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. When the term "algorithmically random" is used to refer to a particular single (finite or infinite) sequence without clarification, it 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 between algorithmic randomness and 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 by (or is the outcome of) 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