Abstract
We introduce, and initiate the study of, average-case bit-complexity theory over the reals: Like in the discrete case a first, naïve notion of polynomial average runtime turns out to lack robustness and is thus refined. Standard examples of explicit continuous functions with increasingly high worst-case complexity are shown to be in fact easy in the mean; while a further example is constructed with both worst and average complexity exponential: for topological/metric reasons, i.e., oracles do not help. The notions are then generalized from the reals to represented spaces; and, in the real case, related to randomized computation.
Supported in part by the Marie Curie International Research Staff Exchange Scheme Fellowship 294962 within the 7th European Community Framework Programme, by the German Research Foundation (DFG) with project Zi 1009/4-1, and by the International Research Training Group 1529. A preliminary version of this work was presented at CCA 2015. Note added in proof: Theorem 9 is conditional to the assertion of Lemma 11(c).
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Schröder, M., Steinberg, F., Ziegler, M. (2016). Average-Case Bit-Complexity Theory of Real Functions. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_43
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DOI: https://doi.org/10.1007/978-3-319-32859-1_43
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