Abstract
We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result (Fritz et al. in Algebra Number Theory 12:1773–1786, 2018). The unusual feature of the use of computers here was that a computer generated but human readable proof was read, understood, generalized and abstracted by mathematicians to obtain the key lemma in an interesting mathematical result.
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Notes
Participants: T. Fritz, S. Gadgil, A. Khare, P. Nielsen, L. Silberman, T. Tao.
https://github.com/siddhartha-gadgil/Superficial/wiki/A-commutator-bound for the full proof as originally posted.
Indeed, an optimal algorithm for \(l_b(g; B)\) for general finite B gives a solution to the word problem for groups, which is known to be algorithmically undecidable. Namely, given relations\(r_1 \in \langle \alpha , \beta \rangle \), \(r_2\in \langle \alpha , \beta \rangle \), ...\(r_m \in \langle \alpha , \beta \rangle \), let B be the set \(\{(r_1, 0), (r_2, 0), \dots , r_n, 0)\}\). Then \(l_b(g; B)= 0\) if and only if g is trivial in the group \(\langle \alpha , \beta ; r_1 = e\), \(r_2 = e, \dots r_m =e \rangle \).
The full code is in the repository https://github.com/siddhartha-gadgil/Superficial. The script is generated from this source. The script uses the same algorithms we originally used, but with modifications to be more robust in memory usage and to avoid concurrency (as the concurrency we implemented leads to non-determinacy, and occasionally to race conditions).
The proof was posted less than two days after we began writing code, and the main question answered less than a day after that.
References
Fritz, T., Gadgil, S., Khare, A., Nielsen, P., Silberman, L., Tao, T.: Homogeneous length functions on groups. Algebra Number Theory 12, 1773–1786 (2018)
Gadgil, S.: Watson–Crick pairing, the Heisenberg group and Milnor invariants. J. Math. Biol. 59, 123–142 (2009)
Khare, A., Rajaratnam, B.: The Khinchin–Kahane inequality and Banach space embeddings for metric groups, preprint (2016)
Khare, A., Rajaratnam, B.: The Hoffmann–Jørgensen inequality in metric semigroups. Ann. Probab. 45, 4101–4111 (2017)
Acknowledgements
I thank the referees and the editors for many valuable comments, which have led to the paper being completely rewritten twice and much improved in the process. It is also a pleasure to thank the rest of the PolyMath 14 team for the collaboration of which the work described here is a part.
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Gadgil, S. Homogeneous Length Functions on Groups: Intertwined Computer and Human Proofs. J Autom Reasoning 64, 677–688 (2020). https://doi.org/10.1007/s10817-019-09523-1
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DOI: https://doi.org/10.1007/s10817-019-09523-1