Exploring the abyss in Kleene's computability theory
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911757862174720 |
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| author | Sanders, Sam |
| author_facet | Sanders, Sam |
| contents | Kleene's computability theory based on the S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $\exists^n$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $\exists^2$, while the former are computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07438 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exploring the abyss in Kleene's computability theory Sanders, Sam Logic Logic in Computer Science 03D75, 03D60 F.1.1; F.4.1 Kleene's computability theory based on the S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier $\exists^n$ and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from $\exists^2$, while the former are computable in $\exists^3$ but not in weaker oracles. Of course, there is a great divide or abyss separating $\exists^2$ and $\exists^3$ and we identify slight variations of our new non-normal functionals that are again computable in $\exists^2$, i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis. |
| title | Exploring the abyss in Kleene's computability theory |
| topic | Logic Logic in Computer Science 03D75, 03D60 F.1.1; F.4.1 |
| url | https://arxiv.org/abs/2308.07438 |