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The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
Revising type2 computation and degrees of discontinuity
 Electronic Notes in Theoretical Computer Science 167
, 2007
"... Abstract. By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS 2000), Brattka (MLQ 2005), and Ziegler (ToCS 2006) have considered different relaxed notions of computability to cover also discontinuous functi ..."
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Abstract. By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS 2000), Brattka (MLQ 2005), and Ziegler (ToCS 2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the jump of a representation: both a TTE–counterpart to the well known recursiontheoretic jump on Kleene’s Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle–computation of discontinuous functions and characterize the computational power of Type2 nondeterminism to coincide with the first level of the Analytical Hierarchy. 1
Real Hypercomputation and Continuity
, 2005
"... By the sometimes socalled 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 t ..."
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By the sometimes socalled 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 superTuring 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; – 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 does provide the required power to evaluate for instance the discontinuous Heaviside function.
Real Hypercomputation and Degrees of Discontinuity
, 2006
"... Abstract. By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS 2000), Brattka (MLQ 2005), and Ziegler (ToCS 2006) have considered different relaxed notions of computability to cover also discontinuous functi ..."
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Abstract. By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS 2000), Brattka (MLQ 2005), and Ziegler (ToCS 2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the jump of a representation, a TTE–counterpart to the well known recursiontheoretic jump on Kleene’s Arithmetical Hierarchy of hypercomputation. We also consider Markov and Banach/Mazur oracle–computation of discontinuous functions and characterize the computational power of Type2 nondeterminism to coincide with the first level of the Analytical Hierarchy. 1