Theory of Computing Systems

, Volume 41, Issue 1, pp 177–206 | Cite as

Real Hypercomputation and Continuity

  • Martin Ziegler


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 \(f\colon \ {\Bbb R}\to{\Bbb R}\). 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 \(\emptyset'\) or its jump \(\emptyset''\); - encoding input \(x\in{\Bbb R}\) and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; - nondeterministic computation. It turns out that any \(f\colon \ {\Bbb R}\to{\Bbb R}\) 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.


Real Function Turing Machine Computable Function Rational Sequence Binary Expansion 
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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.University of PaderbornPaderborn 33095Germany

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