## Abstract

Some authors consider that evolutionary search may be positively influenced by the use of redundant representations, whereas others note that the addition of random redundancy to a representation could be useless in optimization. Given this lack of consensus, two new families of redundant binary representations are developed in this paper. The first family is based on linear transformations and is considered non-neutral. The second family of representations is designed to implement neutrality, and is based on the mathematical formulation of error control codes. A study aimed at assessing the influence of redundancy and neutrality on the performance of a simple evolutionary hillclimber is presented. The (1+1)-ES is modeled using Markov chains and is applied to NK fitness landscapes. The results indicate that the phenotypic neighborhood induced by a redundant representation dominates the behavior of the algorithm, affecting the search more strongly than neutrality, and the representations with better performance on NK fitness landscapes do not exhibit extreme values of any of the indicators of representation quality commonly adopted in the literature.

## 1 Introduction

The existence of redundancy and neutrality in the genetic code which appears in several ways, in particular at the RNA level, and also the neutral theory of molecular evolution proposed by the Japanese scientist Kimura (1968), are considered to be responsible for the evolutionary capacity of living beings. Individuals can be seen as a duality between their genetic code and their characteristics, that is, their behavioral, physiological, and morphological traits. The genotype corresponds to the information stored in chromosomes, which can be used to describe individuals at the gene level, while the phenotype describes the appearance of the individual. Genotype-phenotype mapping uses genotypic information to build the phenotype.

In nature, genotype-phenotype mapping is complex. The influence of a single gene is not restricted to a single cell type, but can simultaneously affect various phenotypic traits. This effect is called pleiotropy. The inverse effect, in which a single phenotypic trait is determined by the interaction of several genes, is called polygeny (Fogel, 1997).

Charles Darwin's theory of natural selection (Darwin, 1859) assumes that natural selection is the driving force of evolution and that randomly advantageous mutations are fixed due to natural selection and can be propagated from generation to generation. On the other hand, the neutral theory of molecular evolution assumes that the driving force of molecular evolution is the random fixation of neutral mutation rather than the fixation of advantageous mutations by natural selection. Kimura (1968) observed that in nature the number of different genotypes which store the genetic material of an individual greatly exceeds the number of different phenotypes which determine the outward appearance, thus, the representation which describes how the genotypes are assigned to the phenotypes must be redundant, and neutral mutations become possible. A mutation is neutral if its application to a genotype does not result in a change of the corresponding phenotype. Because large parts of the genotype have no actual effect on the phenotype, evolution can use them as a store for genetic information that was necessary for survival in the past, and as a playground for developing new properties of the individual that could be advantageous in the future (Rothlauf, 2006).

This background has motivated the proposals of redundant representations in evolutionary computation (EC). Representations are considered redundant if the number of genotypes exceeds the number of phenotypes (Rothlauf, 2006). Thus, the utility of these representations, with or without neutrality, has been analyzed. The notions of neutrality and neutral networks have attracted the attention of several researchers, due to the potential that neutral networks have in establishing alternative ways for the evolution of the population, including the improvement of quality search.

There is no single definition of neutrality, and it is implicitly defined as the presence of neutral networks in the search space. A neutral network can be defined as a path through genotype space via mutations which leave fitness unchanged (Harvey and Thompson, 1996), or as points in the search space that are connected through neutral point-mutations where the fitness is the same for all the points in such a network (Galván-López et al., 2011), or as a set of genotypes connected by single point mutations that map to the same phenotype (Shackleton et al., 2000). The latter perspective is the one used in this paper.

In EC, some researchers have tried to determine the effect of redundancy, neutral or non-neutral, in the behavior of the evolutionary search, but the results were contradictory. While some studies consider that an increase of the evolutionary potential of a population, in other words, the capacity that random mutations can contribute to the improvement of the evolutionary search, can be achieved using redundant representations (Shipman et al., 2000; Shackleton et al., 2000; Ebner, Shackleton, et al., 2001), others note that the addition of random redundancy does not appear to improve that potential (Knowles and Watson, 2002).

Some of the redundant representations used in evolutionary algorithms (EAs) exhibit a random and high degree of redundancy. In this paper, the two types of redundant representations proposed in Fonseca and Correia (2005) and Correia and Fonseca (2007a, 2007b) and detailed in Correia (2009, 2011, 2012) are examined and new developments are presented.

The development of the family of non-neutral representations based on linear transformations is realized using an incremental approach, beginning with a simple noncoding redundant representation and incorporating a different property in each phase. The introduction of redundancy in a controlled and structured way using the mathematical formulation of error control codes, which is applied to detect and to correct errors in communications and digital storage, made possible the development of the family of neutral representations. This family, which exhibits well-defined properties, allows a more formal approach to the questions associated with the role of redundancy and neutrality in evolutionary search than what has been previously achieved.

In addition, this paper aims to enumerate these families and the evaluation of the performance of an (1+1)-ES (evolution strategy) applied to NK fitness landscapes. The (1+1)-ES is modeled using Markov chains and the performance is analyzed in terms of the expected value of the fitness of the best solution found. The evaluation of the performance of the EA showed that some representations systematically have behavior superior to others, suggesting that there are representations that are more appropriate than others for the optimization of fitness landscapes.

The outline of this paper is as follows. In Section 2, the redundancy and neutrality in the genetic code and in the evolutionary computation field are presented and some properties of redundant representations proposed in the literature are explained. The development of the families of redundant binary representations are explained in Section 3, in which redundant representations based on linear transformations and redundant and neutral representations based on error control codes are explained. In Section 4, the properties of the neutral representations are presented as well as the relationships between them. The influence of redundancy and neutrality on the performance of a (1+1)-ES applied to NK fitness landscapes is presented in Section 5. Finally, Section 6 provides conclusions and suggests some guidelines for future research.

## 2 Redundancy and Neutrality

### 2.1 Redundancy and Neutrality in Biology

In DNA (deoxyribonucleic acid), each nucleotide consists of deoxyribose, phosphate, and an organic base: Adenine (A), Guanine (G), Cytosine (C), or Thymine (T). To produce proteins, the cell uses blocks of DNA sequences consisting of three consecutive nucleotides, called triplets, in order to synthesize an amino acid. There are 20 different amino acids; for these to be produced in the cell, another acid, RNA (ribonucleic acid), is needed to produce proteins. Because there are more combinations of nucleotides than amino acids formed, the genetic code is considered redundant and this phenomenon is called degeneracy of the genetic code. Moreover, the genetic code is redundant, but it is not ambiguous, that is, each codon encodes a single amino acid. Due to the degeneracy of the genetic code, some mutations in protein-coding genes are silent, or produce no change in the amino acid sequence of the protein for which they code, since the change from a base in a codon does not always lead to a change of the encoded amino acid. These mutations are called neutral mutations.

There are several levels of organization of the RNA: the primary structure or sequence, the secondary structure or shape, and the tertiary structure. These structures correspond to different ways of describing the structure of RNA at different levels of resolution. The RNA folding predicts the secondary structure based on the primary structure and a large number of secondary structures is possible for a primary structure. However, one or just a few of these structures can meet the minimum energy.

Some studies reveal interesting properties of the RNA secondary structures based on the nucleotide sequences (Schuster, 1997). At this level, it was found that a single secondary structure is generally represented by many different sequences, indicating a highly neutral redundant genotype-phenotype mapping because mutations in different positions in the sequences do not affect the secondary structure. Schuster concluded that the sequence space is “percolated” by long neutral networks that map to identical secondary structures and that there are many different sequences that fold to a small number of structures (common structures) and a few sequences that fold to a large number of structures (rare structures).

