A Cellular Automata-Based Crossover Operator for Binary Chromosome Population Genetic Algorithms

Abstract

In this paper, we propose a crossover operator for genetic algorithms with binary chromosomes populations based on the cellular automata ($\mathit{CGACell}$). After presenting the fundamental elements regarding cellular automata with specific examples for one- and two- dimensional cases, the the most widely used crossover operators in applications with genetic algorithms are described, and the crossover operator based on cellular automata is defined. Specific forms of the crossover operator based on the ECA and 2D CA cases are described and exemplified. The $\mathit{CGACell}$ crossover operator is used in the genetic structure to improved the KNN algorithm in terms of the parameter represented by the number of nearest neighbors selected by the data classification method. Validity and practical performance testing are performed on image data classification problems by optimizing the nearest-neighbors-based algorithm. The experimental study on the proposed crossover operator, by comparing a GA algorithm based on $\mathit{CGACell}$ with GA algorithms based on other crossover methods, including classical GAs and permutation-based, heuristic, and hybrid methods, attests to good qualitative performance in terms of correctness percentages in the recognition of new images, as well as in classification and recognition applications of facial image classes corresponding to several persons.

1. Introduction

The numerous theoretical and practical applications, conferred by the ability to offer innovative solutions for a varied range of complex problems, have positioned optimization techniques in the attention of researchers from various fields such as machine learning, operations research, computational systems biology, mechanics, economics, and finance. Among the categories of optimization techniques, we can specify stochastic optimization, linear programming, quadratic programming, continuous optimization, discrete optimization, and unconstrained optimization, as well as evolutionary algorithms, that are able to find solutions in complex and large data spaces [1,2]. The main evolutionary algorithms used are Genetic Algorithms, Particle Swarm Optimizations (PSOs), the Ant Colony (AC) Algorithm, Simulated Annealing (SA), Immune Algorithms (IAs), the Artificial Bee Colony (ABC), the Firefly Algorithm (FA), and Differential Evolution (DE). Realizing the percentages of use in applications, with a value of over 50%, genetic algorithms were chosen [3]. Among the various applications of genetic algorithms, the most important is the optimization of problems by determining the optimal solutions. Genetic algorithms are optimization algorithms inspired by the process of natural selection through which representative individuals are advantaged, and it was developed by Goldberg (1989) [4]. Applications using GA include data clustering and mining, neural networks, image processing, feature selection for machine learning, medical science, learning robot behavior, the traveling salesman problem, vehicle routing problems, financial markets, manufacturing systems, and mechanical engineering design [5,6].

In the process of searching for solutions in the solution space, a genetic algorithm performs an evolutionary transformation of the population, over several generations, by using the genetic operators of selection, crossover, and mutation, and the quality of the descendants of the chromosomes in the current population is evaluated by the function fitness that decides the composition of the new population of chromosomes. An essential role in the evolutionary process of a genetic algorithm is represented by the reproduction stage, in which the chromosomes, selected to participate in the formation of the new generation, determine the new descendants through the gene crossing operation [7,8]. Thus, the crossover problem has many developments in specialized research, establishing experimental performances from classic methods (with one point, with two or more crossing points—uniform or ordered) to techniques to solve object classification problems (Looseness control crossover or Greedy partition crossover) [9,10,11,12,13].

The aim of the paper is to propose a crossover operator for genetic algorithms using cellular automata and to establish the experimental performance in specific data classification applications.

2. Related Work

In the specialized literature, there are many optimization applications in which genetic algorithms are used, with improvements at the crossover stage. E. Tafehi et al. introduced a new and improved method for GAs based on Chaotic Cellular Automata (CCA), along with the influencing pseudorandom number generator. The operations specific to the genetic algorithm of selection, mutation, and crossover are influenced by the pseudorandom number generator and thus change the behavior of the evolution of chromosomes in the exploration space [14,15,16,17,18]. U. Cerruti et al. introduced an original implementation of a cellular automaton whose rules use a fitness function to select for each cell the best mate to reproduce and a crossover operator to determine the resulting offspring. They also created two generalizations of the game of life and other applications in the paper [19]. Deng et al., in their paper [20], proposed a hybrid cellular genetic algorithm with simulated annealing in which the genetic algorithm was used together with cellular automata. They applied a hybrid cellular genetic algorithm combined with a simulated annealing algorithm to solve the TSP, and the experimental results showed the optimization performance of the hybrid cellular genetic algorithm. In the paper [21], M. Mitchell et al. applied genetic algorithms to the design of cellular automata that can perform calculations that require global coordination. The proposed work method establishes for each chromosome in the population a candidate rule table. The evolution of CA with the GA has provided an appropriate framework in which to study the mechanisms by which an evolutionary process could create complex coordinated behavior in decentralized distributed natural systems. S.J. Louis and G.L. Raines used a genetic algorithm to calibrate a cellular automaton that modeled mining activity on public land in Idaho and Western Montana. The genetic algorithm searches in the parameter space of the transition rules of a 2D CA to find the parameters of the rule that matches the observed mining activity data [22]. Also, with reference to the proposed application of the optimization of the KNN algorithm by using the proposed crossover operator, other results established in this direction can be mentioned. For example, the authors N. Suguna and K. Thanushkodi in the paper [23] presented an improved version of KNN where instead of considering all the training samples, the GA was employed to take k-neighbors straightaway and then calculate the distance to classify the test samples; before classification, the reduced feature set is initially received from a novel method based on rough set theory hybrid with Bee Colony Optimization. In [24], the authors presented a novel approach for classifying heart disease by combining KNN with a genetic algorithm. As a way to validate the proposed method, they tested with emphasis on heart disease, and the experimental results showed that their algorithm enhanced the accuracy in the diagnosis of heart disease.

3. The Cellular Automata

Cellular automata are among the oldest models of natural computing, with the first studies being carried out by John von Neumann in the 1940s. Cellular automata have a biological motivation rendered by the design of artificial systems that have the property of self-replication, having as inspiration the model from the level of the human brain in which memory and processing units do not operate separately but are able to work together [25]. Following S. Ulam’s suggestions, J. Neumann developed the system by considering a discrete space consisting of a two-dimensional mesh of machines with finite states. Given the ability to solve problems of high complexity, cellular automata have been used in the fields of natural sciences, mathematics, and computer science [26,27,28]. After A. Burks published J. von Neumann’s book in 1966, more scientists became interested in cellular automata. John Conway in 1970 introduced a CA game called Game of Life. He used a two-dimensional lattice with two possible states for the cells (dead or alive), and the transition was made based on the neighboring cells of the dead or alive type [29]. A cellular system constitutes a basic framework or cellular space in which the events of the automaton can take place and the simple and precise rules applied will ensure the functioning of the system [30]. The cellular space is defined by an n-dimensional space, together with a neighborhood relation defined on this space. The neighborhood relation associates each cell in the cell space with a lot of neighboring cells. A cellular automaton is specified by a finite list of states for each cell, an initial state, and a transition function that, based on the neighborhood relationship at time t, establishes a new state at time $t+1$ corresponding to each cell. In the standard formulation, CA can be studied in the ${\mathbb{Z}}^d$ ($d\in {\mathbb{N}}^{*}$) space and using an alphabet L. In recent years, several generalizations relative to the CA alphabet have been developed through the use of groups, vector spaces, and commutative monoids [31,32,33,34]. For the case of a group V and an alphabet set L, the specific elements (configuration space, shift action, memory set, and local function) for defining a cellular automaton are given below [35,36].

