September 21, 2015Benoit Monin (UPEC LACL)
The Kolmogorov complexity of a string is, informally, the length of the smallest program that produces this string. We write C(s) = n to mean that the smallest program producing the string s is of length n.
An infinite binary sequence X is said to be C-trivial if the Kolmogorov complexity of its prefixes is minimal, that is, if there is a constant d such that for any n, we have C(X
est n) < C(n) + d (Here X est n denotes the n first bits of X). It is clear that any computable (infinite binary) sequence is C-trivial. Chaitin proved that the converse holds: A sequence X is computable iff it is C-trivial (1). Chaitin also successfully used Kolmogorov complexity to provide a formal definition of the intuitive idea we can have of a random sequence. To do so, he needed a variation of the standard Kolmogorov complexity, called “prefix Kolmogorov complexity” and denoted by K. Using this prefix Kolmorogov complexity K, he defined a sequence Z to be random if for any n we have K(Z est n) > n – d for some constant d. The intuition is that the prefix Kolmogorov complexity should be maximal for prefixes of random sequences. This definition of randomness is still today the most studied, for many reasons that we shall not detail during the talk.
Chaitin conjectured (1) to also be true with prefix Kolmorogov complexity K, that is, X is computable iff X is K-trivial (that is, there is d such that for every n we have K(X
est n) < K(n) + d). Solovay later refuted the conjecture by constructing a non-computable K-trivial sequence A. The notion of K-triviality was born, and had yet to reveal many surprises through its numerous different characterizations, and its connections with algorthmic randomness. After introducing the main concepts with more details, we will try during this talk to give some explanations and intuitions on the work that has been done by various people on K-triviality these last 15 years.