To illustrate the existence of neutral paths, Huynen et al. (1996) performed neutral walks on fitness landscapes that are based on the mapping between the RNA sequence and the RNA secondary structure and that give access to a virtually unlimited number of secondary structures which are a single point mutation from the neutral path.

In biology, there are manifestations of redundancy and neutrality at other levels, but the types discussed here demonstrate the existence of neutral networks and neutral mutations in the biological genetic code. As will be seen throughout this paper, the neutral networks and structures in the context of RNA folding present substantial differences compared to those in the neutral family of representations proposed here. One difference relates to the existence of rare or common RNA structures, indicating that some structures are overrepresented and others underrepresented, while in the neutral family of representations discussed here, that distribution is uniform. Additionally, at the molecular level, the neutral networks can be disconnected and have different sizes and shapes, while in the family of neutral representations under study here, for each representation, all phenotypes are represented by neutral networks with the same shape and size. The fact that the neutral networks and the structures at the molecular level are the result of evolution over millions of years suggests that the genetic code itself is the result of a process of evolution and, therefore, of improvement.

### 2.2 Redundancy and Neutrality in Evolutionary Computation Research

There is great interest in how redundant representations and neutral search spaces influence the behavior and the evolvability of EAs. Diverse redundant representations were used and, with some of these, the performance of the EAs improved and the evolvability, which is defined as the ability of random variations to sometimes produce improvements (Wagner and Altenberg, 1996), is increased (Shackleton et al., 2000; Shipman et al., 2000; Rothlauf, 2006). With others (Ronald et al., 1995; Knowles and Watson, 2002), however, with most of the problems applied, adding redundancy appeared to reduce performance. The loss of diversity in the population, the larger size of the genotypic search space, and the fact that different genotypes that represent the same phenotype compete against each other, are some of the reasons stated for the deterioration of EA performance. On the contrary, the ability of a population to adapt after changes (redundant and neutral representations allow a population to change the genotype without changing the phenotype) is one of the reasons given for the increased performance of EAs.

Banzhaf (1994) proposed a genotype-phenotype mapping called binary genetic programming (BGP) to the solution of constrained optimization problems. He argued that this mapping enables the use of unrestricted search operators in the genotypic space while, at the same time, the feasibility of solutions in the phenotypic space is guaranteed. The author concluded that neutral variants are frequent and play an important role in maintaining genetic diversity. In addition, Banzhaf and Leier (2006) examined the behavior of an evolutionary search on neutral networks in a simple linear genetic programming (GP) system of a Boolean function space problem. The authors drew a parallel between RNA folding problems and GP. They conducted an exhaustive examination of all possible genotypes, showing that there are highly common phenotypes and very few uncommon phenotypes and concluded that neutral networks must be highly intertwined to allow a quick transition from one network to the next.

To analyze the effects of neutrality in EAs, Barnett (1998) introduced a new family of NK fitness landscapes, the NKp fitness landscapes, which has the property of allowing the explicit addition of neutrality in the evolutionary process. In these landscapes, it is not only the roughness of the landscapes that is adjustable, using the parameter *K*, but also the amount of neutrality, parameter *p*, that is present during evolution. The author concluded that at least for a mutation-selection algorithm, avoiding getting stuck in local optima can be achieved in the presence of neutral networks. Also, Barnett (2001) stated that the lack of improvement in fitness is not due to the population being trapped in local optima. The dynamics of evolution must be seen in terms of navigating among neutral networks that eventually will lead to higher-fit neutral networks, such that in problems with a large amount of neutrality, the problem of premature convergence would not exist.

Newman and Engelhardt (1998) developed a model based on the NK fitness landscapes, which Geard et al. (2002) termed as NKq (i.e., quantized NK), that employs different approaches to modeling neutrality. Newman and Engelhardt found that their system was able to produce similar effects of RNA sequence structure maps, of which examples are “the occurrence of ‘common’ structures that occupy a fraction of the genotypic space, which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks that cover the genotypic space in such a way that a member of such a network can be found within a small radius of any point in the space” (Newman and Engelhardt, 1998, p. 1333). They argued that the maximum fitness attained during the adaptive walk of a population evolving on such a landscape increases with increasing degree of neutrality and is directly related to the fitness of the most fit percolating network. On the other hand, Geard et al. compared the NK, NKp, and NKq landscape models and found that there are significantly different structural properties.

To allow neutrality to be measured during evolution, Yu and Miller (2001) introduced the use of explicit neutrality with an integer string coding scheme on a Boolean benchmark problem. The experimental results indicated that there is a positive relationship between neutrality and evolvability: neutrality improves evolvability and neutrality improves success rates. The authors stated that the buffer effect of neutrality protects genotypes from destructive mutations during later stages of evolution. Using the OneMax problem, Yu and Miller (2002) reached the same conclusions.

In terms of the characteristics of the genotype and phenotype representation in Cartesian genetic programming (CGP) and the unexpected benefits of its redundancy to evolutionary search, Miller and Smith (2006) claimed that the best performance was found to employ extremely high levels of redundancy. Vassilev and Miller (2000) also used CGP to evolve digital circuits and they argued that the presence of neutrality in evolutionary search was useful and concluded that neutrality helps to cross wide landscape areas of low fitness.

In the literature, there are a variety of binary genotype-phenotype mappings which exhibit different degrees of redundancy. Shackleton et al. (2000) and Shipman et al. (2000) suggest that the use of highly redundant representations can improve the reliability of search, the ability to deal with changing environments, and the robustness to high mutation rates. These authors proposed five highly redundant genotype-phenotype mappings, where a binary phenotypic search space with cardinality was considered. Table 1 shows the length of the genotype and the degree of redundancy of each of the mappings proposed. As can be verified, the mappings displayed in this table present a degree of redundancy that varies between 14 and 320 redundant bits. On the contrary, in the family of neutral representations analyzed in this paper, it is possible to define representations with a reduced number of redundant bits, 3 or 4. These authors compared the mappings using three different criteria: the number of phenotypes that a given phenotype can reach using one-point mutation within a number of steps (reachability of a given phenotype), the number of new phenotypes encountered as a function of neutral walk length (innovation rate), and the number of different phenotypes which can be reached from a given phenotype by single mutations (connectivity between phenotypes). Ebner, Shackleton, et al. (2001) considered the connectivity between the phenotypes in the following manner: for every phenotype, 100 random genotypes that map to this phenotype were selected. Then, random neutral walks were performed, starting from each of these genotypes, and the phenotypes reachable for this phenotype were marked.

