Abstract
We discuss various ways of representing natural numbers in computations. We are primarily concerned with their computational properties, i.e. which functions each of these representations allows us to compute. We show that basic functions, such as successor, addition, multiplication and exponentiation are largely computationally independent from each other, which means that in most cases computability of one of them in a certain representation does not imply that others will be computable in it as well.
We also examine what difference can be made if we restrict our attention only to those representations in which it is decidable whether two numerals represent the same number. It turns out that the impact of such restriction is huge and that it allows us to rule out representations with certain unusual properties.
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- 1.
The algorithm which is supposed to find out whether \(n \in A\) will have answers for \(n \in \{0,1\}\) explicitly given as special cases.
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Wrocławski, M. (2019). Representations of Natural Numbers and Computability of Various Functions. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_26
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DOI: https://doi.org/10.1007/978-3-030-22996-2_26
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