Abstract

AbstractWe 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. Termination of this rewriting system is equivalent to the Collatz conjecture. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses the automated method of matrix/arctic interpretations and we perform experiments where we obtain proofs of nontrivial weakenings of the Collatz conjecture. Finally, we adapt our rewriting system to show that other open problems in mathematics can also be approached as termination problems for relatively small rewriting systems. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

Highlights

  • Let N = {0, 1, 2, . . .} denote the natural numbers and N+ = {1, 2, 3, . . .} denote the positive integers

  • For all n ∈ N+, there exists some k ∈ N such that Ck(n) = 1. This is a longstanding open problem and there is a vast literature dedicated to its study

  • – We automatically prove various weakenings of the Collatz conjecture and observe that only relatively large matrix/arctic interpretations exist for some generalized Collatz functions

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Summary

Introduction

We describe an approach based on termination of string rewriting to automatically search for a proof of the Collatz conjecture. The invention of the method of matrix interpretations and its variants such as arctic interpretations turns the quest of finding a ranking function to witness termination into a problem that is suitable for systematic search. – We show how a generalized Collatz function can be expressed as a rewriting system that is terminating if and only if the function is convergent. – We automatically prove various weakenings of the Collatz conjecture and observe that only relatively large matrix/arctic interpretations exist for some generalized Collatz functions. Existing termination tools often limit their default strategies to search for small interpretations as they are tailored for the setting where the task is to quickly solve a large quantity of relatively easy problems. – We present adaptations of our rewriting system that allow reformulating several more open problems in mathematics as termination problems of small size

String Rewriting Systems
Interpretation Method
Generalized Collatz Functions
Rewriting in Unary
Rewriting in Mixed Base
Automated Proofs
Convergence of W
Farkas’ Variant
Subsets of T
Odd Trajectories
Collatz Trajectories Modulo 8
More Problems to Approach via Rewriting
Related Work
Future Work
Full Text
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