so on, then C is algorithmically random if and only if A is algorithmically random, and B is algorithmically random relative to A. 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. Stronger than Martin-Löf randomness Relative randomness gives us the first notion which is stronger than Martin-Löf randomness, which is randomness relative to some fixed oracle A. For any oracle, this is at least as strong, and for most oracles, it is strictly stronger, since there will be Martin-Löf random sequences which are not random relative to the oracle A. Important oracles often considered are the halting problem, , and the nth jump oracle, , as these oracles are able to answer specific questions which naturally arise. A sequence which is random relative to the oracle is called n-random; a sequence is 1-random, therefore, if and only if it is Martin-Löf random. A sequence which is n-random for every n is called arithmetically random. The n''-random sequences sometimes arise when considering more complicated properties. For example, there are only countably many sets, so one might think that these should be non-random. However, the halting probability Ω is and 1-random; it is only after 2-randomness is reached that it is impossible for a random set to be . Weaker than Martin-Löf randomness Additionally, there are several notions of randomness which are weaker than Martin-Löf randomness. Some of these are weak 1-randomness, Schnorr randomness, computable randomness, partial computable randomness. Yongge Wang showed that Schnorr randomness is different from computable randomness. Additionally, Kolmogorov–Loveland randomness is known to be no stronger than Martin-Löf randomness, but it is not known whether it is actually weaker. At the opposite end of the randomness spectrum there is the notion of a K-trivial set. These sets are anti-random in that all initial segment is logarithmically compressible (i.e., for each initial segment w), but they are not computable. See also Random sequence Gregory Chaitin Stochastics Monte Carlo method K-trivial set Universality probability Statistical randomness References Further reading Randomness Algorithmic information theory