Abstract
In this paper we use infinitary Turing machines with tapes of length \(\kappa \) and which run for time \( \kappa \) as presented, e.g., by Koepke & Seyfferth, to generalise the notion of type two computability to \({2}^{\kappa }\), where \(\kappa \) is an uncountable cardinal with \(\kappa ^{<\kappa }=\kappa \). Then we start the study of the computational properties of \({{\mathrm{\mathbb {R}}}}_\kappa \), a real closed field extension of \({{\mathrm{\mathbb {R}}}}\) of cardinality \({2}^{\kappa }\), defined by the first author using surreal numbers and proposed as the candidate for generalising real analysis. In particular we introduce representations of \({{\mathrm{\mathbb {R}}}}_\kappa \) under which the field operations are computable. Finally we show that this framework is suitable for generalising the classical Weihrauch hierarchy. In particular we start the study of the computational strength of the generalised version of the Intermediate Value Theorem.
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Notes
- 1.
Carl has also introduced a notion of generalized (strong) Weihrauch reducibility in [3]. Because his goal is to investigate multi-valued (class) functions on \( V \), the space of codes he uses is the class of ordinal numbers, considered with the ordinal Turing machines of Koepke [13]. Therefore his approach is significantly different from ours, and we do not know of any connections between the two.
- 2.
We call an ordinal \( \alpha \) even if \( \alpha = \lambda + 2n \) for some limit \( \lambda \) and natural \( n \), odd otherwise.
- 3.
Note that this can be checked just by looking at the first two bits of p.
- 4.
Note that this is easily computable, it is in fact enough to check that L and R are empty, and this can be done just by checking the first two bits of the first sequence in the left and in the first sequence on the right.
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Acknowledgments
This research was partially done whilst the authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences in the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite. The research benefited from the Royal Society International Exchange Grant Infinite games in logic and Weihrauch degrees. The second author was also supported by the Capes Science Without Borders grant number 9625/13-5. The authors are grateful to Benedikt Löwe and Arno Pauly for the many fruitful discussions and to the Institute for Logic, Language and Computation for the hospitality offered to the first author. Finally, the authors wish to thank the three anonymous referees for the helpful comments which have improved the paper.
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Galeotti, L., Nobrega, H. (2017). Towards Computable Analysis on the Generalised Real Line. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_24
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