Mapping . | Genotype length . | Redundancy degree . |
---|---|---|

Static random mapping | 30 | 2^{30-16}=2^{14}:1 |

Trivial voting mapping | 48 | 2^{48-16}=2^{32}:1 |

Standard voting mapping | 32 | 2^{32-16}=2^{16}:1 |

Cellular automaton mapping (CA) | 144 | 2^{144-16}=2^{128}:1 |

Random Boolean network mapping (RBN) | 336 | 2^{336-16}=2^{320}:1 |

Mapping . | Genotype length . | Redundancy degree . |
---|---|---|

Static random mapping | 30 | 2^{30-16}=2^{14}:1 |

Trivial voting mapping | 48 | 2^{48-16}=2^{32}:1 |

Standard voting mapping | 32 | 2^{32-16}=2^{16}:1 |

Cellular automaton mapping (CA) | 144 | 2^{144-16}=2^{128}:1 |

Random Boolean network mapping (RBN) | 336 | 2^{336-16}=2^{320}:1 |

Using these mappings, Shackleton et al. (2000) and Shipman et al. (2000) observed that the standard voting, the CA, and the RBN mappings were more beneficial than the others and they concluded that the addition of redundancy is advantageous because it allows for an increase in connectivity between phenotypes. Furthermore, they observed that individuals are able to maintain fitness values even in the presence of high mutation rates because a high fraction of mutations are neutral. Ebner, Langguth, et al. (2001) and Ebner, Shackleton, et al. (2001) extended these studies and analyzed the effects of CA and RBN mappings in the context of a population based search. They pointed out that redundant mappings allow the population to spread along the network of neutral mutations and the population is able to quickly recover after a change has occurred. They also concluded that the existence of highly intertwined neutral networks increases the evolvability of a population.

On the contrary, Knowles and Watson (2002) used various optimization problems, among which the H-IFF (hierarchical if and only if), the MAX-SAT (maximum satisfiability problem), and the problem of NK fitness landscapes; and they provided solid evidence that the addition of random redundancy did not prove to be useful in optimization.

Also, Wilson and Kaur (2009) presented a model for the analysis and visualization of genotype to phenotype mappings, where the model groups genes into quotient sets and shows their adjacencies regarding the mapping and then used it to explain the population movements on neutral landscapes. The authors showed the applicability of the approach by applying it to different representations and problems and using it to develop theoretical results on how mapping and neutral evolution affect search in grammatical evolution (GE). The authors studied the two phases of mapping in GE, transcription (the identification of segments of genome with integer values) and translation (the use of the resulting integer value to make a derivation, based on a grammar) and showed that translation and transcription schemes belong to equivalence classes, allowing properties which are derived for specific schemes to be applicable to classes of schemes.

In order to clarify some of the questions in the literature about which types of redundant representations can be beneficial for EAs under which circumstances, Rothlauf (2006) identifies the following properties to characterize the different redundant representations proposed in the literature.

A representation is uniformly redundant if all phenotypes are repre- sented by the same number of genotypes.

A representation has high connectivity if the number of phenotypeswhich are accessible from a given phenotype by single bit mutationsis high.

A representation is synonymously redundant if the genotypes that areassigned to the same phenotype are similar to each other.

A representation has high locality if neighboring genotypes correspond toneighboring phenotypes.

There are other properties such as innovation rate (Ebner, Shackleton, et al., 2001), diameter of connected neutral networks, and accessibility of neighboring neutral networks (Lehre and Haddow, 2005), and others which are examples of properties, but due to space limitations, they will not be discussed in this paper.

Rothlauf (2006) stated that using neutral search spaces where the connectivity between the phenotypes is strongly increased by the use of redundant representation allows many different phenotypes to be reached in one single search step; however, using nonsynonymously redundant representations results in random search, which decreases the efficiency of EAs. The author claimed that when using synonymously redundant representations, the connectivity between the phenotypes is not increased. This conclusion is not consistent with the results obtained with the neutral redundant representations presented in this paper, as can be verified in Section 4.3, where there are synonymously redundant representations that allow the connectivity to be increased between phenotypes, when compared with the nonredundant representation.

Recently, a network-based model called the local optima network was introduced by Ochoa et al. (2008) and Verel et al. (2008) and reexamined by Verel et al. (2011). This model relies on many details of the NK fitness landscapes and compresses the landscape information into a weighted and oriented graph. The authors used two NK variants: probabilistic (NKp) and quantified (NKq), and found some unknown structural differences between the two variants. They attested that the studied network features confirmed that neutrality, in landscapes with percolating neutral networks, may enhance heuristic search.

Galván-López and Poli (2006) studied a simple form of neutrality: a neutral network of constant fitness, identically distributed in the whole search space, applied to a unimodal landscape and a deceptive landscape. The authors analyzed both problem solving performance and population flows to and from neutral networks and the basins of attraction of the optima. They argued that neutrality may be beneficial in some cases, but when it comes at the cost of an increased size of the search space without a corresponding expansion of the solution space, then any benefits it may bring via search bias and others, may be insufficient to compensate for the additional search effort required by a reduced density of solutions.

Also, Poli and Galván-López (2007) presented three different genotype-phenotype mappings which exhibit bitwise neutrality: the majority, the parity, and the truth table encodings. In these mappings, bitwise neutrality is induced by a genotype-phenotype map where each phenotypic bit is obtained by the transformation of a group of genotypic bits via some encoding function. In these three mappings, each phenotypic bit is encoded using *n* genotypic bits. The authors described under which conditions a neutral mapping has the potential to induce big changes in problem hardness and how phenotypic mutation rates change as a function of the genotypic mutation rate for different mappings. They claimed that the performance of a genetic algorithm can change radically with different types of neutrality and mutation rates. In Galván-López et al. (2011), a general overview on the work carried out on neutrality in EAs is provided.

Finally, Poli and Galván-López (2007) argued that the effects of neutrality on evolutionary search are not well understood for several reasons. For instance: there is a lack of a mathematical framework to explain how and why neutrality affects evolution; there is no single definition of neutrality; studies considered problems, representations, and search algorithms that are relatively complex, and as a consequence, the results represent the composition of multiple effects. Furthermore, studies very often focused on particular properties of neutrality without properly defining them. Using the two types of redundant representations proposed in Fonseca and Correia (2005), Correia and Fonseca (2007a, 2007b), and developed in Correia (2009, 2011, 2012), the present paper aims to evaluate the influence of redundancy and neutrality on the performance of a simple evolutionary strategy applied to NK fitness landscapes and in this way to shed some light on how binary redundancy and neutrality affect evolutionary search.

#### 2.2.1 Redundant Binary Representations

When using search algorithms, a metric has to be defined on the search space . Based on the metric, the distance *d*(*x _{a}*,

*x*) between two individuals and describes how similar the two individuals are. The larger the distance, the more different two individuals are (Rothlauf, 2006). In general, different metrics can be defined for the same search space, where different metrics result in different distances and different measurements of the similarity of solutions. In this way, two individuals are neighbors if the distance between two individuals is minimal. For example, when using the Hamming metric for binary strings, the minimal distance between two individuals is

_{b}*d*= 1. Therefore, two individuals

*x*and

_{a}*x*are neighbors if the distance

_{b}*d*(

*x*,

_{a}*x*)=1.

_{b}*f*, and a phenotype-fitness mapping,

_{g}*f*. Denoting as the genotypic space and as the phenotypic space, a genotype-phenotype mapping

_{p}*f*determines which phenotypes result from the decoding of each genotype and can be defined as follows: where denotes a genotype and denotes the corresponding phenotype. The genotype-phenotype mapping

_{g}*f*makes it possible to determine which genotypes represent a particular phenotype, but does not give any information about the similarity between the solutions. The phenotype-fitness mapping assigns a fitness value

_{g}*f*(

_{p}*x*) to each phenotype

_{p}*x*:

_{p}The metrics used may be different in the genotypic search space and in the pheno-typic search space. The metric of the phenotypic search space is determined by the problem to be solved, that is, it depends on the nature of the decision variables. For example, if the problem to be optimized is the NK fitness landscape, the Hamming metric can be used in the phenotypic search space. On the other hand, the genotypic search space metric depends not only on the space, but also on the search operator that is used on the genotypes. The search operator and the metric used in the genotypic search space determine each other and cannot be chosen independently of one another.

