AIXI is a mathematical formalism for a hypothetical (super)intelligent, developed by Marcus Hutter (2005, 2007). AIXI is not computable, and so does not serve as a design for a real-world AI, but is considered a valuable theoretical illustration with both positive and negative aspects (things AIXI would be able to do and things it arguably couldn't do).
See also: Solomonoff induction, Decision theory, AI
The AIXI formalism says roughly to consider all possible computable models of the environment, Bayes-update them on past experiences, and use the resulting updated predictions to model the expected sensory reward of all possible strategies. This is an application of Solomonoff Induction....