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Algorithmic complexity is the shortest and most efficient program capable of producing a binary string. It plays an important role in computational complexity theory and algorithmic information theory. Objects and properties can also have algorithmic complexity, and complexity classes determine the relative difficulty of computing solutions to mathematical and logical problems. The principle of minimum message length is closely related to algorithmic complexity and provides the foundation for statistical and inductive inference and machine learning.
Algorithmic complexity, (computational complexity, or Kolmogorov complexity), is a fundamental idea in both computational complexity theory and algorithmic information theory, and plays an important role in formal induction.
The algorithmic complexity of a binary string is defined as the shortest and most efficient program capable of producing the string. While there are an infinite number of programs that can produce a given string, one program or group of programs will always be the shortest. There is no algorithmic way to find the shortest algorithm that outputs a given string; this is one of the first results of computational complexity theory. Even so, we can make an educated guess. This result, (the computational complexity of a string), turns out to be very important for computability proofs.
Since any physical object or property can in principle be described to its exhaustion by a string of bits, objects and properties can also be said to have algorithmic complexity. Indeed, reducing the complexity of real-world objects to programs that produce the objects as output is one way to look at the enterprise of science. The complex objects that surround us tend to come from three main generation processes; emergence, evolution and intelligence, with the objects produced by each tending towards greater algorithmic complexity.
Computational complexity is a notion frequently used in theoretical computer science to determine the relative difficulty of computing solutions to large classes of mathematical and logical problems. There are more than 400 complexity classes, and additional classes are being discovered all the time. The famous question P = NP concerns the nature of two of these complexity classes. Complexity classes include problems that are much harder than anything you can tackle in math right down to calculus. There are many imaginable problems in computational complexity theory that would take an almost infinite amount of time to solve.
Algorithmic complexity and related concepts were developed in the 1960s by dozens of researchers. Andrey Kolmogorov, Ray Solomonoff and Gregory Chaitin made important contributions in the late 1960s with algorithmic information theory. The principle of minimum message length, closely related to algorithmic complexity, provides much of the foundation for statistical and inductive inference and machine learning.
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