# 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.^{[1]} 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 ^{[2]} based on thermodynamic principles, see also.^{[3]} Subsequently it achieved final form in.^{[4]} 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 ^{[4]} it is proven that
with

where is the Kolmogorov complexity defined by ^{[1]} of the prefix type.^{[5]} This is the important quantity.

### Universality

Let be the class of upper semicomputable distances that satisfy the density condition

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.^{[4]}

### Metricity

The distance is a metric up to an additive
term in the metric (in)equalities.^{[4]}

### Maximum overlap

If , then there is a program of length that converts to , and a program of length such that the program converts to . (The programs are of the self-delimiting format which means that one can decide where one program ends and the other begins in concatenation of the programs.) 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.^{[4]}

### Minimum overlap

The programs to convert between objacts 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^{[6]} 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.^{[7]} Information distance was material in the textbook,^{[8]} it occurs in the Encyclopedia on Distances.^{[9]}

### 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).

## References

- ↑
^{1.0}^{1.1}A.N. Kolmogorov, Three approaches to the quantitative definition of information, Problems Inform. Transmission, 1:1(1965), 1–7 - ↑ M. Li, P.M.B. Vitanyi, Theory of Thermodynamics of Computation, Proc. IEEE Physics of Computation Workshop, Dallas, Texas, USA, 1992, 42–46
- ↑ M. Li, P.M.B. Vitanyi, Reversibility and Adiabatic Computation: Trading Time and Space for Energy, Proc. R. Soc. Lond. A 9 April 1996 vol. 452 no. 1947 769–789
- ↑
^{4.0}^{4.1}^{4.2}^{4.3}^{4.4}C.H. Bennett, P. Gacs, M. Li, P.M.B. Vitanyi, W. Zurek, Information distance, IEEE Transactions on Information Theory, 44:4(1998), 1407–1423 - ↑ L.A. Levin, Laws of Information Conservation (Nongrowth) and Aspects of the Foundation of Probability Theory, Problems Inform. Transmission, 10:3(1974), 30–35
- ↑ A.A. Muchnik, Conditional complexity and codes, Theoretical Computer Science, 271: 1–2(2002), 97–109
- ↑ N.K Vereshchagin, M.V. Vyugin, Independent minimum length programs to translate between given strings, Proc. 15th Ann. Conf. Computational Complexity, 2000, 138–144
- ↑ M.Hutter, Universal Artificial Intelligence: Sequential Decisions Based on Algorithmic Probability, Springer, 1998
- ↑ M.M. Deza, E Deza, Encyclopedia of Distances, Springer, 2009

## Related literature

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