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Computación y Sistemas

versión On-line ISSN 2007-9737versión impresa ISSN 1405-5546

Comp. y Sist. vol.28 no.2 Ciudad de México abr./jun. 2024  Epub 31-Oct-2024

https://doi.org/10.13053/cys-28-2-5026 

Articles of the thematic section

Grammatical Evolution with Codons Selection Order as Intensification Process

Andrés Espinal1 

Marco Aurelio Sotelo-Figueroa1  * 

Jorge Alberto Soria-Alcaraz1 

11 Universidad de Guanajuato, Departamento de Estudios Organizacionales, Mexico.

Abstract:

Grammar Evolution (GE) can be considered a form of Genetic Programming (GP) that has become very popular in the field of Automatic Programming (AP) over the last few years. There has been a lot of research on different aspects of GE, including its parts; the Search Engine, Mapping Process, and Grammar. However, it has been shown that it is possible to select the codons randomly to improve the GE, using a random permutation. This paper introduces a new approach to intensify a solution using permutation heuristics to guide the codon selection order. A non-parametric test was applied to discern between the results obtained by the proposal and those obtained by the canonical GE version and the GE with random permutations.

Keywords: Grammatical evolution; symbolic regression; intensification

1 Introduction

Two important approaches to automatically generate computer programs are Genetic Programming (GP) [23, 26] and Automatic Programming (AP) [15, 3, 39]. These methods have proven invaluable in improving algorithms and resolving challenging issues.

They all have different restrictions and difficulties, and to solve some of these restrictions, a different evolution-inspired method appeared recently, namely Grammatical Evolution (GE) [37]. This method combines the strength of GP with a more methodical strategy based on formal grammar.

GP is an evolutionary algorithm that works based on a population of potential solutions, often represented as tree structures to develop programs [25, 31, 24]. By utilizing evolutionary operators including crossover, mutation, and selection, the goal of GP is to evolve programs that can carry out particular tasks.

Some of the issues GP can present in execution are program bloat, inefficiency, and a lack of control over the search space [32]. There are several proposals to improve GP [5, 10]; one of them [21] includes metaheuristics to improve GP performance.

By establishing a mapping mechanism that relates a given program genotype and phenotype, Grammatical Evolution (GE) offers a solution to these problems. It uses formal context-free grammars to do this [38]. Grammatical Evolution evolves strings of symbols as opposed to program trees directly.

A mapping process between these strings of symbols and computer programs is applied by this method, producing solutions with semantic and syntactic validity. The differentiation of genotype, the evolving string, from the phenotype, the completed program, is one of the major advances of Grammatical Evolution [34].

Because grammars may be created to contain specific structural and syntactical restrictions, this separation improves control over the search space by lowering the possibility that inappropriate or inefficient programs would be constructed.

Moreover, by altering the genotype, the search space may be explored more efficiently, producing a more methodical program evolution. One of the research fields in GE is search engines.

There are several metaheuristics used as search engines like Genetic Algorithm (GA) [16], Differential Evolution (DE) [33, 17], Particle Swarm Optimization (PSO) [36, 42], Estimation Distribution Algorithms (EDA) [40, 29], and Ant Colony Optimization (ACO) [13] among others.

In [44], it was proposed to change the codon selection using a random permutation; this process was used with GE as a search engine, obtaining better results than classic GE. GE has been used successfully in many problems; however, the Symbolic Regression Problem (SRP) has been proposed and utilized as a benchmark problem for GE [1, 4, 43, 30].

The result of applying GE to SRP is an expression or function that fits a given instance of the problem. It is necessary to use formal grammar to ensure that the generated expression or function has syntactic validity.

Finally, GE explores a wide range of possible expressions at execution time, allowing it to discover highly accurate expressions. This paper proposes a methodology to evolve the codon selection order using the 2-opt, 3-opt, and 5-opt, and inversion permute heuristics.

This proposal is tested with instances of the SRP problem. The results obtained by this proposal, the canonical GE version, and the GE with Random Permutations were compared statistically.

2 Grammatical Evolution

The Grammatical Evolution (GE) [37, 38, 34] is a grammar-based form of Genetic Programming (GP) [22, 23]. The concepts of genotype and phenotype are present in both GP and GE.

Originally, the genotype in GP is based on tree representations, which are evaluated directly to obtain the phenotype, whereas GE uses a linear representation, which is employed with a grammar to obtain the phenotype. The four main components, shown in Figure 5.

  • – The Problem Instance defines the problem domain and its conditions. It is used as a measure to guide the search engine and make the optimization process.

  • – The Grammar establishes elements and rules that fit the problem instance’s specifics. In GE, Backus-Naur Form is commonly employed [2], although alternatives such as Attribute Grammar [20, 11] or Christiansen Grammar [7, 35] can also be applied.

