Kolmogorov complexity of enumerating finite sets

Solovay [5] has proved that the minimal length of a program enumerating a set A is upper bounded by 3 times the negative logarithm of the probability that a random program will enumerate A. It is unknown whether one can replace the constant 3 by a smaller constant. In this paper, we show that the constant 3 can be replaced by the constant 2 for finite sets A. We recall first two complexity measures (“information content”) of computably enumerable sets attributed by Solovay in [5] to G. Chaitin (wee keep Solovay’s notations). Let M be a machine with one infinite input tape and one infinite output tape. At the start the input tape contains an infinite binary string ω called the input to M . The output tape is empty at the start. We say that a program p enumerates a set A ⊂ N = {1, 2, . . .} if in the run on every input ω extending p machine M prints all the elements of A in some order and no other elements, and does not move the head on input tape beyond p. We do not require M to halt in the case when A is finite. Let IM (A) denote the minimal length of a program enumerating A. There is a machine M0 (called a universal machine) such that for every other machine M there is a constant c such that