ar X iv : c s / 05 08 06 9 v 1 [ cs . L O ] 1 5 A ug 2 00 5 Real Hypercomputation and Continuity ⋆

نویسنده

  • Martin Ziegler
چکیده

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 : R → 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 H ≡T ∅ ′ or its jump H ′ ≡T ∅ ′′; – encoding input x ∈ R and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; – non-deterministic computation. It turns out that any f : R → R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous Heaviside function.

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تاریخ انتشار 2005