We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary–ternary representations of positive integers. We prove that the termination of this rewriting system is equivalent to the Collatz conjecture. We also prove that a previously studied rewriting system that simulates the Collatz function using unary representations does not admit termination proofs via natural matrix interpretations, even when used in conjunction with dependency pairs. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses natural/arctic matrix interpretations and we find automated proofs of nontrivial weakenings of the Collatz conjecture. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

我们通过字符串重写终止的视角探索 Collat​​z 猜想及其变体。我们构建了一个重写系统，模拟 Collat​​z 函数在对应于正整数的混合二元-三元表示的字符串上的迭代应用。我们证明这个重写系统的终止等价于 Collat​​z 猜想。我们还证明，先前研究的使用一元表示模拟 Collat​​z 函数的重写系统不接受通过自然矩阵解释的终止证明，即使与依赖对结合使用也是如此。为了展示我们的方法在证明数学上有趣的陈述方面的可行性，我们实现了一个使用自然/北极矩阵解释的最小终止证明器，并且我们找到了 Collat​​z 猜想的非平凡弱化的自动证明。尽管我们没有成功证明 Collat​​z 猜想，但我们相信这里的想法代表了一种有趣的新方法。