Next, some of the properties presented in Section 2.2 are explained. In redundant representations, the phenotypic neighborhood of a phenotype corresponds to the phenotypes that are reachable from a particular phenotype by single bit mutations of the genotypes that represent it; and connectivity corresponds to the number of different phenotypes which constitute the phenotypic neighborhood of a phenotype. For a nonredundant binary representation, wherein the phenotype is of length *k* and in that the reachable phenotypes from any phenotype are all different, the connectivity is *k*. For a uniform redundant representation, where genotypes have length and phenotypes have length *k*, if phenotype *x _{p}*=

*f*(

_{g}*x*) is represented by the following

_{g}*m*genotypes, and as each genotype has neighbors, then the set of neighbors of the

*m*genotypes is . As each genotype represents a phenotype, the set of phenotypes of this set is . The number of different ele- ments of this set corresponds to the connectivity. For the neutral redundant representation proposed in this research, the connectivity is equal for all phenotypes of a particular representation, as can be verified in Section 4.1, although the phenotypes can be different.

*x*. In other words, if for all phenotypes the sum over the distances between all genotypes that represent the same phenotype is reasonably small, that is, where , represents the distance between the genotype and genotype and denotes the set of genotypes that represent the phenotype

_{p}*x*.

_{p}*f*, on the metric defined in , and on the metric defined in . Rothlauf (2006) proposes the following expression for the locality

_{g}*d*: where denotes the distance between the and phenotypes, describes the distance between the corresponding genotypes and ,

_{m}*d*

^{g}_{min}is the minimum distance between two neighboring genotypes, and

*d*

^{p}_{min}indicates the minimum distance between two neighboring phenotypes.

Finally, it is fair to say that the importance and the interrelationships between these properties are not yet fully understood. As mentioned, Rothlauf (2006) asserts that, when using synonymously redundant representations, the connectivity between phenotypes does not increase, whereas in the present paper, practical evidence to the contrary is provided.

## 3 Development of Redundant Binary Representations

The absence of conclusive results in the literature on the possible benefits of redundant representations and the existence of redundancy and neutrality in the genetic code were the reasons for the development and analysis of new families of redundant binary representations, which exhibit the properties of redundancy, polygeny, pleiotropy, and neutrality.

The development of the first redundant representations is performed incrementally, starting with a noncoding redundant representation and incorporating one property in each step, first polygeny and then pleiotropy. As the linear transformations do not allow the incorporation of neutrality, unless by using noncoding bits, a second approach is adopted based on error control codes to define neutral redundant binary representation families.

### 3.1 Redundant Representations Based on Linear Transformations

*GF*(2), and

*I*is the identity matrix with size

_{k}*k*.

*GF*(2) is the Galois field , where addition and multiplication operations are modulo-2, which correspond to

*exclusive-or*and

*and*operations, respectively.

#### 3.1.1 Representations with Noncoding Bits

*X*, only one bit is set to 1 in each column and the row where it is in the column indicates the bit of the genotype which encodes the phenotype. For example, a representation where the first

*k*genotype bits correspond to the

*k*phenotype bits and the remaining genotype bits are noncoding bits, the matrix

*X*may take the following form: where represents the zero array.

This representation is uniform because all phenotypes are represented by the same number of genotypes. It also exhibits neutrality because mutating a noncoding bit does not affect the decoded phenotype.

In the context of evolutionary computation, the practical relevance of this type of representation is reduced. As noncoding bits are subject to mutation in the same way that coding bits, but do not produce changes in the phenotype, if the number of redundant bits is not explicitly taken into account in determining the individual mutation rate, the coding bits of the genotype will be less likely to suffer a mutation. This would be equivalent to using a nonredundant representation with a lower mutation rate which, in principle, would slow down the evolutionary process (Fonseca and Correia, 2005).

#### 3.1.2 Representations with Polygeny

In matrix {0, 1}-*Y*, the active bits in each column indicate the location of the genotype bits that interact and contribute to the corresponding bit of the phenotype. As each bit of the genotype may influence only one bit of the phenotype, only one bit may be set to 1 in each row of the *Y* matrix. This representation is non-neutral and the reachable phenotypes from phenotype 0 correspond to the rows of matrix *Y*.

#### 3.1.3 Representations with Polygeny and Pleiotropy

*Z*determine the phenotypic neighborhood induced by the representation, it is the matrix

*Z*that defines if the representation is uniformly or nonuniformly redundant. On the other hand, the order of the rows of the matrix

*Z*defines the position of the genotypic bit that may reach a certain phenotype. Whereas

*v*+

*e*represents the effect of a mutation in the bit

_{i}*i*of genotype

*v*, where

*e*corresponds to a vector with length with a single nonzero bit at position

_{i}*i*, such that:

It is clear that the phenotype reachable from the phenotype *u* by mutation of bit *i* only depends on the phenotype *u* itself and on the row *z _{i}* of matrix

*Z*. Thus, the effect of a mutation, whichever phenotype is considered, does not depend on the particular genotype that represents that phenotype, but only on the bit mutated. This is one of the limitations of this family of representations: all genotypes that decode to a specific phenotype reach the same phenotype by mutation at the same locus. This is one of the limitations resulting from the linearity of the transformation, allowing that connectivity can only grow linearly with the length of the genotype. The (non-neutral network) notation will be used to designate the family of non-neutral representations.

### 3.2 Redundant Representations Based on Error Control Codes

#### 3.2.1 Neutrality

Although the linear representations proposed in the previous section allow specifying the set of phenotypes reachable from a given phenotype by a single mutation, all genotypes that represent the same phenotype reach exactly the same set of phenotypes. According to the neutral theory of evolution, the accumulation of neutral mutations should allow new paths for development to be found. A new approach can be considered if:

Different genotypes representing the same phenotype should reach different sets of neighboring phenotypes through single gene mutations.

Different genotypes that represent the same phenotype need to be connected by a path composed exclusively of neutral single bit mutations.

None of the representations discussed in Section 3.1 have these two properties. Thus, a different approach to representation design is needed in order to obtain neutral representations with the following characteristics.

Split the redundant genotypic space of cardinality into subspaces, each with cardinality 2

^{k}, in such a way that a single bit change in the genotype causes a transition from one subspace to another.Define bijective mappings between each genotypic subspace and the phenotypic space, in such a way that all genotypes which represent the same phenotype form a connected neutral network.

In the binary case, the division of the genotypic space into subspaces in which the minimum Hamming distance between genotypes in the same space is at least two, would guarantee that single bit mutations produce genotypes in subspaces other than that of the original genotype. This is an essential characteristic of error control code design, and results from this area are readily applicable here.

#### 3.2.2 Error Control Codes

The idea behind using error control codes is to add redundancy to the message to be transmitted in such a way that allows the receiver to detect and, if possible, to correct errors that may have occurred during the transmission. Among the many error control codes available, block codes are defined on a binary alphabet which consists of a set of 2^{k} codewords with fixed length , that result from the encoding of the *k* bits of the data to be transmitted, after the addition of check bits and where the codewords form a vector subspace of dimension *k*. In particular, linear block codes may be defined as follows (Lin and Costello, 1983).

