Real Hypercomputation and Continuity |
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Authors: | Martin Ziegler |
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Affiliation: | (1) University of Paderborn, Paderborn 33095, Germany |
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Abstract: | By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We
wonder whether and which kinds of hypercomputation allow for the effective evaluation of also discontinuous
. More precisely the present work considers the following three super-Turing notions of real function computability: - relativized
computation; specifically given oracle access to the Halting Problem
or its jump
; - encoding input
and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; - nondeterministic computation. It turns
out that any
computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation provides
the required power to evaluate for instance the discontinuous Heaviside function. |
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Keywords: | |
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