**The complexity of DNA sequences. Different approaches and definitions**
**Vladimir D. Gusev, Liubov A. Miroshnichenko**
*Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences*

**Abstract**. An important quantitative characteristic of symbolic sequence (texts, strings) is complexity, which reflects at the intuitive level the degree of their "non-randomness". A.N. Kolmogorov formulated the most general definition of complexity. He proposed measuring the complexity of an object (symbolic sequence) by the length of the shortest descriptions by which this object can be uniquely reconstructed. Since there is no program guaranteed to search for the shortest description, in practice, various algorithmic approximations considered in this paper are used for this purpose. Along with definitions of complexity, suggesting the possibility of reconstruction a sequence from its "description", a number of measures are considered that do not imply such restoration. They are based on the calculation of some quantitative characteristics. Of interest is not only a quantitative assessment of complexity, but also the identification and classification of structural regularities that determine its specific value. In one form or another, they are expressed in the demonstration of repetition in the broadest sense. The considered measures of complexity are conventionally divided into statistical ones that take into account the frequency of occurrence of symbols or short “words” in the text, “dictionary” ones that estimate the number of different “subwords” and “structural” ones based on the identification of long repeating fragments of text and the determination of relationships between them.

Most of the methods are designed for sequences of an arbitrary linguistic nature. The special attention paid to DNA sequences, reflected in the title of the article, is due to the importance of the object, manifestations of repetition of different types, and numerous examples of using the concept of complexity in solving problems of classification and evolution of various biological objects. Local structural features found in the sliding window mode in DNA sequences are of considerable interest, since zones of low complexity in the genomes of various organisms are often associated with the regulation of basic genetic processes.

**Key words**: *DNA sequences, algorithms, complexity, entropy, data compression, statistical measures, linguistic measure of complexity, structural measures of complexity.*