Combinatorial information distance

Abstract

Let $|A|$ denote the cardinality of a finite set $A$. For any real number $x$ define $t(x)=x$ if $x\ge 1$ and 1 otherwise. For any finite sets $A,B$ let $\delta (A,B)$ $=$ ${\log }_2(t(|B\cap \bar{A}||A|))$. We define {This appears as Technical Report # arXiv:0905.2386v4. A shorter version appears in the {Proc. of Mini-Conference on Applied Theoretical Computer Science (MATCOS-10)}, Slovenia, Oct. 13-14, 2010.} a new cobinatorial distance $d(A,B)$ $=$ $\max \{\delta (A,B),\delta (B,A)\}$ which may be applied to measure the distance between binary strings of different lengths. The distance is based on a classical combinatorial notion of information introduced by Kolmogorov.