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On the logical depth function
- Publication Year :
- 2013
-
Abstract
- For a finite binary string $x$ its logical depth $d$ for significance $b$ is the shortest running time of a program for $x$ of length $K(x)+b$. There is another definition of logical depth. We give a new proof that the two versions are close. There is an infinite sequence of strings of consecutive lengths such that for every string there is a $b$ such that incrementing $b$ by 1 makes the associated depths go from incomputable to computable. The maximal gap between depths resulting from incrementing appropriate $b$'s by 1 is incomputable. The size of this gap is upper bounded by the Busy Beaver function. Both the upper and the lower bound hold for the depth with significance 0. As a consequence, the minimal computation time of the associated shortest programs rises faster than any computable function but not so fast as the Busy Beaver function.<br />Comment: 11 pages LaTeX; previous version was incorrect, this is a new version with almost the same results
- Subjects :
- Computer Science - Computational Complexity
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1301.4451
- Document Type :
- Working Paper