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Computable real functions of bounded variation and semi-computable real numbers

by Robert Rettinger, Xizhong Zheng, Burchard Von Braunmühl - In Proceedings of COCOON 2002
"... Abstract. In this paper we discuss some basic properties of computable real functions of bounded variation (CBV-functions for short). Especially, it is shown that the image set of semi-computable real numbers un-der CBV-functions is a proper subset of the class of weakly computable real numbers; Two ..."
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Abstract. In this paper we discuss some basic properties of computable real functions of bounded variation (CBV-functions for short). Especially, it is shown that the image set of semi-computable real numbers un-der CBV-functions is a proper subset of the class of weakly computable real numbers; Two applications of CBV-functions to semi-computable real numbers produce the whole closure of semi-computable real num-bers under total computable real functions, and the image sets of semi-computable real numbers under monotone computable functions and CBV-functions are different. 1
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On the Effective Jordan Decomposability

by Xizhong Zheng, Robert Rettinger, Burchard von Braunmühl , 2003
"... The classical Jordan decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper explores the effective version of Jordan decomposition. We give a sufficient and necessary condition for those computable real functi ..."
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The classical Jordan decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper explores the effective version of Jordan decomposition. We give a sufficient and necessary condition for those computable real functions of bounded variation which can be expressed as a difference of two computable increasing functions. Using this condition, we prove further that there is a computable real function which has even a computable modulus of absolute continuity (hence is of bounded variation) but it is not a difference of any two computable increasing functions. The polynomial time version of this result holds too and this gives a negative answer to an open question of Ko in [6].
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Effectively Absolute Continuity and Effective Jordan Decomposability

by Xizhong Zheng , Robert Rettinger , Burchard von Braunmühl , 2002
"... Classically, any absolute continuous real function is of bounded variation and hence can always be expressed as a difference of two increasing continuous functions (socalled Jordan decomposition). The effective version of this result is not true. In this paper we give a sufficient and necessary cond ..."
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Classically, any absolute continuous real function is of bounded variation and hence can always be expressed as a difference of two increasing continuous functions (socalled Jordan decomposition). The effective version of this result is not true. In this paper we give a sufficient and necessary condition for computable real functions which can be expressed as two computable increasing functions (effectively Jordan decomposable, or EJD for short). Using this condition, we prove further that there is a computable real function which has a computable modulus of absolute continuity but is not EJD. The polynomial time version of this result holds accordingly too and this gives a negative answer to an open question of Ko in [6].