  • – The Search Engine oversees optimization and adjusts the genotype based on the quality of the phenotype applied to the problem instance. While the Genetic Algorithm stands as the canonical search engine [16], numerous others have been implemented.

  • – The Mapping Process facilitates the conversion between genotype and phenotype, employing specific strategies such as Depth-First [34], Breadth-First [12], π Grammatical Evolution [36], etc.

Fig. 1 Classical derivation example [45] 

Fig. 2 Derivation example based on permutations [45] 

Fig. 3 Available data 

Fig. 4 Representative expression 

Fig. 5 GE methodology [41] 

Figure 1 shows an example of a classic GE, it uses a sequential codon selection order, and the search engine is applied to the codon values.

Figure 2 shows the proposal [44] example, it uses a permute codon selection order. Figure 1 and 2 uses the same codon values and the results show that is possible to obtain different phenotypes using a selection order.

3 Symbolic Regression Problem

Symbolic Regression Problem (SRP) [1, 4, 18] is the process of obtaining a representative expression from available data that we use when we want to know what was the equation behind our instance.

In SRP, the goal is to seek a model (an equation or a mathematical formula) that describes the relationship between the input variables and the target output variable, without prior knowledge of the functional form.

SRP represents an important problem studied for the GP community [18, 43, 30].

Figure 3 shows the available data from the instance without knowing the function and Figure 4 shows the proposed expression to generate the data.

4 Proposed Approach

Figure 2 shows the proposal from [44] using a codon position random permutation. The proposal is shown in Algorithm 1, the permutation heuristics is applied if the best proposal solution is not improved for n generations.

Algorithm 1 Grammatical evolution algorithm 

In this study, the n value to apply the perturbation heuristic was determined empirically. The permutation heuristics were taken from the state-of-art based on those that can be applied to the Traveler Salesman Problem (TSP) [28, 8] and widely studied permutation problem.

4.1 k-opt

The k-opt heuristic [9, 27, 6] is classified as a local search method because it only slightly modifies the current solution to try to make it better.

It does not ensure the identification of the globally optimal solution, but it remains effective in perturbing an initial solution.

A k-opt heuristic is an attempt to enhance the quality of a solution using iterative swapping of pairs of items. Figure 6 shows an example of 2-opt.

Fig. 6 2-opt example, the first array contains a permutation from 1 to 10, it was chosen two items to interchange it into the second array 

4.2 Inversion

The inversion heuristic [16, 8] involves selecting a subset of the proposal solution and investing the order of its elements. This heuristic neither ensures the identification of the globally optimal but is effective in perturbing an initial solution.

This operation introduces diversity in the population and can potentially explore different regions of the solution space. Figure 7 shows an example of Inversion.

Fig. 7 Inversion example, the first array contains a permutation from 1 to 10, it was chosen a range to invert it into the second array 

5 Experimental Setup

Table 1 shows the Symbolic Regression functions used, the functions F1 to F10 were taken from [18, 45] and the Keijzer from [19, 43, 14]. Mean Root Squared Error (MRSE), Equation 1, was used as a fitness function to discern the quality of each expression proposed by GE:

MRSE=1n∑i=1n(ti−yi)2, (1)

where:

  • – n is the number of data points.

  • – yi is the real value.

  • – ti corresponds to the value obtained.

Table 1 Symbolic regression functions used as instances set 

Function Polynomial
F1 f(x)=X3+X2+X
F2 f(x)=X4+X3+X2+X
F3 f(x)=X5+X4+X3+X2+X
F4 f(x)=X6+X5+X4+X3+X2+X
F5 f(x)=sin (x2) cos (x)−1
F6 f(x)=sin (x)+sin (x+x2)
F7 f(x)=log (x+1)+log (x2+1)
F8 f(x)=x
F9 f(x,y)=sin (x)+sin (y2)
F10 f(x,y)=2sin (x) cos (y)
Keijzer1 f(x)=0.3xsin(2πx)
Keijzer2 f(x)=1+3x+3x2+x3
Keijzer3 f(x,y)=8/(2+x2+y2)
Keijzer4 f(x,y)=x4−x3+y2/2−y
Keijzer5 f(x,y)=x3/5+y3/2−y−x
Keijzer6 f(x1,x2,…,x10)=10.59x1x2+100.5967x3x4−50.59x5x6+20x1x7x9+x3x6x10

Grammars used by each function [45] are the followings:

  • – Grammar 1 for functions F1 to F8.

〈start〉 ⊨ 〈expr〉
〈expr〉 ⊨ (〈expr〉〈op〉〈expr〉)|〈pre〉(〈expr〉)|〈var〉
〈var〉 ⊨ 1|x
〈pre〉 ⊨ exp|log|sin|cos
〈op〉 ⊨ *|/|+|-

Grammar 1. Grammar for the functions F1 to F8

  • – Grammar 2 for functions F9 to F10.

