# CaAIS: Cellular Automata-Based Artificial Immune System for Dynamic Environments
## 3 Cellular Automata  
Cellular automata [47] is a dynamical system with discrete space and time. The CA is a mathematical model with an array of cells with local interactions for investigating sophisticated, complex phenomena. Each cell's behavior is determined based on its neighbor's behavior. CA is a decentralized, discrete, self-organized, and parallel system that enables one to create an ordered structure by starting from a random state. It is shown that the property of CA by applying a CA to a set of structures could not affect the set entropy. In this model, space is specified by a regular grid of cells, each representing a memory of states. In each step, the cell considers neighboring cells, and based on the communication rules, the next state is specified. In addition, each cell can work independently of the other cells.  
Cellular automata consider different neighborhood configurations. Each set of cells is considered to be neighbors in a specific order. The two most well-known neighborhoods are the Von Neumann and Moore neighborhoods. The Von Neumann neighborhood includes four adjacent cells not diagonal to the central cell, while the Moore neighborhood includes all eight surrounding cells. Each cell in the Von Neumann neighborhood has an equal distance from the central cell. This model takes into account a wider range of neighboring cells, allowing for more complex interactions and patterns within the cellular automaton. These neighborhoods are commonly called the nearest neighbors and are illustrated in Figure 1[48]. For the CaAIS, the Von Neumann model may be more suitable for scenarios where a more localized and restricted antibody spread is desired. In contrast, the Moore model allows for a more extensive spread of antibodies across cells.  
The CaAIS is not the first evolutionary algorithm to use CA. In [49], CGA is a cellular evolutionary algorithm that utilizes a decentralized population approach. In this method, tentative solutions are introduced in overlapping neighborhoods. The hybrid CA with particle swarm optimization (PSO), called CPSO, is presented in a study to optimize functions [50]. The CPSO algorithm incorporates a CA mechanism into the velocity update process to modify particle trajectories. The CaAIS is similar to CGA and CPSO in that it is parameter dependent and hybridizes with CA. However, the CaAIS differs from CGA and CPSO models in several aspects. 1: The main evolutionary algorithm is based on AIS. 2: Unlike CGA, the CaAIS model does not use crossover and mutation operators. Based on the AIS algorithm, the CaAIS uses only hypermutation operators. 3: Unlike CGA, in the CaAIS, each antibody interacts only with its pre-defined neighboring antibodies. 4: the CaAIS focuses on optimization for dynamic environments.  
Several hybridizations of CA and evolutionary algorithms are also reported in the literature, including CLA-EC [51], Cellular PSO [52; 53], Cellular DE [19; 54], CLA-DE [55], and Cellular fish swarm [56]. This paper proposes a hybrid model using cellular automata and an artificial immune system for optimization in dynamic optimization.