On the computational properties of ambivalent sets and functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914307975938048 |
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| author | Normann, Dag Sanders, Sam |
| author_facet | Normann, Dag Sanders, Sam |
| contents | Examples of discontinuous functions already appear in the work of Euler, Abel, Dirichlet, Fourier, and Bolzano. A ground-breaking discovery due to Baire was that many discontinuous functions are well-behaved in that they are the pointwise limit of a sequence of continuous functions; the latter form a class nowadays simply called `Baire 1'. We shall study a class strictly between the semi-continuous and Baire 1 functions, called the ambivalent fuctions. In particular, we investigate the computational properties of the class of ambivalent functions and sets, denoted $\bf Δ$, working with Kleene's S1-S9 schemes. Computational equivalences for various standard operations (supremum, Baire 1 representation, \dots) on $\bf Δ$ are established, including the structure functional $Ω_{\bf Δ}$ that decides if a given ambivalent set is non-empty. A selector is shown to be computable relative to $Ω_{\bf Δ}$ and Kleene's quantifier $\exists^{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05620 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the computational properties of ambivalent sets and functions Normann, Dag Sanders, Sam Logic 03B30, 03F35 F.4.1; F.1.1 Examples of discontinuous functions already appear in the work of Euler, Abel, Dirichlet, Fourier, and Bolzano. A ground-breaking discovery due to Baire was that many discontinuous functions are well-behaved in that they are the pointwise limit of a sequence of continuous functions; the latter form a class nowadays simply called `Baire 1'. We shall study a class strictly between the semi-continuous and Baire 1 functions, called the ambivalent fuctions. In particular, we investigate the computational properties of the class of ambivalent functions and sets, denoted $\bf Δ$, working with Kleene's S1-S9 schemes. Computational equivalences for various standard operations (supremum, Baire 1 representation, \dots) on $\bf Δ$ are established, including the structure functional $Ω_{\bf Δ}$ that decides if a given ambivalent set is non-empty. A selector is shown to be computable relative to $Ω_{\bf Δ}$ and Kleene's quantifier $\exists^{2}$. |
| title | On the computational properties of ambivalent sets and functions |
| topic | Logic 03B30, 03F35 F.4.1; F.1.1 |
| url | https://arxiv.org/abs/2602.05620 |