## On the construction of effective random sets (2002)

Venue: | MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2002, LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Merkle02onthe,

author = {Wolfgang Merkle and Nenad Mihailović},

title = {On the construction of effective random sets},

booktitle = {MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2002, LECTURE NOTES IN COMPUTER SCIENCE},

year = {2002},

pages = {2420--568},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We give a direct and rather simple construction of Martin-Löf random and rec-random sets with certain additional properties. First, reviewing the result of Gacs and Kucera, given any set X we construct a Martin-Löf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a Martin-Löf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin real is Martin-Löf random and computably enumerable. Third, by a variant of the basic construction we obtain a rec-random set that is weak truthtable autoreducible. The mentioned results on self- and autoreducibility complement work of Ebert, Merkle, and Vollmer [7-9], from which it follows that no Martin-Löf random set is Turing-autoreducible and that no rec-random set is truth-table autoreducible.

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Citation Context ...t succeeds on the set. So it comes as a slight surprise that in fact any set is wtt-reducible to a Martin-Löf random set. This celebrated result has been obtained independently by Gács [9] and Kučera =-=[11, 12]-=-. They state the result for T-reducibility, however the reductions constructed in their proofs are already wtt-reductions [23, Section 6.1]. In this section, we give an alternate account of the result... |

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Citation Context ...ect value is computed for infinitely many inputs. The concept of i.o. ttautoreducibility is defined accordingly, i.e., we require in addition that the machine performing the reduction is total. Ebert =-=[7]-=- showed that every Martin-Löf random set is i.o. tt-autoreducible. By results of Ebert, Merkle, and Vollmer [8], any Martin-Löf random set can 17sbe i.o.-tt-autoreduced such that the fraction of corre... |

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Citation Context ...re computably enumerable self-reducible. The two latter results complement the known facts that no rec-random set is truth-table autoreducible and that no Martin-Löf random set is Turingautoreducible =-=[8, 24]-=-. 1 Introduction In what follows, we present a comparatively simple way to construct Martin-Löf random and rec-random sets with certain additional properties, which works by diagonalizing against appr... |

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Citation Context ... rational initial value d(λ) is computable if and only if the corresponding betting strategy is rational-valued and computable. Computable martingales are considered in recursion-theoretical settings =-=[1, 20, 21, 23]-=-, while in connection with complexity classes one considers martingales that in addition are computable within appropriate resourcebounds [2, 14, 15, 17]. Definition 2. A set is rec-random if no compu... |

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Citation Context ...ted in order to wttreduce a given set X to a Martin-Löf random set R do not depend on the set X, i.e., there is a single machine that wtt-reduces any given set to some Martin-Löf random set. Hertling =-=[5, 10]-=- investigates general assumptions on a class C that imply that the result of Gács and Kučera as stated in Theorems 10 and 14 holds with C in place of the class of Martin-Löf random sets. He introduces... |

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Citation Context ...ore than a constant [13, Theorem 3.6.1] while a corresponding statement for rec-random sets is false [18]. Remark 8 Given any martingale d, word w, and natural number k, we have d(w) = 1 2k � d(wu) . =-=(3)-=- u∈{0,1} k This follows by an easy inductive argument that uses the fairness condition (1). Conversely, (1) is a special case of (3) where k = 1. 6s3 Every set is reducible to a Martin-Löf random set ... |

1 |
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Citation Context ...rticular, the procedure used to define the words wi is due to him. However, the account in terms of martingales is again considerably less involved than the original one in terms of Martin-Löf covers =-=[5, 9]-=-. Definition 13. A set A is wtt-reducible to a set B with vanishing relative redundancy if A is wtt-reducible to B by a Turing machine M such that the use of M is bounded by a nondecreasing computable... |

1 | Autoreducibility of random sets: a sharp bound on the density of guessed bits - Ebert, Merkle |