Finding descending sequences through ill-founded linear orders
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arXiv
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| Format: | Preprint |
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2020
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| author | Goh, Jun Le Pauly, Arno Valenti, Manlio |
| author_facet | Goh, Jun Le Pauly, Arno Valenti, Manlio |
| contents | In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf{DS}$, despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf{DS}$ and $\mathsf{BS}$ by considering $\boldsymbolΓ$-presented orders, where $\boldsymbolΓ$ is a Borel pointclass or $\boldsymbolΔ^1_1$, $\boldsymbolΣ^1_1$, $\boldsymbolΠ^1_1$. We study the obtained $\mathsf{DS}$-hierarchy and $\mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_03840 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Finding descending sequences through ill-founded linear orders Goh, Jun Le Pauly, Arno Valenti, Manlio Logic Logic in Computer Science Combinatorics 03D30 03D78 06A75 In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf{DS}$, despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf{DS}$ and $\mathsf{BS}$ by considering $\boldsymbolΓ$-presented orders, where $\boldsymbolΓ$ is a Borel pointclass or $\boldsymbolΔ^1_1$, $\boldsymbolΣ^1_1$, $\boldsymbolΠ^1_1$. We study the obtained $\mathsf{DS}$-hierarchy and $\mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. |
| title | Finding descending sequences through ill-founded linear orders |
| topic | Logic Logic in Computer Science Combinatorics 03D30 03D78 06A75 |
| url | https://arxiv.org/abs/2010.03840 |