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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.
Real Numbers: From Computable to Random
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence (equivalently, if its expansion in base b ≥ 2 is random). The aim of this paper is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed. 1 Notation and Background We will use notation that is standard in computability theory and algorithmic information theory; we will assume familiarity with Turing machine computations, computable and computably enumerable (c.e.) sets (see, for example, Soare [48] or Odifreddi [40]) and elementary algorithmic information theory (see,
Randomness Everywhere: Computably Enumerable Reals and Incompleteness
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence. The aim of these lectures is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed.
The Global Power of Additional Queries to pRandom Oracles
, 2001
"... We consider separations of reducibilities by random sets. First, we show a result on polynomial timebounded reducibilities that query their oracle nonadaptively: for every prandom set R, there is a set that is reducible to R with k + 1 queries but is not reducible to any other prandom set with at ..."
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We consider separations of reducibilities by random sets. First, we show a result on polynomial timebounded reducibilities that query their oracle nonadaptively: for every prandom set R, there is a set that is reducible to R with k + 1 queries but is not reducible to any other prandom set with at most k queries. This result solves an open problem stated in a recent survey paper by Lutz and Mayordomo [EATCS Bulletin, 68 (1999), pp. 6480]. Second, we show that the separation result above can be transferred from the setting of polynomial timebounds to a setting of recrandom sets and recursive reducibilities. This extends the main result of Book, Lutz, and Martin [Inform. and Comput., 120 (1995), pp. 4954] who, by using di#erent methods, showed a similar separation with respect to MartinLofrandom sets. Moreover, in both settings we obtain similar separation results for truthtable versus bounded truthtable reducibility.