Abstract
There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch’s Type Two Theory of Effectivity or on domain-theoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysis-oriented line of research is based on Grzegorczyk’s work. All these approaches have turned out to be equivalent.
In this paper it is shown that real numbers as well as real-valued functions are computable in Weihrauch’s sense if and only if they are definable in Escardó’s functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this domain.
For defining the semantics of the language Real. PCF a full subcategory of the category of bounded-complete ω-continuous directed complete partial orders is introduced and it is defined when a domain in this category is effectively given. The subcategory of effectively given domains contains the interval domain and is Cartesian closed.
The paper mainly contains results from the second author’s diploma thesis [19] written under the supervision of the first author.
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Spreen, D., Schulz, H. (2001). On The Equivalence of Some Approaches to Computability on the Real Line. In: Keimel, K., Zhang, GQ., Liu, YM., Chen, YX. (eds) Domains and Processes. Semantic Structures in Computation, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0654-5_5
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DOI: https://doi.org/10.1007/978-94-010-0654-5_5
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