Examining the Impact of Neutral Theory on Genetic Algorithm

Population Evolution

Seamus Hill and Colm O’Riordan

College of Engineering & Informatics, National University of Ireland Galway, Ireland

Keywords:

Neutral Theory, Genetic Drift, Neutrality, Genotype, Phenotype, Genetic Algorithms.

Abstract:

This paper examines the introduction of neutrality as proposed by Kimura (Kimura, 1968) into the genotype-

phenotype mapping of a Genetic Algorithm (GA). The paper looks at the evolution of both a simple GA (SGA)

and a multi-layered GA (MGA) incorporating a layered genotype-phenotype mapping based on the biological

concepts of Transcription and Translation. Previous research in comparing GAs often use performance statis-

tics; in this paper an analysis of population dynamics is used for comparison. Results illustrate that the MGA

population’s evolution trajectory is quite different to that of the SGA population over dynamic landscapes

and that the introduction of neutrality implicitly maintains genetic diversity within the population primarily

through genetic drift in association with selection.

1 INTRODUCTION

Neutral theory as proposed by Kimura (Kimura,

1968), offered an alternative to the Darwinian view;

stating that the mutations involved in the evolutionary

process are neither advantageous nor disadvantageous

to the survival of an individual and that most muta-

tions are causednot by selection, but rather byrandom

genetic drift. However, in (Kimura, 1983), Kimura

pointed out that although natural selection does play a

role in adaptiveevolution, only a tiny fraction of DNA

changes are adaptive. The vast bulk of mutations are

phenotypically silent.

By adopting the principal of Darwinism, simple

genetic algorithms (SGA), can be viewed as imple-

menting the process of evolution without containing

any explicit neutral mutations. In other words, each

mutation is either an advantage or a disadvantage

to the individual in terms of ﬁtness, with selection

then propagating the ﬁtter individuals. As the search

progresses, the exploration and exploitation ratio de-

creases as the population converges. If we are to im-

plement a genetic algorithm (GA) based on the prin-

ciples of neutral theory then neutrality needs to be

introduced. Neutrality can be viewed as a situation

where a number of different genotypes can represent

the same phenotype. Traditionally, GAs are evaluated

and compared in relation to performance measures.

In this paper, in addition to considering performance,

the authors examine the population dynamics associ-

ated with a SGA and a multi-layered GA (MGA). The

motivation is to develop a tunable, synonymous, non-

trivial GA representation which incorporates neutral-

ity, in order to gain an understanding of the effects of

neutrality on population dynamics. The contributions

are as follows: an analysis of the impact of neutrality

on population evolution; an examination of the im-

pact of neutrality on population variation; and ﬁnally

an illustration of the impact of neutrality on pheno-

typic variability.

The paper is laid out as follows: Section 2 gives a

brief background to Neutral theory and the use ofneu-

trality in GAs. Section 3 outlines the MGA used in

the paper, while Section 4 describes the experiments

undertaken. Section 5 outlines and analyses the re-

sults, Section 6 concludes and Section 7 outlines fu-

ture work.

2 BACKGROUND

Neutral Theory as discussed by Kimura (Kimura,

1968), argues that mutation, not selection, is the main

force in evolution. He describes how a mutation from

one gene to another can be viewed as being neutral

if it does not affect the phenotype, as the number of

different genotypes which store genetic information

is far greater than the number of phenotypes. This

implies that the representation from genotype to phe-

notype must incorporate an element of redundancy

196

Hill, S. and O’Riordan, C..

Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 1: ECTA, pages 196-203

ISBN: 978-989-758-157-1

 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

and neutral mutations are possible. Mutations can

be viewed as neutral if they change the genotype but

don’t impact on the phenotype.

Generally, neutrality can be viewed as a situation

where the size of the search space is increased, with-

out an equivalent increase in the solution space. This

results in a situation where a mutated individual, at

the genotypic level, can still represent the same phe-

notype. Neutrality should be beneﬁcial when neutral-

ity changes the search bias in order to improve the

probability of locating the global optimum.

