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

研究了由{0，1，bot}-无限序列表示（格雷码表示）导出的真实的数的可计算结构及其上的不确定多头机（IM 2-machine）的计算、拓扑和Domain理论问题。当我们考虑{0，1，bot}-无限序列上的函数时，GHC中可表示的函数集大于IM 2-可计算函数集，但当只考虑真实的数表示时，它们是相等的。2. IM 2-机不能用函数语言实现，但通过扩展一个带有限制amb算子的函数语言，该算子是用顺序图归约规则而不是项归约规则实现的，可以实现IM 2机器的能力。3.表示拓扑空间所需的底数与拓扑维数相等。在代数整环上得到了一个更一般的结果，并从整环理论上证明了这个结果。引入自然计算结构的编码将引入基于正则开集对的并矢子基。格雷码表示的效率可以通过相应的子基，这是拓扑研究的性质来表征。

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.