Gray-code representation of real number and the induced computability structure

我们研究了与实数可计算结构相关的计算、拓扑和领域理论问题，这些实数可计算结构由{0,1，bot}无限序列（灰码表示）和不确定多头机（IM2-machine）的表示形式推导而来。im2机器可以用带有守卫的逻辑编程语言来实现，比如GHC（守卫角子句）。当我们考虑{0,1，bot}无穷数列上的函数时，GHC中可表示的函数集大于im2可计算的函数集，但当只考虑实数表示时，它们是相等的。函数式语言是无法实现IM2-machines的，但通过对函数式语言进行扩展，用一个受限的amb算子，用顺序图约简规则代替项约简规则来实现IM2-machines的能力。我们将它作为Haskell语言的扩展来实现。表示拓扑空间所需的底部数量与空间的拓扑维数相等。我们在代数定义域上得到了一个更一般的结果，并从定义域理论上证明了它。拓扑空间的编码与子基是密切相关的。生成自然计算结构的编码将生成一个基于正则开集对的二进子基。通过拓扑研究相应子基的性质，可以表征灰度码表示的效率。

We studied computational, topological, and domain-theoretic problems related to the computable structure of real numbers derived form representation with {0,1,bot}-infinite sequences (Gray-code representation) and indeterministic multi-head machine (IM2-machine) operating on them.1.IM2-machines can be realized with logic programming languages with guards like GHC (Guarded Horn Clauses). The set of functions expressible in GHC is larger than the set of IM2-computable functions when we consider functions over {0,1,bot}-infinite sequences but they are equal when only real number representation is considered.2.IM2-machines cannot be realized with functional languages, but by extending a functional language with a restricted amb operator which is realized with sequential graph reduction rules instead of term reduction rules, one can implement the ability of IM2-machines. We have implemented it as an extension of the Haskell language.3.The number of bottoms required to represent a topological space and the topological dimension of the space is equal. We have a more general result on an algebraic domain, which is proved domain-theoretically.4.Codings and subbases of a topological space are closely related. Codings which induce natural computational structure will induce a dyadic subbase, which is based on pairs of regular open sets. The efficiency of the Gray-code representation can be characterized through properties of the corresponding subbase, which are studied topologically.