  • – Grammar 3 for functions Keijzer1 to Keijzer5.

  • – Grammar 4 for function Keijzer6.

The parameters used in the classic GE, Random Permutation GE, and current proposal are shown in Table 3. Those parameters were taken from [44], and the new parameter used to apply the permutation heuristic was chosen empirically. To conduct the comparison, 33 individual runs were executed for each function, using the proposed approach, classical GE, and Random Permutation GE.

Table 2 Median of the results obtained for each instance and GE variant 

Function GA Rnd Inv 2opt 3opt 5opt
F1 0.000 0.000 0.000 0.000 0.000 0.000
F2 0.033 0.012 0.034 0.033 0.031 0.035
F3 0.082 0.082 0.105 0.100 0.065 0.081
F4 0.143 0.112 0.112 0.123 0.130 0.154
F5 0.049 0.044 0.046 0.056 0.046 0.046
F6 0.036 0.043 0.044 0.040 0.023 0.040
F7 0.254 0.256 0.254 0.221 0.223 0.253
F8 0.095 0.103 0.094 0.100 0.094 0.098
F9 0.036 0.046 0.047 0.066 0.036 0.043
F10 0.044 0.044 0.046 0.044 0.044 0.045
Keijzer1 0.061 0.055 0.061 0.059 0.061 0.059
Keijzer2 0.000 0.000 0.000 0.000 0.000 0.000
Keijzer3 0.408 0.408 0.408 0.408 0.404 0.408
Keijzer4 0.677 0.677 0.677 0.677 0.677 0.677
Keijzer5 0.427 0.426 0.426 0.426 0.425 0.428
Keijzer6 2.448 2.233 3.826 3.137 2.943 4.283

Table 3 GA parameters 

Parameter Value
Population Size 300
Dimensions 100
Iterations to apply the heuristic 5
Function Evaluations 250,000
Selection Binary Tournament
Crossover 2-points
Mutation Bit-flip
Mapping Process Depth-First

〈start〉 ⊨ 〈expr〉
〈expr〉 ⊨ (〈expr〉〈op〉〈expr〉)|〈pre〉(〈expr〉)|〈var〉
〈var〉 ⊨ 1|x|y
〈pre〉 ⊨ exp|log|sin|cos
〈op〉 ⊨ *|/|+|-

Grammar 2. Grammar for the functions F9 to F10

〈start〉 ⊨ 〈expr〉
〈expr〉 ⊨ (〈expr〉〈op〉〈expr〉)|〈var〉
〈var〉 ⊨ 1|x|y
〈op〉 ⊨ *|/|+|-

Grammar 3. Grammar for the functions Keijzer from 1 to 5

〈start〉 ⊨ 〈expr〉
〈expr〉 ⊨ (〈expr〉〈op〉〈expr〉)|〈pre〉(〈expr〉)|〈var〉
〈pre〉 ⊨ exp|log|sin|cos
〈var〉 ⊨ 1|x〈c〉
〈op〉 ⊨ +|-|*|/
〈c〉 ⊨ 0|1|2|3|4|5|6|7|8|9

Grammar 4. Grammar for the function Keijze 6

The median of the results was used for statistical comparison among the proposed approach, GE, and Random Permutation GE.

The statistical test was performed using the non-parametric Friedman test, which aimed to establish if any implementation was capable of outperforming the others.

6 Results

Table 2 shows the median of the results of the 33 experiments for each instance. To discern between the results, a non-parametric Friedman test was performed. The value obtained was 11.99107143 with a p-value of 0.034910328.

With this p-value less than 0.1, it was possible to make a post-hoc procedure to determine which GE variant obtained the best results. The results of the post-hoc test are shown in Table 4.

Table 4 Ranking based on medians 

Algorithm Ranking
3opt 2.28125
Rnd 3.03125
GA 3.53125
2opt 4.03125
5opt 4.03125
Inv 4.09375

7 Conclusions

This paper introduced a methodology to intensify the Grammatical Evolution solutions. The intensification was based on the codon position random permutation used previously, where it was shown that a random permutation improves the Grammatical Evolution results.

The permutation was guided by permutation heuristics from the state-of-the-art, using a k-opt and inversion heuristics that have been applied to permutation problems. Symbolic Regression Problems were used because they are widely used in Grammatical Evolution and Genetic Programming to analyze improvement and performance.

The results obtained using the proposal with a 3-opt heuristic are better than the random permutation and the classic Grammatical Evolution. Not all variations of the opt heuristic gave good results, nor did the inversion heuristic. The improvement of n generations was used as a parameter to apply the permutation heuristics; however, it is possible to identify a way to apply it without being an extra parameter.

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Received: January 30, 2024; Accepted: May 06, 2024

* Corresponding author: Marco Aurelio Sotelo-Figueroa, e-mail: masotelo@ugto.mx

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License

 
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