There is a universal constructive martingale d. This martingale is universal in the sense that, given any constructive martingale d, if d succeeds on a sequence, then d succeeds on that sequence as well. Thus, d succeeds on every sequence in RANDc (but, since d''' is constructive, it succeeds on no sequence in RAND). (Schnorr 1971) The class RAND is a subset of Cantor space, where refers to the second level of the arithmetical hierarchy. This is because a sequence S is in RAND if and only if there is some open set in the universal effective null cover that does not contain S; this property can be seen to be definable by a formula. There is a random sequence which is , that is, computable relative to an oracle for the Halting problem. (Schnorr 1971) Chaitin's Ω is an example of such a sequence. No random sequence is decidable, computably enumerable, or co-computably-enumerable. Since these correspond to the , , and levels of the arithmetical hierarchy, this means that is the lowest level in the arithmetical hierarchy where random sequences can be found. Every sequence is Turing reducible to some random sequence. (Kučera 1985/1989, Gács 1986). Thus there are random sequences of arbitrarily high Turing degree. Relative randomness As each of the equivalent definitions of a Martin-Löf random sequence is based on what is computable by some Turing machine, one can naturally ask what is computable by a Turing oracle machine. 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.g., no martingale which is constructive relative to the oracle A succeeds on B) is said to be random relative to A. Two sequences, while themselves random, may contain very similar information, and therefore neither will be random relative to the other. Any time there is a Turing reduction from one sequence to another, the second sequence cannot be random relative to the first, just as computable sequences are themselves nonrandom; in particular, this means that Chaitin's Ω is not random relative to the halting problem. 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