
Algorithmically random sequence

Intuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turing machine. The notion can be applied analogously to sequences on any finite alphabet (e.g. decimal digits). Random sequences are key objects of study in algorithmic information theory.
As different types of algorithms are sometimes considered, ranging from algorithms with specific bounds on their running time to algorithms which may ask questions of an oracle machine, there are different notions of randomness. The most common of these is known as Martin-Löf randomness (K-randomness or 1-randomness), but stronger and weaker forms of randomness also exist. The term "algorithmically random" used to refer to a particular single (finite or infinite) sequence without clarification is usually taken to mean "incompressible" or, in the case the sequence is infinite and prefix algorithmically random (i.e., K-incompressible), "Martin-Löf–Chaitin random". 
It is important to disambiguate algorithmic randomness with stochastic randomness. Unlike algorithmic randomness, which is defined for computable (and thus deterministic) processes, stochastic randomness is usually said to be a property of a sequence that is a priori known to be generated (or is the outcome of) by an independent identically distributed equiprobable stochastic process.
Because infinite sequences of binary digits can be identified with real numbers in the unit interval, random binary sequences are often called (algorithmically) random real numbers. Additionally, infinite binary sequences correspond to characteristic functions of sets of natural numbers; therefore those sequences might be seen as sets of natural numbers.
The class of all Martin-Löf random (binary) sequences is denoted by RAND or MLR.
History.
The first suitable definition of a random sequence was given by Per Martin-Löf in 1966. Earlier researchers such as Richard von Mises had attempted to formalize the notion of a test for randomness in order to define a random sequence as one that passed all tests for randomness; however, the precise notion of a randomness test was left vague. Martin-Löf's key insight was to use the theory of computation to formally define the notion of a test for randomness. This contrasts with the idea of randomness in probability; in that theory, no particular element of a sample space can be said to be random.
Since its inception, Martin-Löf randomness has been shown to admit many equivalent characterizations—in terms of compression, randomness tests, and gambling—that bear little outward resemblance to the original definition, but each of which satisfy our intuitive notion of properties that random sequences ought to have: random sequences should be incompressible, they should pass statistical tests for randomness, and it should be difficult to make money betting on them. The existence of these multiple definitions of Martin-Löf randomness, and the stability of these definitions under different models of computation, give evidence that Martin-Löf randomness is a fundamental property of mathematics and not an accident of Martin-Löf's particular model. The thesis that the definition of Martin-Löf randomness "correctly" captures the intuitive notion of randomness has been called the Martin-Löf–Chaitin Thesis; it is somewhat similar to the Church–Turing thesis.
Three equivalent definitions.
Martin-Löf's original definition of a random sequence was in terms of constructive null covers; he defined a sequence to be random if it is not contained in any such cover. Gregory Chaitin, Leonid Levin and Claus-Peter Schnorr proved a characterization in terms of algorithmic complexity: a sequence is random if there is a uniform bound on the compressibility of its initial segments. Schnorr gave a third equivalent definition in terms of martingales. Li and Vitanyi's book "An Introduction to Kolmogorov Complexity and Its Applications" is the standard introduction to these ideas.
Interpretations of the definitions.
The Kolmogorov complexity characterization conveys the intuition that a random sequence is incompressible: no prefix can be produced by a program much shorter than the prefix.
The null cover characterization conveys the intuition that a random real number should not have any property that is "uncommon". 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. Martin-Löf's idea was to limit the definition to measure 0 sets that are effectively describable; the definition of an effective null cover determines a countable collection of effectively describable measure 0 sets and defines a sequence to be random if it does not lie in any of these particular measure 0 sets. 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. "d" reads a finite string "w" and bets money on the next bit. 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. "d" doubles the money it placed on the bit that actually occurred, and it loses the rest. "d"("w") is the amount of money it has after seeing the string "w". 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.
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 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, formula_24, and the "n"th jump oracle, formula_25, as these oracles are able to answer specific questions which naturally arise. A sequence which is random relative to the oracle formula_26 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 formula_19 sets, so one might think that these should be non-random. However, the halting probability Ω is formula_19 and 1-random; it is only after 2-randomness is reached that it is impossible for a random set to be formula_19.
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., formula_30 for each initial segment w), but they are not computable.