*A block code of length and 2 ^{k} codewords is called a linear code if and only if its 2^{k} codewords form a k-dimensional subspace of the vector space of all the over the field *.

*V*(

*x*) corresponds to the code polynomial,

*U*(

*x*) is the message polynomial consisting of the

*k*bits of the message to be transmitted and

*C*(

*x*) corresponds to the parity-check polynomial. The message polynomial

*U*(

*x*) has degree up to

*k*−1, and because it is a binary code, its coefficients are 0 or 1. The code polynomial

*V*(

*x*) has degree up to .

A cyclic code can be defined using the generator polynomial *g*(*x*) which has to be a factor of and the degree of *g*(*x*) is equal to , the number of parity-check bits. As a consequence of these properties, any code polynomial is a multiple of the generator polynomial and the generator polynomial is also a code polynomial. On the other hand, any polynomial factor of with degree could be used as a generator polynomial; however, not all polynomials in these conditions seem to produce codes with the desirable features. Carlson (2002) mentioned some of the recommended generator polynomials.

*g*(

*x*), which allows the code to be generated in a systematic form (when the message polynomial can be obtained by discarding given components from the corresponding code polynomial),

Table 2 shows the generation of the code polynomials using the generator polynomial *g*(*x*)=*x*^{3}+*x*+1=1011.

U
. | U(x)
. | x^{3}U(x)
. | C(x)=x^{3}U(x)modg(x)
. | V
. |
---|---|---|---|---|

0000 | 0 | 0 | 0 | 0000000 |

0001 | 1 | x^{3} | x+1 | 0001011 |

0010 | x | x^{4} | x^{2}+x | 0010110 |

0011 | x+1 | x^{4}+x^{3} | x^{2}+1 | 0011101 |

0100 | x^{2} | x^{5} | x^{2}+x+1 | 0100111 |

0101 | x^{2}+1 | x^{5}+x^{3} | x^{2} | 0101100 |

0110 | x^{2}+x | x^{5}+x^{4} | 1 | 0110001 |

0111 | x^{2}+x+1 | x^{5}+x^{4}+x^{3} | x | 0111010 |

1000 | x^{3} | x^{6} | x^{2}+1 | 1000101 |

1001 | x^{3}+1 | x^{6}+x^{3} | x^{2}+x | 1001110 |

1010 | x^{3}+x | x^{6}+x^{4} | x+1 | 1010011 |

1011 | x^{3}+x+1 | x^{6}+x^{4}+x^{3} | 0 | 1011000 |

1100 | x^{3}+x^{2} | x^{6}+x^{5} | x | 1100010 |

1101 | x^{3}+x^{2}+1 | x^{6}+x^{5}+x^{3} | 1 | 1101001 |

1110 | x^{3}+x^{2}+x | x^{6}+x^{5}+x^{4} | x^{2} | 1110100 |

1111 | x^{3}+x^{2}+x+1 | x^{6}+x^{5}+x^{4}+x^{3} | x^{2}+x+1 | 1111111 |

U
. | U(x)
. | x^{3}U(x)
. | C(x)=x^{3}U(x)modg(x)
. | V
. |
---|---|---|---|---|

0000 | 0 | 0 | 0 | 0000000 |

0001 | 1 | x^{3} | x+1 | 0001011 |

0010 | x | x^{4} | x^{2}+x | 0010110 |

0011 | x+1 | x^{4}+x^{3} | x^{2}+1 | 0011101 |

0100 | x^{2} | x^{5} | x^{2}+x+1 | 0100111 |

0101 | x^{2}+1 | x^{5}+x^{3} | x^{2} | 0101100 |

0110 | x^{2}+x | x^{5}+x^{4} | 1 | 0110001 |

0111 | x^{2}+x+1 | x^{5}+x^{4}+x^{3} | x | 0111010 |

1000 | x^{3} | x^{6} | x^{2}+1 | 1000101 |

1001 | x^{3}+1 | x^{6}+x^{3} | x^{2}+x | 1001110 |

1010 | x^{3}+x | x^{6}+x^{4} | x+1 | 1010011 |

1011 | x^{3}+x+1 | x^{6}+x^{4}+x^{3} | 0 | 1011000 |

1100 | x^{3}+x^{2} | x^{6}+x^{5} | x | 1100010 |

1101 | x^{3}+x^{2}+1 | x^{6}+x^{5}+x^{3} | 1 | 1101001 |

1110 | x^{3}+x^{2}+x | x^{6}+x^{5}+x^{4} | x^{2} | 1110100 |

1111 | x^{3}+x^{2}+x+1 | x^{6}+x^{5}+x^{4}+x^{3} | x^{2}+x+1 | 1111111 |

Using Equation (10), it is possible to verify that only the code polynomials are divisible by the generator polynomial. When the received polynomial, which corresponds to the word received after transmission, is divided by the generator polynomial, if the remainder is null, the received polynomial corresponds to a code polynomial, otherwise that polynomial does not correspond to a code polynomial and an error occurred during transmission.

*Z*(

*x*) corresponds to the received polynomial, which might not be equal to any of the code polynomials

*V*(

*x*) if some errors occurred during transmission. In this case, where

*e*(

*x*) is the error polynomial. The syndrome polynomial can be written as

e
. | e(x)
. | s
. | S(x)
. |
---|---|---|---|

1000000 | x^{6} | 101 | x^{2}+1 |

0100000 | x^{5} | 111 | x^{2}+x+1 |

0010000 | x^{4} | 110 | x^{2}+x |

0001000 | x^{3} | 011 | x+1 |

0000100 | x^{2} | 100 | x^{2} |

0000010 | x | 010 | x |

0000001 | 1 | 001 | 1 |

e
. | e(x)
. | s
. | S(x)
. |
---|---|---|---|

1000000 | x^{6} | 101 | x^{2}+1 |

0100000 | x^{5} | 111 | x^{2}+x+1 |

0010000 | x^{4} | 110 | x^{2}+x |

0001000 | x^{3} | 011 | x+1 |

0000100 | x^{2} | 100 | x^{2} |

0000010 | x | 010 | x |

0000001 | 1 | 001 | 1 |

To decode using the table, the syndrome polynomial has to be calculated as indicated in Equation (12) and the error polynomial *e*(*x*) is determined from the syndrome polynomial obtained. Using Equation (13), the code polynomial can be calculated. If the code is in systematic form, the message polynomial is directly obtained.

As an example, if the received polynomial is *Z*(*x*)=*x*^{5}+*x*^{4}=0110000, the syndrome polynomial is *S*(*x*)=1=001. Using Table 3, the error polynomial achieved is *e*(*x*)=1=0000001. Adding the error polynomial to the received polynomial, the code polynomial obtained is *V*(*x*)=*x*^{5}+*x*^{4}+1=0110001. Truncating, the message polynomial obtained is *U*(*x*)=*x*^{2}+*x*=0110.

