Information distance is the distance between two finite objects 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
Formally the information distance between and is defined by with a finite binary program for the fixed universal computer with as inputs finite binary strings. In it is proven that with where is the Kolmogorov complexity defined by of the prefix type. This is the important quantity.
Universality
Let be the class of upper semicomputable distances that satisfy the densitycondition This excludes irrelevant distances such as for ; it takes care that if the distance growth then the number of objects within that distance of a geven object grows. If then up to a constant additive term.
Metricity
The distance is a metric up to an additive term in the metric equalities.
Maximum overlap
If, then there is a program of length that converts to, and a program of length such that the program converts to. That is, the shortest programs to convert between two objects can be made maximally overlapping: For it can be divided into a program that converts object to object, and another program which concatenated with the first converts to while the concatenation of these two programs is a shortest program to convert between these objects.
Minimum overlap
The programs to convert between objects and can also be made minimal overlapping. There exists a program of length up to an additive term of that maps to and has small complexity when is known. Interchanging the two objects we have the other program Having in mind the parallelism between Shannon information theory and Kolmogorov complexity theory, one can say that this result is parallel to the Slepian-Wolf and Körner–Imre Csiszár–Marton theorems.
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 result is the normalized compression distance 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 and a means to search the database provides this information. Every search engine on a data base that provides aggregate page counts can be used in the normalized Google distance.
