Entropy and Certainty in Lossless Data Compression

Data compression is the art of using encoding techniques to represent data symbols using less storage space compared to the original data representation. The encoding process builds a relationship between the entropy of the data and the certainty of the system. The theoretical limits of this relationship are defined by the theory of entropy in information that was proposed by Claude Shannon. Lossless data compression is uniquely tied to entropy theory as the data and the system have a static definition. The static nature of the two requires a mechanism to reduce the entropy without the ability to alter either of these key components. This dissertation develops the Map of Certainty and Entropy (MaCE) in order to illustrate the entropy and certainty contained within an information system and uses this concept to generate the proposed methods for prefix-free, lossless compression of static data. The first method, Select Level Method (SLM), increases the efficiency of creating Shannon-Fano-Elias code in terms of CPU cycles. SLM is developed using a sideways view of the compression environment provided by MaCE. This view is also used for the second contribution, Sort Linear Method Nivellate (SLMN) which uses the concepts of SLM with the addition of midpoints and a fitting function to increase the compression efficiency of SLM to entropy values L(x) \u3c H(x) + 1. Finally, the third contribution, Jacobs, Ali, Kolibal Encoding (JAKE), extends SLM and SLMN to bases larger than binary to increase the compression even further while maintaining the same relative computation efficiency

Synchronization of finite automata

A survey of the state-of-the-art of the theory of synchronizing automata is given in its part concerned with the case of complete deterministic automata. Algorithmic and complexity-theoretic aspects are considered, the existing results related to Černý’s conjecture and methods for their derivation are presented. Bibliography: 193 titles. © 2022 Russian Academy of Sciences, Steklov Mathematical Institute of RAS.Russian Foundation for Basic Research, РФФИ, (19-11-50120)Ministry of Education and Science of the Russian Federation, Minobrnauka, (FEUZ-2020-0016)This research was supported by the Russian Foundation for Basic Research under grant no. 19-11-50120 and by the Ministry of Science and Higher Education of the Russian Federation (project no. FEUZ-2020-0016)