*g*(

*x*) is a primitive (or even merely irreducible) polynomial, it is easy to show that the code polynomial

*V*(

*x*) cannot be of type

*x*(otherwise

^{i}*g*(

*x*) itself would have to be of the same type and would not be primitive or even irreducible). Due to the linearity of the code, neither can the difference between two code polynomials have that form, which implies that the minimum distance between different code polynomials must be at least two. Thus, Equation (10) provides a method of mapping arbitrary phenotypes onto a linear class

*C*of code polynomials of degree up to . Furthermore, it suggests that another classes of code polynomials may be defined as follows for every nonzero polynomial

*z*(

_{i}*x*) of degree up to . Clearly, these classes are no longer linear because the sum of two polynomials in class

*C*is not in

_{i}*C*, but since they can be obtained from the linear class

_{i}*C*through the addition of a constant polynomial, they are called the

*cosets*of

*C*. Moreover, changing a single coefficient in a polynomial will have the effect of producing a member of one of the cosets

*C*. This is the property which is explored in error control codes and provides a suitable mechanism for making the sets of reachable phenotypes vary for different genotypic representations of the same phenotype.

_{i}#### 3.2.3 Neutral Representations

*V*(

*x*) of degree of a linear code , generated by the generator polynomial

*g*(

*x*), and the phenotype as the message polynomial

*U*(

*x*) of degree

*k*−1, Equation (11) can define the genotype based on the phenotype. As genotype

*V*(

*x*) belongs to the 0 coset defined by the generator polynomial

*g*(

*x*), the representation of

*U*(

*x*) in each of the cosets can be defined by selecting coset leaders

*z*(

_{i}*x*):

To allow the genotypes *V _{i}*(

*x*), which represent a

*U*(

*x*), to form a neutral connected network, it is enough that the coset leaders

*z*(

_{i}*x*) define themselves a connected graph by links resulting from one bit change. When

*U*(

*x*) corresponds to the 0 phenotype,

*V*(

*x*) is also equal to zero and the polynomials

*V*(

_{i}*x*) correspond to the coset leaders

*z*(

_{i}*x*). In this context, and for this reason, the coset leaders will be designated by the

*zeros*of the representation and the neutral network defined by the coset leaders forms a neutral representation of the (neutral network) family.

*A neutral representation in the family consists in a mapping from a set of binary strings of length (the genotypic space) to the set of binary strings of length*

*k*(the phenotypic space), with , which is characterized by the generator polynomial,*g*(*x*), of degree , and by the set of genotypes:*where each C*

_{i}denotes the subset of all genotypes with syndrome*i*, and each*z*is considered the coset leader of_{i}*C*and is called zero. For each pair (_{i}*z*,_{i}*z*), either_{j}*d*(*z*,_{i}*z*)=1, or there is a set of genotypes ,_{j}*p*=*d*(*z*,_{i}*z*)−1, such that:_{j}The coding and decoding algorithms are explained and presented in Correia (2009).

### 3.3 Enumeration of Neutral Representations

Although each representation of has been defined by the choice of their set of zeros, it is important not to forget that the elements in this set have restrictions of syndrome and connectivity. The algorithm used to generate the families of neutral representations is presented in Correia (2009).

Table 4 presents the number of neutral representations of found for , when the number of redundant bits is in and the generator polynomial *g*(*x*) has a degree of . As observed, the number of neutral representations grows rapidly with *k* and very rapidly with the number of redundant bits . This huge number creates difficulties not only for execution time but also for the storage capacity of the generated representations. For example, for a phenotypic space, if three redundant bits are added, 266 neutral representations are obtained, while 1,845,628 are reached when four redundant bits are joined.

x+1 | x^{2}+x+1 | x^{3}+x+1 | x^{4}+x+1 | g(x) | |

1 | 2 | 4 | 19 | 1,489 | |

2 | 3 | 9 | 266 | 1,845,628 | |

3 | 4 | 18 | 1,828 | ||

4 | 5 | 32 | 8,128 | ||

5 | 6 | 50 | 25,100 | ||

6 | 7 | 75 | 66,750 | ||

7 | 8 | 108 | 158,784 | ||

8 | 9 | 147 | 342,701 | ||

9 | 10 | 196 | 690,748 | ||

10 | 11 | 256 | 1,307,528 | ||

11 | 12 | 324 | 2,350,336 | ||

k |

x+1 | x^{2}+x+1 | x^{3}+x+1 | x^{4}+x+1 | g(x) | |

1 | 2 | 4 | 19 | 1,489 | |

2 | 3 | 9 | 266 | 1,845,628 | |

3 | 4 | 18 | 1,828 | ||

4 | 5 | 32 | 8,128 | ||

5 | 6 | 50 | 25,100 | ||

6 | 7 | 75 | 66,750 | ||

7 | 8 | 108 | 158,784 | ||

8 | 9 | 147 | 342,701 | ||

9 | 10 | 196 | 690,748 | ||

10 | 11 | 256 | 1,307,528 | ||

11 | 12 | 324 | 2,350,336 | ||

k |

## 4 Properties of *NN*_{g}(ℓ, *k*)

_{g}

### 4.1 Uniformity, Phenotypic Neighborhood, and Connectivity

The properties explained in Section 2.2 will now be discussed in the context of neutral representations. The family is considered uniform because all phenotypes are represented by the same number of genotypes. In each representation of , each phenotype is represented by genotypes.

A way to visualize the phenotypic neighborhood of a representation is using hypergraphs. For example, Figure 1 displays a genotypic space with cardinality , where for the sake of simplicity, only the 1100101 genotype (□) and its seven neighbors () are presented, while for all other genotypes, only the first four bits are indicated.

Table 5 presents the genotypes and the phenotypes [] obtained using the neutral representation indicated. As the representation is uniform, each phenotype is represented by genotypes, which corresponds to the number of zeros of the neutral representation.

Syndrome 0 . | Syndrome 1 . | Syndrome 2 . | Syndrome 3 . |
---|---|---|---|

0000 [00] | 0001 [00] | 0010 [00] | 0100 [00] |

0111 [01] | 0110 [01] | 0101 [01] | 0011 [01] |

1001 [10] | 1000 [10] | 1011 [10] | 1101 [10] |

1110 [11] | 1111 [11] | 1100 [11] | 1010 [11] |

Syndrome 0 . | Syndrome 1 . | Syndrome 2 . | Syndrome 3 . |
---|---|---|---|

0000 [00] | 0001 [00] | 0010 [00] | 0100 [00] |

0111 [01] | 0110 [01] | 0101 [01] | 0011 [01] |

1001 [10] | 1000 [10] | 1011 [10] | 1101 [10] |

1110 [11] | 1111 [11] | 1100 [11] | 1010 [11] |

As each phenotype is represented by a neutral network with genotypes, each of which has neighboring genotypes, including, at least, a genotype in the same network, the phenotypic neighborhood of each phenotype has, at most, different phenotypes. The actual number of different phenotypes in this neighborhood determines the connectivity of the representation, which is the same for all phenotypes defined by the representation. The connectivity of is three, because it is possible to reach three different phenotypes (excluding itself). As the topology of the neutral network is the same for all phenotypes, connectivity is also equal for all other phenotypes, although the reached phenotypes are different. Compared to a nonredundant representation with phenotype length of two, where each phenotype can only reach two different phenotypes, the neutral representation can reach a larger number of different phenotypes, in this case, three. In fact, all neutral representations have a connectivity higher than or equal to the connectivity of the corresponding nonredundant representation, that is, higher than or equal to *k*. A table with the maximum connectivity of each can be found in Correia (2009, 2012).

### 4.2 Synonymity, Locality, and Topology

*d*(

*z*,

_{i}*z*) corresponds to the Hamming distance between zero

_{j}*z*and zero

_{i}*z*. This expression gives a normalized measure of synonymity, since the total number of distances was considered. A representation is considered synonymously redundant when the value of is low and nonsynonymously redundant when this value is high.