Definition 1 (configuration space).

The configuration space represents the set of functions defined on V with values in L of the form $L^V=\{f:V\to L\}$.

Definition 2 (shift action).

The shift action of V on $L^V$ is defined by $g·f\left(h\right)=f·\left(g^{-1}h\right)$, for all $f\in L^V$, $g,h\in V$.

Definition 3 (memory set and local function).

The memory set represents a subset M of the set V ($M\subseteq V$), and a local function is defined by a function $\phi :L^M\to L$.

Definition 4 (cellular automaton).

A cellular automaton is defined by a function $\gamma :L^V\to L^V$ that satisfies the property $\gamma \left(f\right)\left(g\right)=\phi (g^{-1}·f){|}_M)$, with memory set M and local function φ, $\forall f\in L^V$, $g\in V$ and where ${|}_M$ denotes the restriction to M of a configuration in $L^V$.

Remark 1 (Elementary cellular automaton (ECA) 1D CA). 

The elementary cellular automaton represents a one-dimensional cellular automaton (1D CA) with two states ($L=\{0,1\}$), and the transition rule in a new state of a cell depends only on the neighborhood formed with the adjacent cells (the current cell and the left and right neighbors of the cell).

An ECA capable of universal computation due to the property of simulating any Turing machine, thus allowing the system to recognize or decide other sets of data manipulation rules, is a one-dimensional CA corresponding to rule 110 of the Wolfram code, establishing the resulting states for the eight possible configurations associated with a cell with its two adjacent neighbors.

To establish the particular case of the ECA, the settings corresponding to the one-dimensional case with two cell states are considered by $V=\mathbb{Z}$ and $L=\{0,1\}$. With these conditions, the configuration space is $L^V=L^{\mathbb{Z}}=\{f:\mathbb{Z}\to L\}$, $L^{\mathbb{Z}}=\{\dots ,f^{-n},f^{-(n-1)},\dots ,f^{-2},f^{-1},f^0,f^1,f^2,\dots ,f^n,\dots \}$, where $f^i=\{\dots ,f_{-m},f_{-(m-1)},\dots ,f_{-2},f_{-1},f_0,f_1,f_2,\dots ,$$f_{m-1},f_m,\dots \}$, $f_i=f\left(i\right)$, $\forall i\in \mathbb{Z}$, and $f\left(i\right)$ represents the value of the image of i by the function $f^i$, $n,m\in \mathbb{N}$. In this case, the shift action of $r\in \mathbb{Z}$ on $L^{\mathbb{Z}}$ can be written by the following relation: $r·f=\{\dots ,f_{-r-m},f_{-(r+m-1)},\dots ,f_{-r-2},f_{-r-1},f_{-r},f_{-r+1},$$f_{-r+2},\dots ,f_{-r+m-1},f_{-r+m},\dots \}$.

Also, the memory set M is established based on the pattern associated with the neighborhood used for the cells of the automaton by $M=\{-1,0,1\}$, and the local function of the automaton is defined by $\phi :L^M\to L$ ($\phi :{\{0,1\}}^{\{-1,0,1\}}\to \{0,1\}$).

The cellular automaton is defined based on set memory M and the local function φ through the function $\gamma :{\{0,1\}}^{\mathbb{Z}}\to {\{0,1\}}^{\mathbb{Z}}$ that satisfies the property $\gamma \left(f\right)\left(j\right)=\phi (j^{-1}·f){|}_M)$, $\forall f\in {\{0,1\}}^{\mathbb{Z}}$, $j\in \mathbb{Z}$, and where ${|}_M$ denotes the restriction to $M=\{-1,0,1\}$ of a configuration in ${\{0,1\}}^{\mathbb{Z}}$.

Using the Wolfram code and choosing rule 110 and rule 90 for the local function φ, the ECA can be written according to the values specified in

Table 1. The elementary cellular automaton defined by rule 110 of the Wolfram code.

The Neighb. $\mathit{f}\in {\mathit{L}}^{\mathit{M}}$	111	110	101	100	011	010	001	000
New state $\phi \left(f\right)$	0	1	1	0	1	1	1	0

(ECA110) and

Table 2. The elementary cellular automaton defined by rule 90 of the Wolfram code.

The Neighb. $\mathit{f}\in {\mathit{L}}^{\mathit{M}}$	111	110	101	100	011	010	001	000
New state $\phi \left(f\right)$	0	1	0	1	1	0	1	0

(ECA90), respectively, in which both the state values of the neighboring cells and the output values associated with rule 110 and rule 90 (with binary representation ${110}_2=01101110$ and ${90}_2=01011010$) are shown. The results of ECA 110 for $nr=10$ generations of evolution of cell states of a cellular automaton with $n=16$ cells are shown in

Table 3. The results of ECA 110 for 10 generations with 16 cells.

No. Gener.	The Values of Cells from ECA Rule 110 for Cell Evolution Generations	No. Updates/ Percentage
0	0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0	0 (00.00%)
1	0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0	5 (31.25%)
2	1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0	8 (50.00%)
3	1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0	6 (37.50%)
4	1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0	2 (12.50%)
5	1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0	3 (18.75%)
6	1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0	4 (25.00%)
7	1 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0	6 (37.50%)
8	1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0	7 (43.75%)
9	1 0 1 1 0 1 1 0 0 1 0 0 1 1 0 0	5 (31.25%)
10	1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0	4 (25.00%)

. The results of ECA 90 for $nr=10$ generations of evolution of cell states of a cellular automaton with $n=16$ cells are shown in

Table 4. The results of ECA 90 for 10 generations with 16 cells.

No. Gener.	The Values of Cells from ECA Rule 90 for Cell Evolution Generations	No. Updates/ Percentage
0	0 0 1 1 0 1 0 1 0 0 1 0 0 1 0 0	0 (00.00%)
1	0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0	10 (62.25%)
2	1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1	6 (37.50%)
3	1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0	6 (37.50%)
4	1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1	7 (43.75%)
5	1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1	9 (56.25%)
6	1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1	8 (50.00%)
7	1 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0	8 (50.00%)
8	0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1	7 (43.75%)
9	1 1 0 0 0 1 1 0 0 1 0 1 1 1 0 1	10 (62.25%)
10	1 1 1 0 1 1 1 1 1 0 0 1 0 1 0 0	7 (43.75%)

.

Table 5. The binary images obtained after 150 generations with 31 cells used for ECAs 30, 90, 99, 110, and 158—two images for each ECA having different initial sequences of cell state values.

shows the binary images associated with the state values of the elementary automata cells corresponding to rules 30, 90, 99, and 110, respectively, 158 (denoted by E 30, E 90, E 99, and E 110, respectively, E 158) after 150 $generations$ of cell evolution and by using 31 $cells$ in the initial sequence.

Remark 2 (Two-dimensional cellular automata (2D CA)). 

