Abstract
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.
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Acknowledgements
We thank Johannes Waldmann for insightful discussions regarding arctic matrix interpretations, for pointing us to [53] and the rewriting system in Example 3.3, for responding to our challenge to solve Farkas’ variant with Matchbox, and for feedback on an early draft. We thank Carsten Fuhs and Jürgen Giesl for responding to our challenge to solve Farkas’ variant with AProVE. We additionally thank Carsten Fuhs for his thorough explanations of the dependency pair framework and AProVE’s strategies. We thank Florian Frohn for responding to the challenge to solve the subsystems from Section 4.3 with AProVE. We thank Jeffrey Lagarias for discussions regarding the problems in Section 4.5. We thank Luke Schaeffer and Chris Lynch for discussions on alternative rewriting systems that simulate the Collatz map. We thank Jeremy Avigad and Jasmin Blanchette for their detailed comments on an early draft. Finally, we thank the reviewers of CADE for their comments on the preliminary version of this paper and the reviewers of the Journal of Automated Reasoning for their comments on the journal version.
This material is based upon work supported by the National Science Foundation under grant CCF-2006363.
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Yolcu, E., Aaronson, S. & Heule, M.J.H. An Automated Approach to the Collatz Conjecture. J Autom Reasoning 67, 15 (2023). https://doi.org/10.1007/s10817-022-09658-8
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DOI: https://doi.org/10.1007/s10817-022-09658-8