Although originally viewed as being anti-

Darwinian, Kimura (Kimura, 1983) stated that al-

though natural selection is important in evolution, the

number of DNA changes which are adapted in evo-

lution are small, with the vast majority of mutations

being phenotypically silent. Following Kimura, work

by King and Dukes (King and Dukes, 1969) describes

how much of the evolution of proteins is down to neu-

tral mutations and genetic drift. A number of studies

focused on neutral theory (Schuster et al., 1994; Huy-

nen et al., 1996; Huynen, 1996; Schuster, 1997) etc.

and illustrated that by introducing redundant repre-

sentation and thus, neutral mutation, the connectivity

in ﬁtness landscapes can be altered. In other words,

when a number of genotypes represent the same phe-

notype, they can be viewed as a neutral set and in turn

alter the way in which a population explore the search

space.

Previous work on GA genotype-phenotype map-

pings which introduce neutrality, has produced results

indicating the neutrality may prove useful in chang-

ing environments and over more difﬁcult landscapes.

Ebner et al. (Ebner et al., 2001) outlined how high

levels of mutation could be sustained by having neu-

tral networks present. They also identiﬁed that neutral

networks assist in maintaining diversity in the popula-

tion, which may be advantageous in a changing envi-

ronment. Similar ﬁndings were found in (Grefenstette

and Cobb, 1993). Toussaint and Igel (Toussaint and

Igel, 2002) argued that approaches to self-adaption in

evolutionary algorithms can be viewed as an example

of the beneﬁts of neutrality, because with non-trivial

neutrality, different genotypes which are part of the

same neutral set may have different phenotypic distri-

butions.

3 MULTI-LAYERED GENETIC

ALGORITHM (MGA)

The concepts of Variation and Variability need to be

differentiated. Variation can be described as the dif-

ference between individuals in a population and can

be seen as relating to a collection. Variability, on

the other hand, can be described as the leaning to

vary and the “variability of a phenotypic trait de-

scribes the way in which it changes in response to

environmental and genetic inﬂuences” (Wagner and

Altenberg, 1996). When exploring the phenotypic

space, it is critical to gain an understanding of the

variational topology in trying to determine the shape

of the landscape (Toussaint, 2003b). Many evolu-

tionary algorithms are created using a ﬁxed variation

topology, in other words you don’t need to track the

neutrality to explain the evolution trajectory. How-

ever, in nature, phenotypic variation landscapes are

not ﬁxed. These non-ﬁxed phenotypic variation land-

scapes can be referred to as non-trivial in terms of

their genotype-phenotype map (Toussaint, 2003b). A

non-trivial genotype-phenotype map can be viewed

as having the following characteristics: ﬁrstly, a phe-

notype can be encoded by many genotypes and sec-

ondly, the phenotypic variability of a number of phe-

notypes will depend on their genotypes (Toussaint,

2003a).

The primary inspiration for the multi-layered GA

can be found in the biological processes of transcrip-

tion and translation. At a very basic level, the bio-

logical process of transcription involves the copying

of information stored in DNA into an RNA molecule,

which is complementary to one strand of the DNA.

