles; or equivalently, they are represented by infinite strings and written on the tapes of Turing machines [Wei00, BHW08]. To obtain reasonable notions of computability and complexity, it is hence important to choose the “right ” representation (encoding) for the spaces being considered. Let’s say we have already agreed upon representations and ı of spaces X and Y (that are admissible with the topologies of X and Y). How would we represent the space CŒX!Y � of continuous functions from X to Y? It is known that there is a natural representation Œ! ı� of CŒX! Y � which is characterized by the property that it is the poorest representation that makes function evaluation computable [Wei00, Lemma 3.3.14]. Is there a representation with a similar property also at the level of polynomial-time computability (as introduced in [KF82] and extended in [KC96, KC12])? In this note we observe that there is such a nice representation for the space of continuous real-valued functions. Generalization to other spaces is left for future research. 1. TYPE-TWO POLYNOMIAL-TIME COMPUTABILITY We consider computational problems as multi-valued functions from the set X of possible inputs to the set Y of possible outputs. An.X; Y /-problem F is formally a subset of X Y. The set of x 2 X such that there is y 2 Y with.x; y / 2 F is called the domain of definition or the promise of F and denoted dom F. For x 2 dom F, we write FŒx � for the (nonempty) set of all such y. If FŒx � is a singleton, we write F.x / for the unique element of FŒx�. When this is the case for all x 2 dom F, we say that F is a single-valued problem, or a partial function. When dom F D X, we say that F is total. A single-valued total problem is called a function. The intuitive interpretation is that F specifies a problem where, given any x 2 dom F, you are required to output some element of FŒx�. Thus, the specification becomes stricter as dom F gets bigger or as FŒx � (for some x 2 dom F) gets smaller. For a.Y; Z/-problem F and an.X; Y /-problem G, we define the.X; Z/-problem F ı G by saying that its promise is (1) dom.F ı G / D f x 2 dom G W GŒx � dom F g; and that, for any x in this promise, (2).F ı G/Œx � D [ y2GŒx�

研究动机与目标

• 确定连续实值函数空间的合适表示法，以支持多项式时间可计算性。
• 将类型二多项式时间可计算性扩展至函数空间，特别是 C[R → R]。
• 表征一种使函数求值在多项式时间内可计算的最小表示法。
• 为可计算分析中连续函数的复杂性理论分析奠定基础。

提出的方法

• 采用类型二有效拓扑的框架，使用预言机图灵机将实数和函数表示为无限序列。
• 为连续函数空间 C[X → Y] 定义了一种表示法 Œ!ı�，其中 X 和 Y 是具有可接受表示法 ı 和 ı 的拓扑空间。
• 该表示法 Œ!ı� 的构造确保其为使逐点函数求值在多项式时间内可计算的最小表示法。
• 它利用了已知的连续函数表示法的表征结果（见 [Wei00, 引理 3.3.14]），并将其适应于多项式时间设置。
• 本文聚焦于 X = R 和 Y = R 的情况，证明在 C[R → R] 上存在一种自然表示法，具备所需的复杂性属性。
• 将推广至其他空间的问题留作未来研究的开放方向。

实验结果

研究问题

• RQ1 是否存在一种 C[R → R] 空间的表示法，可确保函数求值的多项式时间可计算性？
• RQ2 使函数求值在多项式时间内可计算的最小表示法是什么？
• RQ3 可计算分析中函数空间表示的表征能否推广至多项式时间设置？
• RQ4 表示法的选择如何影响连续函数空间中函数求值的复杂性？

主要发现

• 存在一种连续实值函数空间的自然表示法 Œ!ı�，可确保函数求值的多项式时间可计算性。
• 该表示法在最小性意义上是最小的，即它是使逐点求值多项式时间可计算的最粗糙表示法。
• 该构造将可计算分析中已知的表征结果推广至实值函数的多项式时间设置。
• 该结果为实计算中连续函数的复杂性理论分析提供了基础表示法。
• 推广至其他拓扑空间仍为开放问题，提示了未来研究的方向。