# Application of Genetic Algorithms for Periodicity Recognition and Finite Sequences Sorting
## 1 Introduction  
Unpredictable strings, often associated with chaos theory, represent data sequences that exhibit complex and disorderly behavior, making them an intriguing subject of study in various scientific disciplines. In chaos theory, unpredictable strings are characterized by their sensitivity to initial conditions, where even small changes in the initial values can lead to entirely different outcomes, emphasizing the nonlinear nature of chaotic systems [1]. Research on unpredictable strings and randomly determined unpredictable functions has significant potential advantages in various domains today. Specifically, this research field plays a significant role in cryptography and information security for generating reliable encryption keys, random initialization vectors, and secure random numbers. It also plays a crucial role in ensuring the confidentiality and integrity of communication channels, which can enhance communication protocols protecting confidential data from interception and tampering. For example, in [2], a method for synchronizing two fast random bit generators (RBG), based on coupled chaotic lasers, was developed. This research addresses the issue of ensuring security in synchronizing RBGs in fast physical systems, where predictability is eliminated. Furthermore, the study of unpredictable strings has introduced a new concept of unpredictable sequences of a finite number of symbols [3]. The first and second laws of large sequences for random processes in discrete time were proven, with the latter being closely related to [PERSON]'s theorem. If we consider an infinite sequence \(a_{i}\), then an unpredictable string of length \(k\) is defined as a finite array \(a_{s}\), \(a_{s+i}\ldots a_{s+k}\), where \(s\) and \(k\) are positive integers, subject to the following conditions: \(a_{i}=a_{s+i}\) for \(k=0,1,2,\ldots,k-1\), and \(a_{i}\
eq a_{s+i}\). In other words, we consider a section of a sequence of length \(k\), starting from position \(s\), where the values are repeated in increments of \(k\), but the next element after this section differs from the element standing at the position of the first element of the repeating sequence. This definition allows you to identify a part of the sequence with predictable repetition and the unpredictable part following it. These unpredictable strings play an important role in defining unpredictable sequences with a limited number of characters. The sequence \(a_{i}\) itself is considered unpredictable if it allows the existence of unpredictable strings of arbitrary length. This definition describes an array with a repeating sequence of length \(k\), starting at position s, and it is considered unpredictable if the element following the repeated sequence \(a_{k}\) is different from the element at the same position as the first element of the sequence \(a_{s+k}\). This means that the array has a predictable part followed by an unpredictable part. Unpredictable strings are used to define unpredictable sequences of a finite number of symbols. A sequence \(a_{i}\) is unpredictable if it allows unpredictable strings of arbitrarily large length. Stochastic processes with discrete time and finite state spaces allow for a countable set of realizations that are unpredictable sequences [3]. Researching this topic will allow for a better understanding of fundamental concepts related to chaos in statistics, random processes, probability theory, and dynamic processes [4]. In the paper [5], the application and evaluation of a cryptosystem based on chaotic mappings are investigated. An analysis was conducted to assess the impact of the chosen chaotic mapping on the properties of messages and the security of the cryptosystem using methods from statistical mechanics and information theory. The paper [6] explores a synchronization scheme for complex networks of chaotic systems. Chaotic systems, including Rossler, Chen, Lorenz, and Lu, are considered as complex chaotic systems within complex networks.