Confusion about what's a measure? What's a distribution? Simple but important! For bitstrings x:
M measure: M(empty string)=1 M(x) = M(x0)+M(x1) nonnegative for all finite x. P probability distribution: Sum_x P(x) = 1; P(x) nonnegative --- M semimeasure - replace "=" by ">=": M(x) >= M(x0)+M(x1) P semidistribution - replace: Sum_x P(x) <= 1 --- Examples: 1. Distribution: E.g., integers n: P(n) = 6/(Pi n^2) 2. Semidistribution: m(x) = probability of guessing a halting program for x (BTW, Hal, this was first published by Levin in 1974, not by Chaitin in 1975) 3. Measure: E.g., each x of size n gets weight 2^-n 4. Semimeasure: E.g., mu^M(x) = probability of guessing a halting or nonhalting monotone TM program whose output starts with x Check out: Measures and Probability Distributions Section 4 of "Algorithmic TOEs" http://www.idsia.ch/~juergen/toesv2/node15.html Juergen Schmidhuber http://www.idsia.ch/~juergen/ http://www.idsia.ch/~juergen/everything/html.html http://www.idsia.ch/~juergen/toesv2/
