On the Diophantine problem related to power circuits
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914453060059136 |
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| author | Rybalov, Alexander |
| author_facet | Rybalov, Alexander |
| contents | Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that support addition and the operation $(x,y) \mapsto x \cdot 2^y$ on integers. They also posed the question of decidability of the Diophantine problem over the structure $\langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$, which is closely related to power circuits. In this paper, we prove that the Diophantine problem over this structure is undecidable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Diophantine problem related to power circuits Rybalov, Alexander Logic Group Theory Number Theory Rings and Algebras 11U05, 03D35 Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that support addition and the operation $(x,y) \mapsto x \cdot 2^y$ on integers. They also posed the question of decidability of the Diophantine problem over the structure $\langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$, which is closely related to power circuits. In this paper, we prove that the Diophantine problem over this structure is undecidable. |
| title | On the Diophantine problem related to power circuits |
| topic | Logic Group Theory Number Theory Rings and Algebras 11U05, 03D35 |
| url | https://arxiv.org/abs/2601.00835 |