Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class

We study a reverse problem - given a reduced dynamics or partial dynamics, can we compute a residue class who presents that dynamics.We design a dedicated algorithm that takes as input a dynamics with length t consists of “I” or “O” can output a residue class who present this dynamics in the first t transformations. We thus design computer program that can output a reside class by inputting a reduced dynamics or partial dynamics. That is, inputting c∈{I,O}^L, CntO(s) ≤ ceil( CntI(s)*lambda), lambda=ln1.5/ln2=0.58469250, s = Substr(c,1,i), i=1,2,..., L. In other words, the dynamics is above of or cutting ratio line in our proposed Collatz graph.Note that, the algorithm is quit lightweight and designed from our formal proof of Partition Theorem - We prove that all natural numbers are partitioned regularly corresponding to ongoing dynamics. Given any natural number x that is i module 2^t (i is an odd integer), the first t transformations in terms of I or O can be determined and identical with the first t transformations of of i. Once current value after t (t is greater or equal to 2) transformations of I or O, is less than x, then reduced dynamics of x is obtained. Otherwise, the residue class of x (namely, i module 2^t) can be partitioned into two halves (namely, i module 2^{t+1} and i+2^t module 2^{t+1}), and either half presents I or O in intermediately forthcoming (t+1)-th transformation.

本研究探讨了一种逆向问题——给定一种简化的动力学或部分动力学，我们能否计算出一个表示该动力学的剩余类。我们设计了一种专用的算法，该算法以长度为t的动力学作为输入，其中包含“I”或“O”，能够输出在最初的t次变换中呈现该动力学的剩余类。因此，我们设计了一款计算机程序，通过输入简化的动力学或部分动力学来输出剩余类。具体而言，输入为c∈{I,O}^L，且满足CntO(s) ≤ ceil(CntI(s)*lambda)，其中lambda=ln1.5/ln2=0.58469250，s = Substr(c,1,i)，i=1,2,..., L。换言之，该动力学位于我们提出的Collatz图的上侧或切割比例线。值得注意的是，该算法极为轻量，其设计基于我们对划分定理的正式证明——我们证明所有自然数均按照持续的动力学的规律进行划分。对于任何自然数x，若其为i模2^t（i为奇整数），则可以确定其前t次变换（以“I”或“O”表示），并且与i的前t次变换相同。一旦当前值在经过t次（t大于等于2）的“I”或“O”变换后小于x，则得到x的简化动力学。否则，x的剩余类（即i模2^t）可以被划分为两个部分（即i模2^{t+1}和i+2^t模2^{t+1}），且任一部分在接下来的(t+1)次变换中均会呈现“I”或“O”。