On the relation between descriptional complexity and algorithmic probability (Q1057064)

It has been conjectured that the descriptional complexity (or algorithmic entropy) of a string differs from the negative logarithm of its a priori probability (the probability that a string is the output of an optimal universal computer with random input) by at most an additive constant. This conjecture is disproved, and in fact a previously known upper bound on the difference is shown to be the best possible. The proof uses a two- person memory-allocation game between players called User and Server. User sends incremental requests for memory space, which Server allocates in a write-once memory; for each item, some of the allocated space must be in one piece, in order to give a short address. Related results are presented.