Notes on facticity and effective complexity

The discovery that some Turing Machines are universal computers that can emulate any other Turing Machine, and thus, according to the Church-Turing thesis, any effective computational procedure available to the human mind, led to the development of a powerful theory of the complexity of individual objects in the work of Solomonoff, Kolmogorov and Chaitin. Intuitively this theory identifies the "complexity" of an object described by a target string of characters from some alphabet (typically binary 1s and 0s), x, with the length of the shortest input program, x, on which some reference universal Turing Machine U will halt with x as output. This concept identifies the complexity of the target with its incompressibility. As we will see below, while compressibility is a central issue in understanding the problem of complexity, incompressibility itself is not a good measure for our intuitive notion of complexity. A completely random, structureless string (representing, for example, the state of a perfect gas in thermodynamic equilibrium) is incompressible, but intuitively is a simple, not a complex, object. Thus we will refer to the length of the shortest input string that generates the target string, as the target string's Kolmogorov complexity, or algorithmic information content. In formal terms, let the (countably infinite) set of all finite binary strings, including the null string be 80, 1<, and denote the length of a string x as x . If a Turing Machine T with input z ' halts with string z we write T@z 'D = z. Given a reference universal Turing Machine V , the (plain) Kolmogorov complexity of a string x is C@xD = x where V@xD = x, and < x £ x ' " x ' V@x 'D = x, or, more compactly, C@xD = argmin x' 8x ' V@x 'D = x< .