_{j}Locality does not vary with different phenotypes using the neutral representations and can also be calculated on the basis of a single phenotype, leading to a simplification of Equation (2) in Section 2.2.1. The locality for the family can be calculated using the average of the distances from a phenotype to the corresponding neighboring phenotypes, including the phenotype in question.

The topology of the neutral networks was also studied using the MDS (multi-dimensional scaling) technique (de Leeuw, 2000). This technique consists of determining the positions of a set of points in the Euclidean space, such that the distances between them approximate a given measure of dissimilarity of the objects that are to be visualized. The minima and maxima synonymity and locality for each , as well as the list of all topologies for the family are available in Correia (2009).

### 4.3 Relationship Between Properties

Figure 2 presents scatter graphs showing the relationship between connectivity, locality, and synonymity of the family, which corresponds to the family of representations with the highest value of *k* that was completely enumerated. Although the scatter graphs only show the properties for the family, the results are similar for the remaining families of . These scatter graphs show that there are representations with low locality that reach high connectivity values. The chart on the top-right-hand side of Figure 2 contradicts the idea presented in Rothlauf (2006) that synonymously redundant representations do not allow the connectivity between the phenotypes to increase in comparison with the corresponding nonredundant representations. The circle in this chart shows that there is a set of synonymously neutral representations that have high values of connectivity. The family has a variety of representations with different properties that provide an important basis for the study of the relationships between them and will enable the study of to what extent these properties can affect the performance of an evolutionary algorithm.

## 5 Performance of Redundant Binary Representations

In order to study the influence of redundancy and neutrality on evolutionary algorithm performance, the application of a (1+1)-ES (Rudolph, 2000) to the NK fitness landscapes with different degrees of roughness was chosen. The and representations were considered. Since neutral representation families involving more than three redundant bits could not be fully enumerated, the study was based on the family. This provided a genotypic space on nontrivial size which was still sufficiently small to model the behavior of the (1+1)-ES using Markov chains, side-stepping the difficulties associated with the performance evaluation of the (1+1)-ES on such a small search space (from a practical perspective).

### 5.1 NK Fitness Landscapes

The NK fitness landscapes proposed by Kauffman (1995) are based on the fitness landscapes proposed by Wright (1932), who found that when there are fitness interactions between genes, the genetic composition of a population can evolve into multiple domains of attraction. On the other hand, genes can suppress or enable the effects of other genes, and this interaction is called epistasis.

The NK fitness landscapes are a way to explore how epistasis can control the roughness of the landscape. For this purpose, a fitness function family whose roughness could be shaped by a single parameter was developed. fitness landscapes are defined as stochastically generated fitness functions of bit strings with length *N* and interactions between groups of *K*+1 bits. In order to compute the overall fitness of one string, each bit contributes a component to the total fitness based on its own value and the values of *K* other bits. Fitness contributions come from a uniform distribution ranging from 0.0 to 1.0. The fitness of a string is computed as the sum of bit contribution at all *N* loci divided by *N* for normalization to the range [0; 1].

The roughness of the landscape is controlled by the parameter *K* and reaches a maximum when *K*=*N*−1 and a minimum when *K*=0. When *K*=0, there is no epistasis, the landscape has a unique local optimum (the global optimum), and the fitness of different chromosomes is highly correlated with the Hamming distance between them. When *K*=*N*−1, the number of interactions between bits is at a maximum and the function is purely random, so the landscape has many local optima, and the fitness of the chromosomes is not correlated with the Hamming distance between them. The choice of the *K* bits is known to affect the computational complexity of the NK fitness landscape optimization problem.

### 5.2 Markov Chain Modeling of a (1**+**1)-ES

The modeling of an evolutionary algorithm, specifically a (1+1)-ES, may be done through Markov chains. This strategy corresponds to stochastic hill climbing (Mitchell, 1996), where an individual parent generates an individual child and the best of both becomes the parent in the next iteration. In the case of a (1+1)-ES, the current state corresponds to the parent in each iteration, and the state space matches the search space. In the strategy used, the child is generated by mutation of a single bit of the parent genotype, replacing it in the next iteration if its fitness is not less than the fitness in the current iteration. The transition probabilities depend both on the mutation operator and on the fitness function. Since the neighborhood is finite and small, the transition matrix is sparse. This is useful because it facilitates the storage of the matrix in memory, and reduces the computation time of the state distribution. The transition matrices for each type of representation are presented in Correia (2009).

### 5.3 Performance Measure

As a first approach, the probability of achieving a global optimum was considered. As was reported by Correia and Fonseca (2007a, 2007b), in the long term, sometimes it is the representation which has a higher probability of reaching the global optimum than the corresponding representation or vice versa.

These results indicated that the neutral and non-neutral representations are not exactly equivalent. As the phenotypic neighborhood of each non-neutral representation was built to be equal to that of a neutral representation, one would expect the probability of reaching the global optimum to be the same in both cases. In fact, the probability of reaching the global optimum depends on the probability of the first individual being in its basin of attraction. However, it is also possible that the first individual is at the border of two different basins of attraction. In this case, the probability of convergence for each of the optima will depend not only on the phenotypic neighborhood, but also on whether the representation is neutral or not, and if it is neutral, on the topology of the neutral network. In addition, it was found that the convergence of the neutral representations was slower than the corresponding non-neutral representations. This result was not surprising, as the complete scanning of the neighborhood of a neutral network requires it to be crossed at the expense of neutral mutations.

As the probability of achieving the global optimum does not appear to be an appropriate indicator, the evolution of the expected value of fitness of the NK landscape as a function of time was considered to be a good indicator of representation performance. This value is calculated from the probability distributions provided by the Markov chain.

Determine the expected value of fitness for each pair representation-instance of the NK fitness landscape.

Rank all representations based on that value, separately, for each instance of the NK fitness landscape, since the value of fitness is not comparable between different instances.

To ensure that the number of local optima of the NK fitness landscapes does not increase with the redundant representations used, representations in which phenotypic neighborhood contains the nonredundant representation neighborhood were the only ones considered. This set of representations matches 12,819 representations of .

To know whether the behavior of the representations are similar to the behavior of the representations, using a (1+1)-ES and with knowledge of how the behaviors of these two types of representations are when compared to the random search, the expected values of fitness obtained with these representations on an instance of the NK fitness landscapes for each one of the values of *K* were analyzed. Random search was used to quantify the problem difficulty.

Figure 3 presents scatter graphs of the expected values of fitness obtained. The black and the gray spots correspond to the expected values of fitness after 100 and 500 iterations, respectively. The + and * symbols indicate the expected value of fitness for the purely random search, using the nonredundant representation, for 100 and 500 iterations, respectively.

Analyzing the first chart which corresponds to an instance of the fitness landscape presenting a single global optimum, all neutral and non-neutral representations seem to achieve the global optimum at the end of 500 iterations. This can be easily observed by the concentrated gray spots. However, after 100 iterations, there are many representations that have not yet achieved the global optimum. It is also possible to verify that random search, after 100 and 500 iterations, presented a worse behavior than the evolutionary strategy. This behavior is gradually reversed with the increase of the value of *K*, which can be visualized in the remaining charts of Figure 3. As is expected, the more difficult the problem, the worse the behavior of the evolutionary strategy compared to the random search. Moreover, the fact that in every chart of Figure 3 the black spots are always situated below the diagonal indicates that the neutral representations converge more slowly than the corresponding non-neutral representations. However, at the end of 500 iterations, the behaviors of the two types of representations are similar, since the gray spots are close to the diagonal chart.

