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Combining Determinism and Indeterminism
arXiv - CS - Computational Complexity Pub Date : 2020-09-02 , DOI: arxiv-2009.03996
Michael Stephen Fiske

Our goal is to construct mathematical operations that combine indeterminism measured from quantum randomness with computational determinism so that non-mechanistic behavior is preserved in the computation. Formally, some results about operations applied to computably enumerable (c.e.) and bi-immune sets are proven here, where the objective is for the operations to preserve bi-immunity. While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group (Sym$(\mathbb{N})$) on the natural numbers $\mathbb{N}$. This new uncountable subgroup is called the bi-immune symmetric group. We show that the bi-immune symmetric group contains the finitary symmetric group on the natural numbers, and consequently is highly transitive. Furthermore, the bi-immune symmetric group is dense in Sym$(\mathbb{N})$ with respect to the pointwise convergence topology. The complete structure of the bi-immune symmetric group and its subgroups generated by one or more bi-immune rearrangements is unknown.

中文翻译:

结合决定论和非决定论

我们的目标是构建数学运算,将量子随机性测量的不确定性与计算确定性相结合,以便在计算中保留非机械行为。正式地,这里证明了有关应用于可计算可枚举 (ce) 和双免疫集的操作的一些结果,其目标是使操作保持双免疫。在对自然数进行重排操作时,我们发现双免疫重排会在自然数 $\mathbb{N}$ 上生成无限对称群 (Sym$(\mathbb{N})$) 的不可数子群。这个新的不可数子群称为双免疫对称群。我们表明双免疫对称群包含自然数上的有限对称群,因此具有高度的传递性。此外,相对于逐点收敛拓扑,双免疫对称群在 Sym$(\mathbb{N})$ 中是密集的。由一种或多种双免疫重排产生的双免疫对称基团及其亚基的完整结构是未知的。
更新日期:2020-11-20
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