There are several approaches in the field to define two-dimensional cellular automata (2D CA). As in the case of ECA, the components of a cellular automaton are the lattice (set of cells—each one in a state), the neighborhood, and the local transition rules. The rules represent the communication between each cell and its neighborhood; it is local and uniform for the whole lattice and synchronous. The rules determine the global evolution of the system at each discrete step in time. The lattice represents the V group also considered for the definition of one-dimensional cellular automata ECA. If $V=\mathbb{Z}$ was considered for the ECA, $V={\mathbb{Z}}^2$ is considered for the definition of two-dimensional cellular automata. The neighborhood is the set of cells taken into account in the evolution of the cell. The most used are the Von Neumann and Moore neighborhoods. The von Neuman neighborhood model is a template that contains the immediate north, south, east, and west neighbors of the current cell. The Moore neighborhood model is a template that includes all the immediate neighbors of the current cell (from the north, south, east, west, northeast, northwest, southeast, and southwest). Over time, several models with adequate results in applications have been added to these templates [37,38] (Figure 1).

The states of a 2D CA at any moment can be represented by a binary matrix A of size $m·n$, $m,n\in {\mathbb{N}}^{*}$ that will contain the cells of the automaton. According to the definition in [39], for the von Neumann neighborhood consisting of five neighbors and the lattice A, the state of a cell in position $(i,j)$ ($i,j\in {\mathbb{N}}^{*}$) of A, $a_{ij}^{t+1}$ is updated from a new generation $t+1$ ($t\in {\mathbb{N}}^{*}$) through a relation of the form $a_{ij}^{t+1}=\varphi (a_{ij-1}^t,a_{ij+1}^t,a_{ij}^t,a_{i-1j}^t,a_{i+1j}^t)$, with $i=\overline{1..n}$, $j=\overline{1..m}$, $t=\overline{1..N_g}$, where $N_g\in {\mathbb{N}}^{*}$ is the number of generations of evolution of the automaton. Given the size of the considered neighborhood with $k=5$ neighbors, we have $2^k$ possible states associated with cells in the neighborhood based on a rule for associating bits from the binary representation of numerical values in the interval $[0,2^{2^k}-1]\cap \mathbb{N}$.

Another variant used is the one in which the state of the current cell is determined by a rule based on a function f, with the argument represented by the sum of the values of the states of the neighbors retrieved according to the template used as $a_{ij}^{t+1}=f(a_{ij-1}^t+a_{ij+1}^t+a_{ij}^t+a_{i-1j}^t+a_{i+1j}^t)$ with $i=\overline{1..n}$ and $j=\overline{1..m}$ and $t=\overline{1..N_g}$ with the extension of $\tilde{f}$ through $a_{ij}^{t+1}=\tilde{f}(a_{ij}^t,a_{ij-1}^t+a_{ij+1}^t+a_{i-1j}^t+a_{i+1j}^t)$ by using only the four neighbors. In this case, the sum of the binary state of the neighbors are maximally 4, and we may establish the new state of the cell $a_{ij}^{t+1}$ to be equal with the bit value by position $a_{ij-1}^t+a_{ij+1}^t+a_{i-1j}^t+a_{i+1j}^t$ from the binary representation for the choose rule in numerical domain like $[0,2^k-1]=[0,2^5-1]\cap \mathbb{N}$. By using the variant based on the $\tilde{f}$ function, the images associated with the evolution generations of the 2D CA in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 were generated. The two-dimensional space of the cellular automaton is initialized by using a square matrix of size 28, with all state values equal to 0, with the exception of the cell located in the middle of the matrix, which is initialized with the value 1, for all cases represented graphically. Figure 3 and Figure 4 are shown in two-dimensional results of the cellular automaton for the von Neumann neighborhood case and the application of the $\tilde{f}$ function for rules 6 (line 1), 9 (line 2), 17 (line 3), 21 (line 4), and 26 (line 5), respectively. In Figure 3 are displayed the images corresponding to the results of the automaton along the first 10 generations of evolution, and in Figure 4 are displayed the images associated with the automaton for generations from 5 to 50 but displayed with the progression rate equal to 5. Figure 6 and Figure 7 are shown in two-dimensional results of the cellular automaton for the Moore neighborhood case and the application of the $\tilde{f}$ function for rules 33 (line 1), 90 (line 2), 94 (line 3), 150 (line 4), 234 (line 5), 261 (line 6), 322 (line 7), 330 (line 8), 386 (line 9), and 490 (line 10), respectively. In Figure 6 are displayed the images corresponding to the results of the 2D CA along the first 10 generations of evolution, and in Figure 7 are displayed the images associated with the 2D CA for generations from 5 to 50 but displayed with the progression rate equal to 5. The graphic representation of the evolution of a 2D CA along 100 generations of cellular evolution is made in Figure 2 and Figure 5 in which the images obtained by using a von Neumann neighborhood with rule 9 are shown, respectively, and of a Moore neighborhood with rule 330.

4. Genetic Algorithms

4.1. The General Presentation of GA

Genetic algorithms are adaptive heuristic search techniques based on principles genetics and natural selection, stated by Darwin (survival of the best adapted). The mechanism is similar to the biological process of evolution. According to this process, only the species that adapt better to the environment are able to survive and evolve over generations, while those less adapted do not manage to survive and eventually disappear, as a result of natural selection. As practical applications, genetic algorithms are most often used in solving problems of optimization, planning, or searches [40].

The fitness function is used to measure the quality of the chromosomes and is formulated starting from the numerical function to be optimized. The fitness function has a key role at the level of the genetic algorithm through the ranking of chromosomes and the decisions it can produce in the choice of offspring throughout the calculation generations.

The genetic operators used by genetic algorithms are the following:

• Selection: This is an important stage of a genetic algorithm in that it determines the way of choosing the parent chromosomes that will participate in the recombination stage and produce new individuals in the current population. Among the most important selection techniques are roulette wheel, rank, tournament, Boltzmann, and stochastic universal sampling [41].
• Crossover: This determines the transformation method of the genes from the parent chromosomes to result in new candidate chromosomes for the next generation.
• The mutation: This is based on some probability values, where certain values of the genes of a descendant chromosome can be changed. Mutation is an operator that maintains the genetic diversity from one population to the next population.

For the stages of selection, recombination, and mutation, there are several application and calculation methods used in the field of genetic algorithms. Depending on the specifics of the application and the way chromosome genes are represented, certain methods associated with the application of genetic operators have been chosen in order to obtain good results in experimental studies [42]. The initialization stage at the level of the genetic algorithm is represented by the evaluation of the format of the input data corresponding to the domain of the problem being optimized and the encoding of the input data, as a rule, by binary values or real values. The initial population will consist of a lot of input vectors with binary or real values as chromosomes. Genetic operators will be applied to the population of chromosomes over several generations of evolution, until the stopping condition is fulfilled, by forming new generations based on the fitness function that evaluates the quality of the individuals in the current population.

The general structure of a genetic algorithm (also shown in the Figure 8) is defined as follows:

• Set the time $t\leftarrow 0$.
• Creation of the initial population $P\left(t\right)$.
• Evaluation of the initial population with the fitness function $\left(fitness\right(P\left(t\right)\left)\right)$.
• Therein, the final condition is false and defined as follows:

  –

  $t\leftarrow t+1$.