The process of translation then converts the RNA, us-

ing a predeﬁned translation table, to manufacture pro-

teins by joining amino acids. These proteins can be

viewed as a manifestation of the genetic code con-

tained within DNA and act as organic catalysts in

anatomy. The MGA includes a layered genotype-

phenotype map which adopts a basic interpretation

of the transcription and translation processes and al-

lows for the implementation of a missense mutation

operator. The genotype search space is represented

by Φ

where Φ

= {0, 1}

and l is the genotype

length. The transcription phase of the MGA maps

the binary genotype to the DNA search space Φ

where Φ

= {A,C, G,T}

l/2

, with the following map-

pings: 00 → A; 01 → C; 10 → G and 11 → T. Fol-

lowing this, a bijective mapping takes place, map-

ping the DNA space to an RNA space Φ

, where

= {A,C,G,U}

l/2

. U is included for biological

plausibility and has no impact on the evolution un-

less we include operators at this level. Following tran-

scription, the translation phase takes place, mapping

→ Φ

, where Φ

represents the phenotype space

and Φ

= {0,1}

l/c

, where c is the cardinality chosen

at initialisation to create a translation table. The level

of redundancy is determined by c and in this paper

c = 6 (see Figure 1), and implies |Φ

| > |Φ

| where

Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

197

c > 1. Missense mutation in nature is carried out at the

RNA level. In relation to the MGA, the Missense mu-

tation mapping is as follows: A → U, C → G, G → A

and U → C. To summarise the variation operators,

one-point crossover and single-point mutation occur

at the genotype level prior to transcription and mis-

sense mutation takes place before translation.

The MGA introduces a tunable multi-layered GP-

map, which allows a haploid GA to exhibit, some of

the characteristics normally associated with a diploid.

That is a mechanism for allowing alleles or combina-

tions of alleles which proved useful in previous gen-

erations (Goldberg and Smith, 1987) and thus, main-

taining a form of long term memory without the need

to develop a dominance scheme. The MGA popu-

lation consists of a population of haploid individu-

als, which allows for the use of traditional crossover

and mutation variation operators on the genotype.

This differs from the approach used by diploid GAs

(DGAs) i.e. (Yang, 2006), where each individual

has two chromosomes and crossover is divided into

two steps and mutation is viewed as being neutral.

Another difference between the MGA mapping and

that of a DGA, is that in the DGA, a phenotype al-

lele is made up from a single genotype allele which

is expressed. In the MGA, a single phenotype al-

lele is made from the cardinality incorporated in the

genotype, in this paper we use 6-bits. Although the

MGA’s GP-map is non-deterministic, the approach

differs from that of real-coded binary representation,

which incorporate a gene-strength adjustment mech-

anism (Kubalik, 2005). Real-coded binary represen-

tations can use standard crossover operators, but mu-

tation is implicit due to the gene-strength adjustment

mechanism (Kubalik, 2005).

Figure 1: 6-bit MGA Representation Mapping.

4 EXPERIMENTATION

Deception is often used in testing GAs and im-

plies that the search strategy can be misled (Whitley,

1991a). As noted in (Morrison and DeJong, 2002),

diversity is critical for GAs, particularly when the

landscape is evolving as recombining a homogeneous

population will not enable the GA to locate the new

optimum. Hamming difference is used as a measure

of diversity both for the genotypic and phenotypic

diversity. In order to examine the population evo-

lution for both the SGA and the MGA, experiments

were conducted over a 4-bit fully deceptive landscape

(Whitley, 1991b) which is reversed after generation

50, allowing the local optimum to become the global

optimum.

-2

-1

-2

-1

Fitness

4-bit Deceptive Landscape

Fitness

Figure 2: 4-bit Deceptive Landscape.

Figure 2 graphically illustrates the landscape of the

4-bit deceptive problem (Whitley, 1991b), with the x

and y co-ordinates indicating the location on the grid.

To analyse the adaptive qualities of both GAs, the

landscape reverses after generation 50, Figure 3, illus-

trates the landscape after the change, where the global

optimum becomes the local optimum and visa-versa.

-2

-1

-2

-1

Fitness

Reversed 4-bit Deceptive Landscape

Fitness

Figure 3: Reversed 4-bit Deceptive Landscape.

5 RESULTS

The analysis that follows is broken into three parts:

the impact of neutrality on population evolution; the

impact of neutrality on variation and the impact of

neutrality on phenotypic variability. However, the re-

sults begin by comparing both GAs in a conventional

manner based on performance. Figure 4, illustrates

the off-line (averaged best ﬁtness) and on-line (av-

eraged ﬁtness) performance for both the SGA and

the MGA. The results indicate that the changing 4-

bit deceptive landscape initially proved equally easy

for both the SGA and the MGA. However, after the

landscape changes, the SGA becomes trapped on the

local optimum, while the MGA succeeds in locating

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

198

the global optimum. A Wilcoxon Rank sum test in-

dicates that the results shown in Figure 4, that is both

the off-line and on-line comparison between the SGA

and MGA, were statistically signiﬁcant. The remain-

ing analysis examines the evolution of both popula-

tions, in an attempt to gain a better understanding of

the impact on neutrality on population dynamics over

the course of an evolutionary time period and to illus-

trate the performance of both population in a dynamic

environment.

