The Kolmogorov complexity characterization conveys the intuition that a random sequence is incompressible: Each measure 0 set can be thought of as an uncommon property. It is not possible for a sequence to lie in no measure 0 sets, because each one-point set has measure 0.
Since the union of a countable collection of measure 0 sets has measure 0, this definition immediately leads to the theorem that there is a measure 1 set of random sequences. Note that if we identify the Cantor space of binary sequences with the interval [0,1] of real numbers, the measure on Cantor space agrees with Lebesgue measure. The martingale characterization conveys the intuition that no effective procedure should be able to make money betting against a random sequence. A martingale d is a betting strategy. It bets some fraction of its money that the next bit will be 0, and then remainder of its money that the next bit will be 1.
Since the bet placed after seeing the string w can be calculated from the values d w , d w 0 , and d w 1 , calculating the amount of money it has is equivalent to calculating the bet. The martingale characterization says that no betting strategy implementable by any computer even in the weak sense of constructive strategies, which are not necessarily computable can make money betting on a random sequence.
For a fixed oracle A , a sequence B which is not only random but in fact, satisfies the equivalent definitions for computability relative to A e. Two sequences, while themselves random, may contain very similar information, and therefore neither will be random relative to the other. An important result relating to relative randomness is van Lambalgen 's theorem, which states that if C is the sequence composed from A and B by interleaving the first bit of A , the first bit of B , the second bit of A , the second bit of B , and so on, then C is algorithmically random if and only if A is algorithmically random, and B is algorithmically random relative to A.
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A closely related consequence is that if A and B are both random themselves, then A is random relative to B if and only if B is random relative to A. A sequence which is n -random for every n is called arithmetically random. The n -random sequences sometimes arise when considering more complicated properties.
Some of these are weak 1-randomness, Schnorr randomness, computable randomness, partial computable randomness.
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Yongge Wang showed [2] that Schnorr randomness is different from computable randomness. At the opposite end of the randomness spectrum there is the notion of a K-trivial set. These sets are antirandom in that all initial segment have the least K-complexity up to a constant b.
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