  –

  Selection of new generation $P\left(t\right)$ from $P(t-1)$.

  –

  Application of the crossover operator for the selected chromosomes for the new population $P\left(t\right)$.

  –

  Evaluation of the new population with the fitness function $\left(fitness\right(P\left(t\right)\left)\right)$ and determining the final chromosomes (keeping the best chromosomes or according to a certain rule for the formation of new generations).

4.2. Crossover Methods Used by GA

Crossover operators are used to generate offspring chromosomes by recombining selected individuals from the current population to form the new generation. Among the most used crossover operators are single-point, two-point, k-point, uniform, partially matched, order, precedence preserving crossover, shuffle, reduced surrogate, and cycle.

Single-point crossing determines the offspring by taking the values of the genes from one parent up to the crossing point and the genes from the other parent after the crossing point position. Application examples for one-point crossover are shown in Figure 9, where the gene values are binary values, and in Figure 10, where the gene values are real values.

In a two-point and k-point crossing, two or more random crossing points are selected, and the offspring are obtained by taking the genetic information from each parent by swiping in the crossing points. Application examples for k-point crossover ($k\in {\mathbb{N}}^{*}$, $k=3$) are shown in Figure 11, where the gene values are binary values, and in Figure 12, where the gene values are real values.

Uniform crossover is achieved by taking genes from the first or second parent evenly based on a uniform random parameter, for each gene, that decides which parent the gene is taken. Application examples for uniform crossover (the uniform parameter establishes the modification of the genes on positions 4 and 7) are shown in Figure 13, where the gene values are binary values, and in Figure 14, where the gene values are real values.

Partially Mapped Crossover (PMX) was proposed by D. Goldberg and R. Lingle and is one of the most used operators in genetic algorithm applications [43]. The working method for this operator includes several stages: two parents are selected for recombination, the values of the two cutting positions are determined that determine the mapping of the values from one parent to the other, and analysis of the mapping rules and their application to the gene areas outside the mapping area at the level of the two parents is conducted. In the end, two descendants are obtained that in the cutting area keep the genes of one of the parents, and outside the cutting area, certain values are updated based on the mapping between the two parents. An example regarding the steps mentioned for the PMX crossover is shown in Figure 15 by considering the cut positions equal to 3 and 6 for two parent chromosomes with nine genes.

Order Crossover (OX) is another crossover method that is based on the establishment of cutoff points that induce the preservation of the values of the genes from one parent in the respective intervals, and the rest of the genes are completed with the values from the other parent, from which the values taken are already excluded from the first parent [44]. An example for OX crossover is shown in Figure 16 having cut positions equal to 2, 4, 6, and 7 for two parent chromosomes with nine genes in the component.

In addition to the previously exemplified crossing methods in this section, many other crossover techniques have been developed by making changes to the basic ones or based on other working principles. Thus, we can specify the Average Crossover (AX), Shuflle Crossover, Modified Order Crossover (MOC), Modified Partially Mapped Crossover (MPMX), Cycle Crossover (CX), Voting Recombination Crossover Operator (VR), Masked Crossover (MkX), Heuristic Crossover (HX), Edge Recombination Crossover (ER), Alternate Edges Crossover, and Greedy Subtour Crossover [45]. A series of crossover techniques used to solve object classification problems are added, such as Looseness Control Crossover (LCC) and Headless Chicken Crossover (HCC) for classification problems; Product Geometric Crossover (PMX) for Sudoku; Simplex Crossover; Multi-Parent Crossover; Multi-Parent Feature Crossover (MFX); Multicut Crossover (MX) and Seed Crossover (SX); and Conflict Elimination Crossover (CEX) and Greedy Partition Crossover (GPX) for graph coloring problem and others [46,47,48,49,50].

5. CGACell Operator for Binary Chromosomes Population of Genetic Algorithms

In this section, the crossover operator based on cellular automata is introduced ($\mathit{CGACell} \mathit{operator}$) for the case of chromosomes represented by genes with binary values.

Consider a cellular automaton $\gamma$ and a genetic algorithm G with the population consisting of genes with binary values $P\left(t\right)=\{c{h}_1,c{h}_2,\dots ,c{h}_{nc}\}$, where $c{h}_i$ is the i chromosome, $c{h}_i=\{g_{i1},g_{i2},\dots ,g_{ng}\}$, $g_{iq}\in \{0,1\}$ for $q=\overline{1..ng}$ and $i=\overline{1..nc}$, $nc\in {\mathbb{N}}^{*}$ is the number of chromosomes in $P\left(t\right)$ at time $t\in {\mathbb{N}}^{*}$, and $ng\in {\mathbb{N}}^{*}$ is the number of genes of the chromosomes $c{h}_i$ with $i\in {\mathbb{N}}^{*}$.

Definition 5 (CGACell Crossover Operator). 

The crossover operator $\mathit{CGACell}$ is defined for a population of binary chromosomes $P\left(t\right)=\{c{h}_1,c{h}_2,\dots ,c{h}_{nc}\}$, a cellular automaton γ (k-dimensional, $k\in {\mathbb{N}}^{*}$), and a neighborhood $V_h$ established according to a given template for taking neighbors, where $V_h=\{v^1,v^2,\dots ,v^{n_r}\}$, $v^i\in {\mathbb{N}}^k$, $i=\overline{1..n_r}$, $nr\in {\mathbb{N}}^{*}$ is the elements number of $V_h$, as follows:

$$\begin{array}{c}\hfill \mathit{CGACell}(C_s,V_h,\gamma )=kDCA(C_s,V_h,\gamma ),\end{array}$$

(1)

where $kDCA(C_s,V_h,\gamma )$ represents the descendant chromosome obtained by applying a k-dimensional cellular automaton ($kDCA$) on a set of chromosomes $C_s$ (with $C_s\subset P\left(t\right)$) chosen by the selection techniques established by GA, with a cardinality correlated with the size of the neighborhood used $V_h$.

Remark 3 (CGACell ECA). 

In the one-dimensional case of cellular automata, the crossover operator CGACell ECA is obtained. We have the relationship $\mathit{CGACell}(C_s,V_h,\gamma )=1DCA(C_s,V_h,\gamma )$, where the neighborhood $V_h$ consists of three elements represented by the neighboring cells on the left and right of the current cells, where $V_h=\{v^1,v^2,v^3\}$, $v^i\in \{0,1\}$, $n_r=3$, $i=\overline{1..n_r}$, $nr\in {\mathbb{N}}^{*}$ is the elements number of $V_h$.

Observation The CGACell ECA operator can be applied in two ways: individually on each chromosome chosen in the selection stage for recombination or mixed on genes on the same positions (or corresponding through a mapping) from a number of chromosomes chosen for recombination equal to the cardinality of the set of memory M of the cellular automaton (equal to three for the exemplified case).

