The Non-normal Abyss in Kleene’s Computability Theory

Sanders, Sam

doi:10.1007/978-3-031-36978-0_4

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2024

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Cited by 2 publications

References 22 publications

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On the Computational Properties of Weak Continuity Notions

Sanders

2024

Twenty Years of Theoretical and Practical Synergies

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On the Computational Properties of Weak Continuity Notions

Sanders

2024

Twenty Years of Theoretical and Practical Synergies

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Exploring the abyss in Kleene’s computability theory

Sanders

2024

COM

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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 ∃ 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 ∃ 2 , while the former are computable in ∃ 3 but not in weaker oracles. Of course, there is a great divide or abyss separating ∃ 2 and ∃ 3 and we identify slight variations of our new non-normal functionals that are again computable in ∃ 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.

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The Non-normal Abyss in Kleene’s Computability Theory

Cited by 2 publications

References 22 publications

On the Computational Properties of Weak Continuity Notions

On the Computational Properties of Weak Continuity Notions

Exploring the abyss in Kleene’s computability theory

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