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Representation Theorems for Analytic Machines and Computability of Analytic Functions

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Abstract

We present results concerning analytic machines, a model of real computation introduced by Hotz which extends the well-known Blum, Shub and Smale machines (BSS machines) by infinite converging computations. The well-known representation theorem for BSS machines elucidates the structure of the functions computable in the BSS model: the domain of such a function partitions into countably many semi-algebraic sets, and on each of those sets the function is a polynomial resp. rational function. In this paper, we study whether the representation theorem can, in the univariate case, be extended to analytic machines, i.e. whether functions computable by analytic machines can be represented by power series in some part of their domain. We show that this question can be answered in the negative over the real numbers but positive under certain restrictions for functions over the complex numbers. We then use the machine model to define computability of univariate complex analytic (i.e. holomorphic) functions and examine in particular the class of analytic functions which have analytically computable power series expansions. We show that this class is closed under the basic analytic operations composition, local inversion and analytic continuation.

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Notes

  1. Indeed, a machine that computes χ ℚ enumerates the rationals and compares its input to each of the enumerated numbers. If the input is equal to the current number of the enumeration, it outputs 1 and halts. Otherwise, it outputs 0 and proceeds with the enumeration.

  2. For example, consider the function f defined by the power series \(f(z):=\sum_{k=0}^{\infty}(\frac{1}{2})^{k}z^{k},\,|z|<2\). If we now consider z 0=−1, the power series expansion of f at z 0 is given by \(\sum _{k=0}^{\infty}\frac{2}{3^{k+1}}(z+1)^{k}\), which has a convergence radius of 3. Therefore, the radius of convergence of the expansion at −1 exceeds the boundary of the original domain {z:|z|<2}.

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Acknowledgements

We would like to thank the referees for their helpful suggestions, which have greatly improved the quality of the paper.

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  1. Fachbereich für Informatik, Universität des Saarlandes, Saarbrücken, Germany

    Tobias Gärtner & Günter Hotz

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  1. Tobias Gärtner
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  2. Günter Hotz
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Correspondence to Tobias Gärtner.

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Gärtner, T., Hotz, G. Representation Theorems for Analytic Machines and Computability of Analytic Functions. Theory Comput Syst 51, 65–84 (2012). https://doi.org/10.1007/s00224-011-9374-z

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