20 40 60 80 100 120 140 160 180 200

Fitness

Generations

SGA & MGA 4-bit Deceptive Problem - Online/Offline Performance Analysis

MGA Off-line Performance

MGA On-line Performance

SGA Off-line Performance

SGA On-line Performance

Figure 4: SGA & MGA On-line/Off-line Performance.

5.1 Neutrality & Population Evolution

5.1.1 Analysis Before the Landscape Change

In relation to the evolution of the SGA’s population,

Figure 5 gives an overview of the population distri-

bution over the landscape at generation 0, with the

initial population of 20 individuals randomly spread

over the landscape. Figure 6 shows the MGA popu-

lation distribution over the problem landscape. The

initial randomly generated population distribution for

the MGA is quite similar to that of the SGA.

-2

-1

-2

-1

Fitness

4-bit Reversed Deceptive Landscape SGA Population Generation 0

Fitness

Figure 5: SGA - Fitness Landscape Generation 0.

Examining the populations for both GAs at gen-

eration 50, which is the last generation before the

landscape changes, we see that the SGA’s popula-

tion has converged, apart from the impact of mutation

(Figure 7). The MGA’s population has also located

-2

-1

-2

-1

Fitness

4-bit Deceptive Landscape MGA Population Generation 0

Fitness

Figure 6: MGA - Fitness Landscape Generation 0.

-2

-1

-2

-1

Fitness

4-bit Deceptive Landscape SGA Population Generation 50

Fitness

Figure 7: SGA - Fitness Landscape Generation 50.

-2

-1

-2

-1

Fitness

4-bit Deceptive Landscape MGA Population Generation 50

Fitness

Figure 8: MGA - Fitness Landscape Generation 50.

the global optimum. However, due to the genotype-

phenotype mapping, the population doesn’t converge

as much as the SGA. Figure 8, shows the many-to-one

representation present in the population, with differ-

ent genotypes having the same ﬁtness represented by

different colours and shapes.

The MGA population’s evolutionary trajectory

differs considerably by not converging on the global

optimum. Therefore the population consists of a num-

ber of neutral networks, which is a result of gene ﬂow

due to the presence of neutrality.

5.1.2 Analysis After the Landscape Change

Figure 9, illustrates the SGA population distribution

when the landscape changes at generation 51 and

shows the population, which now converges on the

local optimum as the landscape has reversed in rela-

tion to the ﬁtness function. It also indicates all but

one member of the population are located on the local

optimum.

The MGA population can be seen distributed on

Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

199

-2

-1

-2

-1

Fitness

4-bit Reversed Deceptive Landscape SGA Population Generation 51

Fitness

Figure 9: SGA - Fitness Landscape Generation 51.

-2

-1

-2

-1

Fitness

4-bit Reversed Deceptive Landscape MGA Population Generation 51

Fitness

Figure 10: MGA - Fitness Landscape Generation 51.

the problem landscape in Figure 10, with the bulk of

the population located on the local optimum. The ﬁg-

ure also indicates that due to the presence of neutrality

in the representation, the population is dispersed over

a wider number of ﬁtness plateaus.

Examining the population evolution through to

generation 200 for both GAs. The SGA population

remains trapped on the local optimum (see Figure 11),

while the MGA population remains clustered around

the global optimum, see Figure 12. This appears to in-

dicate that the MGA, through the genotype-phenotype

mapping, implicitly maintains a level of genetic di-

versity within the population and is resistant to con-

vergence, thereby offering the ability to adapt in a dy-

namic environment. The impact of neutrality on pop-

ulation evolution is described in Sub-section 5.1 and

illustrated the population distribution for the SGA and

MGA, both before and after the landscape changed.