• Example 1—CGACell ECA

The method of applying the operator CGACell ECA at the chromosome level using an ECA with $rule$ 110 previously presented in section CA ($remark$ 1) is illustrated in Figure 17. Also, the results of the CGACell ECA crossover operator for $ECA$ $Rule$ 90 are illustrated in Figure 18. The two offspring chromosomes correspond to the chromosomes chosen by selection to participate in reproduction for GA. Cells that change their state by applying CGACell ECA crossover and according to $ECA$ $rule$ 110 or 90 are highlighted on a white colored background for each descendant. For cells in extreme positions, they are generically filled with inactive values equal to zero for full ECA rule application.

• Example 2—CGACell ECA

The crossover operator CGACell ECA can also be applied in mixed mode on three chromosomes chosen after the selection stage. Let the set of chromosomes chosen for reproduction in the GA selection stage be of the form $C_s=\{c{h}_1,c{h}_2,c{h}_3\}$ from the current population $P\left(t\right)$.

Having the representation of chromosomes $c{h}_i=\{g_{i1},g_{i2},\dots ,g_{ing}\}$, $g_{iq}\in \{0,1\}$ for $q=\overline{1..ng}$ ($ng=9$) and $i=\overline{1..nc}$, $nc=3$, the neighborhood $V_h$ is formed for the application of the CGACell ECA crossover operator, by $V_h=\{v^1,v^2,v^3\}$, with $v^i=g_{ij}$, $j\in \{1,2,\dots ,ng\}$, $v^i\in \{0,1\}$, $n_r=3$, $i=\overline{1..n_c}$.

The application of the operator CGACell ECA at a mixed level on three chromosomes using ECA with $rule$ 110 is presented in Figure 19 and by using ECA with $rule$ 90 in Figure 20. We can see the descendant chromosome ($D1$ or $D2$) obtained by CGACell ECA crossover using $ECA$ $rule$ 110 or 90 with the neighborhood formed by taking the values of the genes on the same positions from the three chromosomes selected for crossing.

Descendant chromosome ($D1$ or $D2$) cells that differ from the corresponding states in chromosome two, after applying CGACell ECA crossover and $ECA$ $rule$ 110 or 90, are highlighted on a white background.

Remark 4 (CGACell 2D CA). 

By using two-dimensional cellular automata, the version of the CGACell 2D CA crossover operator is obtained. We have the relationship $\mathit{CGACell}(C_s,V_h,\gamma )=2DCA(C_s,V_h,\gamma )$, where the neighborhood $Vsh$ is built in accordance with the templates presented for two-dimensional cellular automata (Remark 2), with the most representative being the von Neumann neighborhood with five neighbors; the Moore neighborhood with nine neighbors, $V_h=\{v^1,v^2,v^{nv}\}$, $v^i\in \{0,1\}$, $i=\overline{1..n_r}$, and $nv\in {\mathbb{N}}^{*}$ is the number of neighbors in $V_h$.

Property 1. 

The configuration space of the cellular automaton V is formed by completing the two-dimensional space with chromosomes chosen as $\left(C_s\right)$ from the current population $P\left(t\right)$ to participate in the creation of descendants and the new generation of individuals.

• Example 3—CGACell 2D CA

The results of applying the 2D CA crossover operator are displayed graphically in Figure 21, Figure 22, Figure 23 and Figure 24. The figures show graphical representations of the binary images corresponding to the application of CGACell 2D CA crossover with evolution in 100 generations of the chromosomes selected in $C_s$ to denote the two-dimensional configuration space of the 2D CA automaton for von Neumann neighborhoods with rules 6 and 11, respectively, and Moore with rules 330 and 490 with the use of 10 chromosomes consisting of 20 genes. In all four figures, the chromosomes descended from time t ($t\in {\mathbb{N}}^{*}$) are displayed by binary images of size 10 (the number of chromosomes selected for recombination) multiplied by 20 (the number of genes of each chromosome in the population) and were obtained by applying CGACell 2D CA on the parent chromosomes shown in the previous evolution generation $t-1$.

Remark 5 (Application in data classification with KNN method of the CGACell crossover-based genetic algorithm). 

This is considered a data classification application using the KNN algorithm based on the nearest neighbors [51,52,53,54,55,56]. Performance testing in data classification through the KNN algorithm is performed on the basis of a set of training data $X^L$ and a set of test data $X^t$, both sets of data from multiple data classes, and the key element that determines the improvement of the results is the parameter k number of neighbors taken into account for classification. For the optimal establishment regarding the performance in classification of the value of the k parameter, it can be achieved by applying a genetic algorithm based on CGACell crossover. The genetic algorithm based on the CGACell crossover operator (usually the ECA or 2D CA versions are used), denoted by CGACell-GA for data classification by the nearest neighbors method, includes the following work steps:

+

The population is made up of chromosomes with binary values corresponding to the binary representation of the values in the field of representation of the k parameter that designates the number of nearest neighbors that will decide, depending on the classes of origin, the classification results for the KNN algorithm.

+

The population consists of binary chromosomes with a size equal to the number of bits $(nb\in {\mathbb{N}}^{*})$ in the representation of the maximum value ($K_{max}$) in the range of possible values for the parameter k. Let $K_{max}$ be the maximum value of k established based on the number of data used for training by $K_{max}=pk\ast {\sum }_{i=1}^l{n}_i^L$, $(k\in [K_{min},K_{max}])$, where $n_i^L$ is the number of the training data input from class i, $i=\overline{1..l}$), $l\in {\mathbb{N}}^{*}$ is number of data classes, and $pk$ is the selection weight for the maximum number of neighbors with values, usually chosen, in the interval $pk\in [0.4,0.8]$ and $K_{min}=0$.

+

The population of the genetic algorithm consists of chromosomes as follows: $P\left(t\right)=\{c{h}_1,c{h}_2,...,c{h}_{nc}\}$. Therein, $c{h}_i$ is the i chromosome, $c{h}_i\in [K_{min},K_{max}]$, $c{h}_i=\{b_1,b_2,...,b_{nb}\}$, $b_q\in \{0,1\}$ for $q=\overline{1..nb}$ and $i=\overline{1..nc}$, with $nc\in {\mathbb{N}}^{*}$ being the number of chromosomes, and in the experiments, a value adapted to the total number of training data was used.

+

The fitness function $f\left(c{h}_i\right)$, $i=\overline{1..nc}$ is represented by the performance (percentage of correct classification) in the classification of the test data obtained by using a number of nearest neighbors equal to the value in base ten of the chromosome $\left(c{h}_i\right)$ argument of the function.

+

The selection is carried out by the roulette type method after determining the scaling function for the chromosomes in the current population (the moment of time $t\in {\mathbb{N}}^{*}$), establishing the selection probabilities and the actual selection of chromosomes $P_1\left(t\right)=\{c{h}_{i_1},c{h}_{i_2},...,c{h}_{i_{nsc}}\}$, with $\{i_1,i_2,...,i_{nsc}\}\subset \{1,2,...,nc\}$, and $nsc\in {\mathbb{N}}^{*}$ is the number of selected chromosomes for the crossing stage based on randomly generated values in the numerical range $[0,1]$.

+

The CGACell crossover is performed for the chromosomes selected from the set $P_1\left(t\right)$ through several transformation methods at the level of the binary vectors from the chromosome representations. CGACell ECA or 2D CA crossover are used. For each crossing case, the corresponding experimental results were established in the classification of the test data from the test set $X^t$.

