Real hypercomputation and continuity

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By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of hype rcomputation allow for the effective evaluation of also discontinuous f: R ->-> R. More precisely the present work considers the following three superTuring notions of real function computability: - lativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0"; - encoding input x epsilon R and/or output y = f (x) in weaker ways also related to the Arithmetic Hierarchy; - nondeterministic 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 provides the required power to evaluate for instance the discontinuous Heaviside function.
Publisher
SPRINGER
Issue Date
2007-07
Language
English
Article Type
Article; Proceedings Paper
Keywords

COMPUTABILITY; HIERARCHY; NUMBERS; COMPLEXITY; MACHINES

Citation

THEORY OF COMPUTING SYSTEMS, v.41, no.1, pp.177 - 206

ISSN
1432-4350
DOI
10.1007/s00224-006-1343-6
URI
http://hdl.handle.net/10203/203397
Appears in Collection
CS-Journal Papers(저널논문)
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