5.2 Neutrality & Variation

As a traditional SGA maps directly from the genotype

to the phenotype, both the genotypic search space

and the phenotypic search space are identical. With

the MGA mapping, neutrality increases the geno-

typic search space in comparison to the phenotypic

search space. Using normalised Hamming distances

between individuals, Figure 13 and Figure 14, il-

lustrates the population diversity for both the SGA

search spaces and the MGA genotypic search spaces,

respectively, at generation 0. From the ﬁgures, it ap-

pears that at generation 0, there is quite a large level

-2

-1

-2

-1

Fitness

4-bit Reversed Deceptive Landscape SGA Population Generation 200

Fitness

Figure 11: SGA - Fitness Landscape Generation 200.

-2

-1

-2

-1

Fitness

4-bit Reversed Deceptive Landscape MGA Population Generation 200

Fitness

Figure 12: MGA - Fitness Landscape Generation 200.

0.2

0.4

0.6

0.8

Hamming Distance

SGA Generation 0 - Population Hamming Distance

Genotypes/Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 13: SGA Genotypic/Phenotypic Diversity Genera-

tion 0.

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 0 - Genotype Population Hamming Distance

Genotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 14: MGA Genotypic Diversity - Generation 0.

of diversity present in both populations. This can

be explained as evolution has not yet begun and the

populations have been randomly generated. Also, as

neutrality is introduced the MGA’s genotypic search

space increases, while the MGA’s phenotypic space

(Figure 15) is quite similar to that of the SGA.

As the populations evolve to generation 50, con-

vergence has occurred in the SGA (apart from the in-

ﬂuence of mutation) and the population is shown in

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

200

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 0 - Phenotype Population Hamming Distance

Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 15: MGA Phenotypic Diversity - Generation 0.

0.2

0.4

0.6

0.8

Hamming Distance

SGA Generation 50 - Population Hamming Distance

Genotypes/Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 16: SGA Genotypic/Phenotypic Diversity Genera-

tion 50.

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 50 - Genotype Population Hamming Distance

Genotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 17: MGA Genotypic Diversity - Generation 50.

Figure 16. With the MGA population, genetic diver-

sity is maintained within the genotypic search space,

see Figure 17. The MGA phenotypic search space

(Figure 18) illustrates the diversity at a phenotypic

level.

At generation 200, the SGA population remains

on the local optimum and contains little diversity to

allow it to escape. This situation is shown in Fig-

ure 19, reﬂecting the lack of diversity within the pop-

ulation when viewed as a function of Hamming dis-

tance.

The MGA genotypic population diversity at gen-

eration 200, is shown in Figure 20 and the pheno-

typic diversity is illustrated in Figure 21. The level of

phenotypic diversity present, relates to the many-to-

one genotype-phenotype mapping and to individuals

of the population being part of other neutral networks.

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 50 - Phenotype Population Hamming Distance

Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 18: MGA Phenotypic Diversity - Generation 50.

0.2

0.4

0.6

0.8

Hamming Distance

SGA Generation 200 - Population Hamming Distance

Genotypes/Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 19: SGA Genotypic/Phenotypic Diversity Genera-

tion 200.

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 200 - Genotype Population Hamming Distance

Genotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 20: MGA Genotypic Diversity - Generation 200.

0.2

0.4

0.6

0.8

Hamming Distance

MGA Generation 200 - Phenotype Population Hamming Distance

Phenotypes

Hamming Distance

0.2

0.4

0.6

0.8

Figure 21: MGA Phenotypic Diversity - Generation 200.

5.3 Neutrality & Phenotypic Variability

The ﬁnal examination of the effects of neutrality on

the evolutionary trajectory of the MGA population re-

lates to the impact of mutation. In order to examine

the effect of single-bit mutation and missense muta-

tion, we randomly selected an individual from both

Examining the Impact of Neutral Theory on Genetic Algorithm Population Evolution

201

the SGA and MGA populations from various gen-

erations. We then ﬂipped all of the bits, one at a

time, in sequence and measured the impact on ﬁtness.