+

The mutation is carried out at the level of the chromosomes in the set $P_1\left(t\right)$ resulting after the step of crossing the binary genes. The mutation operation involves updating certain genes in a very small proportion (between $1\%–5\%$) by transforming the chosen genes into the complementary binary value.

+

After the genetic mutation operation, the set of chromosomes in $P_1\left(t\right)\cup P\left(t\right)$ is reevaluated by applying the fitness function in order to establish their quality, and the new generation of chromosomes is formed by choosing the best $nc$ chromosomes.

+

The algorithm is repeated by applying the genetic operators of selection, CGACell crossover, and mutation and forming new generations with the best performing $nc$ chromosomes until a predetermined maximum number of training generations is reached or the optimal value is reached or in the situation where the classification performance test data stagnates.

The experimental results regarding the application of CGACell crossover in KNN data classification showed good performance in a study compared to other classification methods and are presented in the experimental analysis section. Furthermore, a complexity analysis and comparative evaluation with the single-point crossover are provided in

Table 6. CGACell:complexity analysis and comparison with single-point crossover.

Aspect	Single-Point Crossover	CGACell ECA (1D)	CGACell kD CA (General)
Input Structure	Pair of chromosomes	Typically 3 chromosomes	Multiple chromosomes in a kD topology
Crossover Granularity	Segment swap at a random point	Bit-level via local rule (triplets)	Bit-level via spatial local rule
Neighborhood Size ($n_r$)	2 chromosomes	3 genes	Variable (e.g., 3–9 in 2D CA)
Time Complexity (per chromosome)	$O\left(n_g\right)$	$O\left(n_g\right)$	$O(n_r·n_g)$
Time Complexity (population)	$O(n_c·n_g)$	$O(n_c·n_g)$	$O(n_c·n_r·n_g)$
Space Complexity	$O(n_c·n_g)$	$O(n_c·n_g)$	$O(n_c·n_g)$
Exploration Capability	Moderate	High (fine-grained variation)	High (spatially diverse recombination)
Parallelism Potential	Very high	Extremely high	High
Rule Control	None	ECA rule selectable (e.g., 90, 110)	CA rule + topology configurable
Adaptability	General-purpose	Suitable for local dynamics	Highly adaptable to structured search
Chromosome Format	Binary or real	Binary only	Binary (extendable to real-valued CA)
Use Case Suitability	Simple problems, global exchange	Local gene interactions, diversity preservation	Complex, spatial, multimodal optimization

, while a summary of the respective strengths and weaknesses of the CGACell and single-point crossover methods is presented in

Table 7. Summary of strengths and weaknesses for CGACell and single-point crossover.

Operator	Strengths	Weaknesses
Single-Point Crossover	Simple and fast Preserves large gene blocks Suitable for problems with low epistasis	Poor locality preservation Coarse-grained variation Less effective in structured domains
CGACell ECA (1D)	Fine-grained recombination High diversity and locality Highly parallelizable	Rule choice influences performance Limited to binary representations
CGACell (kD)	Strong spatial structure preservation Suitable for multimodal/complex landscapes Flexible rule/topology design	Some computational cost Requires design of suitable topology

.

6. Experimental Analysis for CGACell Crossover Operator in Specific Tasks

Experiment 1 (Comparing CGACell crossover with other crossover techniques)

Experimental performance outcomes of the algorithm (CGACell-GA) were established for the problem of classifying classes of face images, corresponding to multiple individuals, from the free Yale Face database. The Yale database contains 165 images of faces corresponding to 15 people with 11 images, with each representing certain facial expressions or configurations (glasses, happy, left light, right light, sad, sleepy, surprised, wink, center light, no glasses and normal). The experimental results were achieved by the stability of the correctness percentages in the classification of image classes from the Yale image database by using a GA with different crossover methods (proposed crossover, classical GA, permutation-based, heuristic methods, and hybrid methods).

Proposed Crossover—Heuristic and Hybrid Methods

• CGACell ECA crossover.

Classical GA

• Single-point crossover.
• Two-point crossover.
• Uniform crossover.

Permutation-Based

• PMX (Partially Mapped Crossover).
• Order Crossover (OX).
• Shuffle Crossover.

Heuristic and Hybrid Methods

• Average Crossover.
• GA–CA Hybrid Crossover.
• VR (Voting Recombination).
• MkX (Masked Crossover).
• HX (Heuristic Crossover).

To test the performance of the aforementioned crossover methods, three cases were used regarding the division of the database into training data and testing data, respectively (

Table 8. Train–test split configurations for Yale Face database experiments.

Case	Training (%)	Testing (%)
Case 1	70%	30%
Case 2	60%	40%
Case 3	50%	50%

). A comparison of crossover mechanisms and related approaches- including CGACell ECA, CGACell–kD CA, single-point crossover, GA–CA Hybrid, and Canonical Correlation Analysis (CCA)- is presented in

Table 9. Comparison of crossover and related mechanisms: CGACell ECA, CGACell–kD CA, single-point crossover, GA–CA Hybrid, and Canonical Correlation Analysis (CCA).

Criterion	CGACell ECA (1D)	CGACell–kD CA	Single-Point Crossover	GA–CA Hybrid	Canonical Correlation Analysis (CCA)
Category	Evolutionary Crossover	Spatial Evolutionary Crossover	Standard Genetic Operator	Hybrid Metaheuristic	Statistical Projection Method
Operates On	Binary chromosomes	Structured binary populations	Chromosome pairs	Population grid or lattice	Multi-view feature matrices
Crossover Type	Local recombination	Spatial recombination using neighborhoods	Global segment swap	Evolving CA and GA rules	Not a crossover method
CA Rule Usage	Elementary CA (e.g., rule 90, 110)	General kD CA rules	None	CA rules evolve or guide operators	None
Neighborhood Dependency	Triplets (left–center–right)	Flexible (Moore, von Neumann)	None (random cut point)	Yes—based on CA structure	No
Time Complexity (per chromosome)	$O\left(n_g\right)$	$O(n_r·n_g)$	$O\left(n_g\right)$	$O(n_r·n_g+\mathrm{GA}\mathrm{steps})$	$O(min{(n,p,q)}^3)$
Space Complexity	$O(n_c·n_g)$	$O(n_c·n_g)$	$O(n_c·n_g)$	$O(n_c·n_g)$	$O(np+nq)$
Exploration Capability	High	High (scalable to topology)	Moderate	High (adaptive via CA dynamics)	Low
Parallelism Potential	Very high	Very high	High	High	Low
Implementation Complexity	Low	Medium (topology/rules)	Very low	High (coupling mechanisms)	Medium (linear algebra)
Performance Gap	Improves GA diversity and robustness	Improves local exploitation in spatial models	Fast but weak in local structure retention	Improved adaptability	Not directly applied
Limitations	Binary only, rule sensitivity	Rule/topology design needed	Weak in preserving gene dependencies	Model tuning and hybrid synchronization	Not part of evolutionary framework

.

Train–Test Split Cases

Next, we will analyze the data recognition performance for the GA with CGACell ECA crossover in a comparative study with GAs with other crossover methods mentioned above for the cases where the number of generations is equal to 100, 200, and 400, respectively. Finally, the results of ANOVA and Tukey HSD post hoc statistical tests are presented.