As the representations were revealed to be a vast family of representations with different properties, it was imperative to verify whether some of the 12,819 selected representations systematically presented better behavior than others. In order to do so, the average of their ranks based on sets of 20 instances of NK fitness landscapes for several values of *K* was determined. Note that if there is total agreement between these ranks, the average rank would simply be the integer numbers from 1 to 12,819. On the other hand, if there is no relationship between these ranks, the distribution of values of the average rank would be identical to that obtained by the calculation of the average of an equal number of random ranks. This is the null hypothesis of this study.

Figure 4 shows that there is indeed some correlation between ranks for each set of instances of fitness landscapes, where *K*=0, 1, 2, 3, 4, 5, 6, 10, since the distribution of the average ranks is distant from the distribution under the null hypothesis, in the direction of the diagonal of the chart. Although no value for the significance is calculated, it is necessary to note that the differences are much higher than the variation between different estimates of the distribution of ranks under the null hypothesis. Furthermore, the results vary with the value of *K*. When *K*=0, 4, 5, 6, 10, the differences related to the random case seem to be more pronounced than when *K*=1, 2, 3. Excluding case *K*=0, the higher the *K*, the higher the number of local optima and the greater the difference.

To know whether there is an agreement between the behavior of the neutral representations in the fitness landscapes with different values of *K*, the average ranks obtained by the neutral representations for seven pairs of values of *K* were considered. Figure 5 shows the average ranks obtained. The black dots correspond to the 20 neutral representations with better aggregated average ranks.

In general, a positive correlation between the average ranks obtained for each representation in landscapes with similar values of *K* is visible. In particular, the 20 best representations presented good performance for every value of *K* considered, with the exception of the extreme cases, *K*=0 and *K*=10. For *K*=0, there is a set of representations with a clearly superior performance to the 20 representations indicated. This set consists of the representations with lower connectivity which in unimodal landscapes lead to a rapid convergence for optimum. The remaining representations also lead to the convergence for the optimum, but more slowly. For *K*=10, the function is purely random and the best representations are those that exhibit the highest connectivity.

Finally, to verify how the relationship is between the performance of the neutral representations in the and the neutral representation properties, Figure 6 is displayed. This figure shows how the connectivity, locality, and synonymity of the selected representations relate to their performance. The results show that the 20 best representations present high values, but not extreme values of connectivity and show intermediate values of locality and synonymity. The same can be seen in Figure 7, which illustrates that in the set of the representations considered, the connectivity, locality, and synonymity are positively correlated.

In conclusion, this research showed how the phenotypic neighborhood induced by a representation influences the performance of an evolutionary algorithm. Using a (1+1)-ES and the expected value of fitness, it was possible to conclude that the phenotypic neighborhood induced by the representation seems to dominate the behavior of the algorithm, affecting the search more significantly than neutrality. In general, neutrality delayed the convergence of the strategy, but did not seem to significantly affect the long-term behavior of the algorithm.

Among neutral representations in which phenotypic neighborhood contains the phenotypic neighborhood of the nonredundant representation, some of them presented a performance systematically superior than others. This observation suggests that there are representations that are more appropriate than others for the optimization of NK fitness landscapes.

## 6 Conclusions

The existence of redundancy and neutrality in the genetic code and the neutral theory of molecular evolution proposed by Kimura (1968) have motivated the use of redundant and neutral representations in EC. The main reasons for developing the family of neutral redundant binary representations analyzed in this paper were twofold. First, the lack of consensus in the literature about the performance of neutral representations when used with EAs and, second, the verification that some of the representations proposed in the literature are highly redundant and use weakly structured genotype-phenotype mappings.

The mathematical formulation of the error control codes was the basis for the definition of the family of neutral redundant binary representations . The linear block codes, in particular the Hamming codes and their formulation as cyclic codes, were studied. The , for were generated, when the number of redundant bits was in the range . Each representation of this family exhibits different properties, namely uniformity, connectivity, locality, and synonymity, which were the properties analyzed.

As seen throughout the paper, the neutral networks and structures in the context of the RNA folding present substantial differences compared to those of the family. One difference relates to the existence of rare or common RNA structures, where some structures are overrepresented and others underrepresented, while in the family, that distribution is uniform. Additionally, for each representation of , all phenotypes are represented by neutral networks with the same shape and size, while at the molecular level, the neutral networks can be disconnected and have different shapes and sizes.

A family of non-neutral redundant binary representations based on linear transformations was also presented. The development of this family was performed incrementally, starting with a noncoding redundant representation, adding polygeny, followed by pleiotropy. These representations have proved to be highly redundant, when compared to the , but still less redundant than some of the representations that have been proposed in the literature (Shipman et al., 2000; Shackleton et al., 2000; Ebner, Shackleton, et al., 2001). The family of non-neutral representations allows us to set the desired phenotypic neighborhood, using the matrix that defines it. This simple and straightforward way to set the desired neighborhood is what compensates for the high level of redundancy. The increase of connectivity between phenotypes is achieved at the expense of the increase in the corresponding genotypic search space.

The enumeration of all representations for each family was achieved. The results were diversified representation families, with a range of different properties, which corroborated some previous results in the literature and rejected others. Thus, this investigation shows that even synonymously redundant representations may increase the connectivity between phenotypes in comparison with the nonredundant representation, contesting the opinion of some authors and corroborating the view that representations with high locality allow an efficient evolutionary search in easy problems (Rothlauf, 2006).

In order to study the role of redundancy and neutrality in the behavior of a simple evolutionary algorithm, using the families of representations proposed, a study was conducted with a set of representations with different properties. For this, a (1+1)-ES was used and applied to NK fitness landscapes, and its behavior was modeled by Markov chains. The performance of a (1+1)-ES was considered as the expected value to achieve the best solution.

For each neutral representation, a non-neutral representation of the family with the same phenotypic neighborhood was considered. Thus, the influence of neutrality could be separated from the effects of the neighborhood common to both representations. Thus, neutrality seems to less significantly affect the search than the phenotypic neighborhood, delaying the convergence of the algorithm, compared with the corresponding non-neutral representation, but not significantly changing the algorithm's long-term behavior.

In the family proposed, there are some representations that are more appropriate than others for the optimization of fitness landscapes. Furthermore, it was found that these representations did not present extremes values of connectivity, synonymity, or locality, which is in opposition to what one would expect taking into account the recommendations proposed in the literature. This might help to elucidate the current difficulty in obtaining demonstrably successful redundant representations in EC.

A similar investigation was developed for the 0-1 knapsack problem, but due to lack of space, it is not presented or discussed in this paper. However, the results were consistent with those obtained for the NK fitness landscapes.

The results achieved in this study open multiple paths for future work. One of them is to apply the representations proposed to the 0-1 quadratic programming and to NK fitness landscapes with a random neighborhood. Additionally, as there are many neutral representations with different phenotypic neighborhoods, but which present the same neutral network topology, it would be interesting to classify neutral representations on the basis of neutral networks topology. This classification would allow a faster enumeration and would facilitate the generation of families with a bigger phenotypic space.

## References

*p*family of fitness landscape

*p*and NK

*q*