Abstract
A c.e. real x is Solovay reducible (S-reducible) to another c.e. real y if y is at least as difficult to be approximated as x. In this case, y is at least as random as x. Thus, the S-reducibility classifies relative randomness of c.e. reals such that the c.e. random reals are complete in the class of c.e. reals under the S-reducibility. In this paper we investigate extensions of the S-reducibility outside the c.e. reals. We show that the straightforward extension does not behave satisfactorily. Then we introduce two new extensions which coincide with the S-reducibility on the c.e. reals and behave reasonably outside the c.e. reals. Both of these extensions imply the rH-reducibility of Downey, Hirschfeldt and LaForte [6]. At last we show that even under the rH-reducibility the computably approximable random reals cannot be characterized as complete elements of this reduction.
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Zheng, X., Rettinger, R. (2004). On the Extensions of Solovay-Reducibility. In: Chwa, KY., Munro, J.I.J. (eds) Computing and Combinatorics. COCOON 2004. Lecture Notes in Computer Science, vol 3106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27798-9_39
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DOI: https://doi.org/10.1007/978-3-540-27798-9_39
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