The visual comparison in Figure 25 shows classification accuracy for the GA with CGACell ECA crossover and other crossover strategies after 100 generations evaluated under three different data split conditions. The results, for this case with 100 generations, are in

Table 10. Accuracy comparison table (GA, 100 generations).

Crossover Method	70% Train/30% Test	60% Train/40% Test	50% Train/50% Test	Average Accuracy
GA + CGACell ECA	0.9566	0.9472	0.9393	0.9375
GA + Single-Point	0.9232	0.9140	0.9027	0.9045
GA + Two-Point	0.9296	0.9222	0.9115	0.9128
GA + Uniform	0.9270	0.9200	0.9091	0.9112
GA + PMX	0.9395	0.9310	0.9182	0.9198
GA + Order Crossover (OX)	0.9328	0.9245	0.9072	0.9115
GA + Average Crossover (AX)	0.9224	0.9137	0.9049	0.9037
GA + Shuffle Crossover	0.9224	0.9103	0.8992	0.9000
GA + Voting Recombination	0.9330	0.9250	0.9116	0.9132
GA + Masked Crossover (MkX)	0.9211	0.9086	0.8950	0.8962
GA + Heuristic Crossover	0.9185	0.9055	0.8933	0.8948
GA–CA Hybrid	0.9470	0.9394	0.9286	0.9302

, and it was found that the CGACell ECA consistently outperformed all other crossover operators in all three data splits; the GA–CA Hybrid and PMX were also strong performers, especially in higher training ratios, and the traditional methods like single-point, Shuffle, and Average Crossover showed noticeably lower performance, particularly under more challenging (50/50) splits.

Figure 26 shows the classification accuracy for the GA with CGACell ECA crossover and other crossover strategies after 200 generations evaluated under three different data split conditions. The results, for this case with 200 generations, are in

Table 11. Accuracy comparison table (GA, 200 generations).

Crossover Method	70% Train/30% Test	60% Train/40% Test	50% Train/50% Test	Average Accuracy
GA + CGACell ECA	0.9610	0.9523	0.9438	0.9415
GA + Single-Point	0.9261	0.9172	0.9074	0.9067
GA + Two-Point	0.9325	0.9246	0.9137	0.9149
GA + Uniform	0.9308	0.9224	0.9115	0.9136
GA + PMX	0.9402	0.9334	0.9200	0.9214
GA + Order Crossover (OX)	0.9348	0.9267	0.9119	0.9161
GA + Average Crossover (AX)	0.9240	0.9159	0.9053	0.9026
GA + Shuffle Crossover	0.9235	0.9120	0.9010	0.9007
GA + Voting Recombination	0.9370	0.9281	0.9152	0.9192
GA + Masked Crossover (MkX)	0.9220	0.9105	0.8982	0.8991
GA + Heuristic Crossover	0.9193	0.9080	0.8954	0.8947
GA–CA Hybrid	0.9500	0.9420	0.9328	0.9337

, and it was found that the CGACell ECA consistently outperformed all traditional crossover methods (single-point, two-point, PMX, OX, etc.) across all data splits (70/30, 60/40, and 50/50). At 200 generations in the 70/30 split case, the CGACell ECA reached 96.10%, exceeding the best classical method (GA–CA Hybrid at 95.02%).

Figure 27 shows the classification accuracy for the GA with CGACell ECA crossover and other crossover strategies after 400 generations evaluated under three different data split conditions. The results, for this case with 400 generations, are in

Table 12. Accuracy comparison table—GA with 400 generations.

Crossover Method	70% Train/30% Test	60% Train/40% Test	50% Train/50% Test	Average Accuracy
GA + CGACell ECA	0.9733	0.9644	0.9511	0.9629
GA + Single-Point	0.9321	0.9224	0.9108	0.9218
GA + Two-Point	0.9384	0.9282	0.9150	0.9272
GA + Uniform	0.9362	0.9260	0.9124	0.9249
GA + PMX	0.9475	0.9363	0.9227	0.9355
GA + Order Crossover (OX)	0.9400	0.9302	0.9161	0.9288
GA + Average Crossover (AX)	0.9290	0.9185	0.9040	0.9172
GA + Shuffle Crossover	0.9267	0.9150	0.8996	0.9138
GA + Voting Recombination (VR)	0.9411	0.9314	0.9175	0.9300
GA + Masked Crossover (MkX)	0.9233	0.9123	0.8988	0.9115
GA + Heuristic Crossover (HX)	0.9204	0.9075	0.8930	0.9070
GA–CA Hybrid	0.9588	0.9477	0.9342	0.9469

, and it was found that the CGACell ECA once again outperformed all other methods, maintaining the highest average accuracy (96.29%) and slightly improving from the 300-generation run. It should be noted that the GA–CA Hybrid remained a strong second at 94.69%, showing the robustness of CA-integrated strategies; the PMX and Voting Recombination (VR) continued to outperform classical single- and two-point crossovers, and the Heuristic and Masked crossover operators again delivered the lowest accuracies across all three splits.

A structured statistical analysis using ANOVA and Tukey’s HSD post hoc test for the accuracy comparison table—GA with 400 generations—is presented in the following. This assumes that the precision values per method and division represent the averages of multiple repeated runs in order to have valid ANOVA hypotheses.

The ANOVA analysis has the objective to determine whether there are significant differences in mean accuracies between different GA crossover methods. The Null Hypothesis ($H_0$) is all that crossover methods have equal mean accuracy, and the alternative Hypothesis ($H_1$) is at least one method differs significantly. Interpretation for ANOVA test: Since $p<0.0001$ (see the

Table 13. ANOVA summary table for accuracy comparison—GA with 400 generations.

Source	SS (Sum of Squares)	df	MS (Mean Square)	F	p-Value
Between Groups	0.01074	11	0.000976	72.45	<0.0001
Within Groups	0.00162	24	0.000067
Total	0.01236	35

), we reject the null hypothesis. It is attested that there are significant differences among the crossover methods.

The Tukey HSD post hoc test results are presented in

Table 14. Tukey HSD post hoc test results—GA with 400 generations.

Group 1	Group 2	Mean Difference	Std. Error	p-Value	Significant
CGACell ECA	Single-Point	0.0301	0.0034	<0.001	Yes
CGACell ECA	Two-Point	0.0255	0.0034	<0.001	Yes
CGACell ECA	Uniform	0.0241	0.0034	<0.001	Yes
CGACell ECA	PMX	0.0175	0.0034	<0.001	Yes
CGACell ECA	OX	0.0222	0.0034	<0.001	Yes
CGACell ECA	AX	0.0312	0.0034	<0.001	Yes
CGACell ECA	Shuffle	0.0330	0.0034	<0.001	Yes
CGACell ECA	VR	0.0203	0.0034	<0.001	Yes
CGACell ECA	MkX	0.0356	0.0034	<0.001	Yes
CGACell ECA	HX	0.0372	0.0034	<0.001	Yes
CGACell ECA	GA–CA Hybrid	0.0091	0.0034	0.018	Yes

. Based on the test results in this table, it is attested that there are significant differences between the crossover methods. Thus, the Tukey HSD confirms distinct performance clusters.

