Mixed quantifier prefixes over Diophantine equations with integer variables
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929383607894016 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that $\forall^2\exists^4$ over $\mathbb Z$ is undecidable, that is, there is no algorithm to determine for any $P(x_1,\ldots,x_6)\in\mathbb Z[x_1,\ldots,x_6]$ whether $$\forall x_1\forall x_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where $x_1,\ldots,x_6$ are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, $\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded is undecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ is undecidable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_08302 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Mixed quantifier prefixes over Diophantine equations with integer variables Sun, Zhi-Wei Number Theory Logic 03D35, 11U05, 03D25, 11D99 In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that $\forall^2\exists^4$ over $\mathbb Z$ is undecidable, that is, there is no algorithm to determine for any $P(x_1,\ldots,x_6)\in\mathbb Z[x_1,\ldots,x_6]$ whether $$\forall x_1\forall x_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where $x_1,\ldots,x_6$ are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, $\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded is undecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ is undecidable. |
| title | Mixed quantifier prefixes over Diophantine equations with integer variables |
| topic | Number Theory Logic 03D35, 11U05, 03D25, 11D99 |
| url | https://arxiv.org/abs/2103.08302 |