Correction to: “On the Rate of Decrease in Logical Depth,” Theor. Comput. Sci., 702(2017), 60–64, by
L.F. Antunes, A. Souto, and P.M.B. Vit´ anyi
P.M.B. Vit´anyi1
CWI and University of Amsterdam
In section 4 of the paper mentioned in the title it is assumed that, for all x∈ {0,1}∗, the string x∗ is the only incompressible string such thatU(x∗) = x. However, this assumption is wrong in that for many x there may be an incompressible stringpwith|x| ≥ |p|>|x∗|such thatU(p) =x. Moreover, the computation ofU(p) =xmay be faster than that ofU(x∗) =x. For example, the function from x ∈ {0,1}∗ to the least number of steps in a computation U(p) =xfor an incompressible stringpmay be computable. The argument in the paper is correct if we use
Definition1. Letxbe a string andba nonnegative integer. The logical depth, version 2, ofxat significance levelb, is
depth(2)b (x) = min
d:p∈ {0,1}∗∧Ud(p) =x∧ |p| ≤K(x) +b , the least number of steps to computexby a program which isb-incompressible with respect tox∗.
As a further correction: In the first line of the proof of Theorem 2 the string xn is more properly denoted byx; the remaining changes are self-evident.
Email address: Paul.Vitanyi@cwi.nl(P.M.B. Vit´anyi)
1CWI, Science Park 123, 1098XG Amsterdam, The Netherlands. (CWI is the National Research Institute for Mathematics and Computer Science in the Netherlands.)
Preprint submitted to Theoretical Computer Science August 20, 2018
Acknowledgement
The problem addressed here was raised by Marlou Gijzen.
2
