This paper presents an algorithmic method that, given a positive integer $j$, generates the $j$-th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly $j$ applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the $j$-th convergence stair for any $j > 0$. The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. To the best of our knowledge, this is the first time that such a general specification is provided, which can have significant applications in analyzing and testing the behaviors of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.

翻译：本文提出了一种算法方法，给定正整数$j$，可生成第$j$个收敛阶梯，其中包含所有通过恰好在$j$次应用Collatz函数后满足Collatz猜想的自然数。为此，我们提出了一种新颖的Collatz猜想表述形式，将其视为一个并发程序，并给出了任意$j>0$时第$j$个收敛阶梯的一般情形规范。所提出的规范以无限组无限二叉树的层次化线性排列方式对Collatz数进行组织。据我们所知，这是首次提供此类通用规范，该规范对分析和测试复杂非线性系统的行为具有重要应用价值。我们已将该方法实现为软件工具，可生成各阶梯的Collatz数。此外，我们证明：从任意收敛阶梯中的任何值出发，该猜想均成立。然而，要证明该猜想，必须证明每个自然数都会出现在某个阶梯中，即所有阶梯的并集等于自然数集，这仍是一个未解难题。