The aim of this approach is to examine how the pres-

ence of neutrality impacts on mutation in GAs which

have synonymous representations. A representation

is viewed as synonymously redundant if the geno-

types representing the same phenotype have the same

properties and are next to one another in the muta-

tion space. With the MGA population, a genotype,

when mutated, can produce either a silent or an adap-

tive single-bit mutation and a silent or adaptive mis-

sense mutation.

Figure 22 shows the impact of mutation in an SGA

population. Figure 23 visualises the impact of single-

bit mutation on an individual from the MGA popu-

lation. The phenotypic distribution for both GAs is

similar as the neutrality associated with the MGA rep-

resentation is synonymous.

-2

-1

-2

-1

Fitness

Genotype 2 - 1111 - One Bit Mutation Neighbours - Generation 0

Fitness

Figure 22: SGA 1-Point Mutated Individual.

-2

-1

-2

-1

Fitness

MGA Genotype 3 - 001100101000110110001000 - One Bit Mutation Neighbours - Generation 50

Fitness

Figure 23: MGA 1-Point Mutated Individual.

The MGA representation includes missense muta-

tion which operates within the layers of the genotype-

phenotype mapping and allows the phenotypic vari-

ability to differ for individuals having the same level

of ﬁtness. Figure 24 illustrates the phenotypic distri-

bution for individual 010000011000010000111000,

showing that the phenotypic distribution for the MGA

is greater than that of the SGA, as individuals are

occupying more ﬁtness plateaus. Figure 25, il-

lustrates the phenotypic distribution for individual

100010011110010000001010, which resides on the

same neutral network with a ﬁtness level of 30. They

-2

-1

-2

-1

Fitness

MGA Genotype - 010000011000010000111000 - Fitness 30 - Bit Mutation & Missense Mutation

Fitness

Figure 24: MGA 1-Point & Missense Mutated Individual.

-2

-1

-2

-1

Fitness

MGA Genotype - 100010011110010000001010 - Fitness 30 - Bit Mutation & Missense Mutation

Fitness

Figure 25: MGA 1-Point & Missense Mutated Individual.

show that the MGA population allows individuals re-

siding on the same ﬁtness plateau, have different phe-

notypic distributions. This section highlighted the ef-

fect of mutation, with results illustrating that the SGA

can only access local plateaus, while the MGA, which

can be phenotypically silent, has greater variability.

6 CONCLUSION

A population’s ability to survive in dynamic environ-

ments often depends on a level of diversity to be main-

tained within the population. As a GA search involves

a mapping between the genotype and the phenotype,

a SGA, through it’s one-to-one genotype-phenotype

mapping, quickly eliminates diversity from the popu-

lation through its selection policy and low mutation

rates. The results presented, illustrate that through

the implementation of Neutral theory, as proposed

by Kimura (Kimura, 1968), the genotype-phenotype

mapping of the MGA allows for a tunable, non-trivial,

many-to-one relationship. The contribution of this

form of mapping is the implicit maintenance of re-

lated genetic diversity within the population, which

allows the occupation by the population, of a greater

number of ﬁtness plateaus. By adopting this approach

convergence at a phenotypic level can be achieved,

but genetic diversity is maintained at a genotypic

level. Neutral theory (Kimura, 1968), would suggest

that where genetic changes spread across a popula-

tion, changes may or may not alter the phenotype and

are a result of genetic drift. The results indicated

that neutrality, as introduced by the MGA mapping,

ECTA 2015 - 7th International Conference on Evolutionary Computation Theory and Applications

202

impacts on the search space by increasing its size

and population variation. Finally, the results showed

that adaptive mutations operate in a similar manner to

SGA mutations, but allow a greater number of ﬁtness

plateaus to be reached. Silent mutations, on the other

hand, offer the ability to search the landscape without

impacting on ﬁtness. This increases the size of the

genetic pool and impacts on gene ﬂow.

7 FUTURE WORK

Future work includes comparing the performance of

the MGA with that of other diversity maintenance

techniques over a number of problem landscapes.

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