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Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
What Is a Random String?
, 1995
"... Chaitin's algorithmic definition of random strings  based on the complexity induced by selfdelimiting computers  is critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an adequate model for nite rando ..."
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Chaitin's algorithmic definition of random strings  based on the complexity induced by selfdelimiting computers  is critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an adequate model for nite random objects. It is a better model than the original (Kolmogorov) proposal. Finally, some open problems will be discussed.
What Is a Random String? (Extended Abstract)
"... # Cristian Calude + Abstract Chaitin's algorithmic definition of random stringsbased on the complexity induced by selfdelimiting computersis critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an ..."
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# Cristian Calude + Abstract Chaitin's algorithmic definition of random stringsbased on the complexity induced by selfdelimiting computersis critically discussed. One shows that Chaitin's model satisfy many natural requirements related to randomness, so it can be considered as an adequate model for finite random objects. It is a better model than the original (Kolmogorov) proposal. Finally, some open problems will be discussed. Keywords: Blankendmarker complexity, Chaitin (selfdelimiting) complexity, random strings. 1 Motivation Suppose that persons A and B give us a sequence of 32 bits each, saying that they were obtained from independent coin flips. If A gives the string u = 01001110100111101001101001110101 and B gives the string v = 00000000000000000000000000000000, then we would tend to believe A and would not believe B: the string u seems to be random, but the string v does not. Further on, if we change the value of a bit (say, from 1 to 0) in a (non) "random" stri...