# Information distance

Information distance is the distance between two finite objects (represented as computer files) expressed as the number of bits in the shortest program which transforms one object into the other one or vice versa on a universal computer. This is an extension of Kolmogorov complexity. The Kolmogorov complexity of a single finite object is the information in that object; the information distance between a pair of finite objects is the minimum information required to go from one object to the other or vice versa. Information distance was first defined and investigated in  based on thermodynamic principles, see also. Subsequently it achieved final form in. It is applied in the normalized compression distance and the normalized Google distance.

## Properties

$ID(x,y)=\min\{|p|:p(x)=y\;\&\;p(y)=x\},$ $E(x,y)=\max\{K(x\mid y),K(y\mid x)\},$ ### Universality

$\sum _{x:x\neq y}2^{-D(x,y)}\leq 1,\;\sum _{y:y\neq x}2^{-D(x,y)}\leq 1,$ This excludes irrelevant distances such as $D(x,y)={\frac {1}{2}}$ for $x\neq y$ ; it takes care that if the distance growth then the number of objects within that distance of a geven object grows. If $D\in \Delta$ then $E(x,y)\leq D(x,y)$ up to a constant additive term.

## Applications

### Theoretical

The result of An.A. Muchnik on minimum overlap above is an important theoretical application showing that certain codes exist: to go to finite target object from any object there is a program which almost only depends on the target object! This result is fairly precise and the error term cannot be significantly improved. Information distance was material in the textbook, it occurs in the Encyclopedia on Distances.

### Practical

To determine the similarity of objects such as genomes, languages, music, internet attacks and worms, software programs, and so on, information distance is normalized and the Kolmogorov complexity terms approximated by real-world compressors (the Kolmogorov complexity is a lower bound to the length in bits of a compressed version of the object). The result is the normalized compression distance (NCD) between the objects. This pertains to objects given as computer files like the genome of a mouse or text of a book. If the objects are just given by name such as Einstein' or table' or the name of a book or the name `mouse', compression does not make sense. We need outside information about what the name means. Using a data base (such as the internet) and a means to search the database (such as a search engine like Google) provides this information. Every search engine on a data base that provides aggregate page counts can be used in the normalized Google distance (NGD).