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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.08422 |
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| _version_ | 1866913350350274560 |
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| author | Karpov, Vladimir E. |
| author_facet | Karpov, Vladimir E. |
| contents | It is well known that whenever a class of structures $\mathcal{K}_1$ is interpretable in a class of structures $\mathcal{K}_2$, then the hereditary undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the hereditary undecidability of (a suitable fragment of) the theory of $\mathcal{K}_2$. In the present paper, we construct a $Σ_1$-interpretation of the class of all finite bipartite graphs in the class of all pairs of equivalence relations on the same finite domain; from this we obtain the hereditary undecidability of the $Σ_2$-theory of the second class. Next, we construct a $Σ_1$-interpretation of the class of all pairs of equivalence relations on the same finite domain in the class of all pairs consisting of a linear ordering and an equivalence relation on the same finite domain; this gives us the hereditary undecidability of the $Σ_2$-theory of the second class. The corresponding results are, in a sense, optimal, since the $Π_2$-theories of the classes under consideration are decidable.
Keywords: undecidability, elementary theories, prefix fragments |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08422 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hereditary undecidability of fragments of some elementary theories Karpov, Vladimir E. Logic It is well known that whenever a class of structures $\mathcal{K}_1$ is interpretable in a class of structures $\mathcal{K}_2$, then the hereditary undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the hereditary undecidability of (a suitable fragment of) the theory of $\mathcal{K}_2$. In the present paper, we construct a $Σ_1$-interpretation of the class of all finite bipartite graphs in the class of all pairs of equivalence relations on the same finite domain; from this we obtain the hereditary undecidability of the $Σ_2$-theory of the second class. Next, we construct a $Σ_1$-interpretation of the class of all pairs of equivalence relations on the same finite domain in the class of all pairs consisting of a linear ordering and an equivalence relation on the same finite domain; this gives us the hereditary undecidability of the $Σ_2$-theory of the second class. The corresponding results are, in a sense, optimal, since the $Π_2$-theories of the classes under consideration are decidable. Keywords: undecidability, elementary theories, prefix fragments |
| title | Hereditary undecidability of fragments of some elementary theories |
| topic | Logic |
| url | https://arxiv.org/abs/2405.08422 |