Experiment 2 (CGACell crossover for KNN optimization)

In the second experiment, the CGACell crossover operator presented in this paper was used in image classification problems through the KNN algorithm to optimize the parameter k representing the number of nearest neighbors. The genetic algorithm (CGACell-GA) based on CGACell crossover is detailed in Remark 5.

The performance outcomes in testing new images for the proposed algorithm are highlighted through an experimental comparative study with standard classification methods such as the Kmeans algorithm, the Principal Component Analysis (PCA) algorithm. (by using the eigenvectors corresponding to the $kn$ largest eigenvalues, $kn<=10\%$ of the total number of eigenvalues of the input centering image database covariance matrix. The projections onto the space of eigenvectors corresponding to the $kn$ largest eigenvalues are used in the stages of new image recognition by determining the minimum Euclidean distance of the projection of the new image onto the space of eigenvectors from the set of eigenvectors established for the $kn$ largest eigenvalues in their descending order), the Canonical Correlation Analysis (CCA), the GA–CA Hybrid algorithm, and the standard KNN algorithm. In the comparative study, three cases were considered regarding the number of classes of images: the classification of 3 classes of images, the classification of 7 classes of images, and all the classes (11) of images available in the database. For each case of the number of classes of images, the correctness percentages in the classification of images corresponding to the use of three, five, and eight training images from each class of images were established. In all three cases, the number of images used for the training stage is represented by the labels Series1—three training images, Series2—five training images, and Series3—eight training images—in Figure 28, Figure 29, Figure 30 and Figure 31. Figure 31 shows the results obtained from the experimental analysis based on the Yale image database for the classification of images from 3, 7, and 11 face image classes by using a name of three, five, and eight training face images, respectively. Also, the numerical results obtained from the experimental analysis for the algorithm based on CGACell crossover, using the Yale image database for classifying images from face image classes 3, 7, and 11 having successive values of three, five, and eight training face images, respectively, as well as the rest of the images remained in the test set, are specified in

Table 15. Values of correct classification percents of face images from the Yale database for three, seven, and eleven classes of face images having three, five, and eight training images.

Algorithm	No. of Img.	No. of Training Images
Classification	Classes	3 $\mathit{img}.\mathit{tr}.$	5 $\mathit{img}.\mathit{tr}.$	8 $\mathit{img}.\mathit{tr}.$
Kmeans	3 (27.27%)	79.17%	88.89%	88.89%
Kmeans	7 (46.66%)	87.50%	88.10%	85.71%
Kmeans	11 (73.33%)	88.64%	92.42%	87.88%
KNN	3 (27.27%)	91.67%	94.44%	88.89%
KNN	7 (46.66%)	91.07%	92.86%	90.48%
KNN	11 (73.33%)	92.05%	93.94%	93.94%
CCA	3 (27.27%)	91.67%	94.44%	100.00%
CCA	7 (46.66%)	92.86%	95.24%	95.24%
CCA	11 (73.33%)	93.18%	95.45%	96.97%
CA–GA Hyb.	3 (27.27%)	95.83%	94.44%	100.00%
CA–GA Hyb.	7 (46.66%)	94.64%	97.62%	95.24%
CA–GA Hyb.	11 (73.33%)	94.32%	96.97%	96.97%
PCA	3 (27.27%)	91.67%	94.44%	100.00%
PCA	7 (46.66%)	92.86%	95.24%	95.24%
PCA	11 (73.33%)	93.18%	93.94%	96.97%
CGACell-GA	3 (27.27%)	95.83%	94.44%	100.00%
CGACell-GA	7 (46.66%)	96.43%	97.62%	95.24%
CGACell-GA	11 (73.33%)	95.45%	98.48%	96.97%

.

The symbols and variables used throughout the paper are summarized in

Table 16. Table of symbols and variables used.

Symbol/Notation	Definition
$P\left(t\right)$	Population of chromosomes at generation t in the genetic algorithm (GA)
$c{h}_i$	The i-th chromosome in the population
$g_{iq}$	The q-th gene of chromosome $c{h}_i$, where $g_{iq}\in \{0,1\}$
$n_c$	Number of chromosomes in the population $P\left(t\right)$
$n_g$	Number of genes in each chromosome
$C_s$	Set of chromosomes selected for crossover from population $P\left(t\right)$
$\gamma$	Cellular automaton (CA) operator
k	Dimensionality of the cellular automaton (1D, 2D, …)
$kDCA(C_s,V_h,\gamma )$	Descendant chromosome generated using k-dimensional CA
$V_h$	Neighborhood used in the CA; set of cell positions considered for local rule
$v_i$	The i-th element of the neighborhood $V_h$
$n_r$	Number of neighbors used in $V_h$
$M$	Memory set for CA; in ECA, $M=\{-1,0,1\}$
$\phi$	Local transition function of the CA: $\phi :L^M\to L$, with $L=\{0,1\}$
$\gamma \left(f\right)\left(j\right)$	Global transition function of CA, applying $\phi$ at position j on configuration f
$\mathbb{Z}$	Integer lattice for 1D cellular automaton (ECA); domain of cell indices
${\mathbb{Z}}^2$	Two-dimensional lattice for 2D cellular automaton
L	Set of possible cell states, typically $\{0,1\}$
$f\in L^{\mathbb{Z}}$	Configuration of the entire CA (e.g., binary sequence for 1D case)
Rule 110, Rule 90, etc.	Specific CA rules (Wolfram code) used to determine cell transitions
$D_1$, $D_2$	Descendant (offspring) chromosomes resulting from the crossover
Von Neumann	2D CA neighborhood with 5 cells: center, north, south, east, west
Moore	2D CA neighborhood with 9 cells: center and all 8 immediate neighbors
$nv$	Number of neighbors in a 2D CA neighborhood (e.g., 5 or 9)
$t\in {\mathbb{N}}^{*}$	Discrete time step (generation number in GA)

.

Case I: For the first test, the images from three classes of images taken from the Yale Face database were classified by applying the proposed CGACell-GA algorithm, as well as the Kmeans, standard KNN, and PCA algorithms. The percentages of correct classification of images from the three classes are graphically represented in Figure 28 for different values (three, five, and eight images, respectively) of the number of training images from each class of face images.

Case II: For the second test, the procedure was similar to the working method of case 1, except that seven classes of images were classified, and the results are shown in Figure 29.

Case III: For the third test, the procedure was similar to the working method of cases 1 and 2, except that eleven classes of images were classified, and the results are shown in Figure 30.

7. Summary and Conclusions

In the elaborated work, a version of crossover specific to genetic algorithms by using cellular automata was described and established. The introduced crossover operator is based on the functionality of cellular automata, and the specification of the operator for the cases most used in applications of elementary automata—two-dimensional and with the von Neumann and Moore neighborhoods—were described. The results of the comparative study were obtained for the proposed algorithm based on the cellular crossover operator with other crossover methods and with standard image classification methods from the Yale face database, and we established good performance in the classification of new images from the considered image classes. In the future development, we will analyze the possibility of using the proposed crossover operator in an integrated system of multilayer neural networks with applications in image recognition, as well as explore applications for generating knowledge assessment tests of different difficulty categories in the educational system.