F’roc. 1996 Australian New Zealand Conf. on Intelligent Information Systems, 18-20 November 1996, Adelaide, Australia Editors, Naraslmhan and Jam

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Evolutionary Computation and The Principle of Natural Selection

Pierre A. I. Wijkman

Stockholm University and Royal Institute of Technology Department of Computer and Systems Sciences

Electrum 230, 164 40 Kista, Sweden E-mail: pierre@dsv.su.se

Abstract We present an alternative model in the field of evo- lutionary computation. The presented model is, like other models in evolutionary computation, based on the principle of natural selection. The difference between the presented model and the other models is a difference in the interpretation of the principle of natural selection. Traditional models in evolutionary computation provides only a partial interpretation of the principle of natural selection, while the presented model is based on a more complete interpretation, As a consequence, the presented model can deal with the problem of local optima in a novel way.

1 Introduction Nature is extremely superior to man in designing systems of high complexity. The process in nature that is responsible for this design process is called evolution. Two fields that both are closely related to evolution are evolutionary biology and evolutionary computation. Evolutionary biology is a field that from observations in nature build models of evolu- tion. Evolutionary biology tries, in short, to achieve a better understanding of evolution. Evolutionary computation is a field that as a first step tries to inter- pret the models from evolutionary biology into more abstract forms and as a second step tries to apply these abstract models in the construction of artificial, or man made, systems. Evolutionary computation tries, in short, to apply the design principles of evo- lution. Evolutionary computation is accordingly directly dependent on the results from evolutionary biology. The results of evolutionary computation can, in tum, give valuable information back to evolu- tionary biology. Researchers within evolutionary biology can use evolutionary computation to test their models in simulations with artificial environ- ments and artificial systems. In spite of the fact that the two fields of evolutionary biology and evolution- ary computation are very closely related they are cur- rently treated as two separate fields, especially from the viewpoint of the biologists Perhaps will the two fields become united as computers become faster and more realistic models in evolutionary biology can be simulated and tested

2 Evolutionary Computation This paper belongs to the area of evolutionary com- putation. As a first step, we present a new interpreca- tion of the in evolutionary biology main principle of natural selection. As a second step, we apply the resulting abstract model in the solution of a number of problems. Before we give a more specific defini- tion of the field of evolutionary computation we need to say some words of the concept of system. A sys- tem (see [B79] for a more formal definition and [GLSIZ] for a introduction to system analysis) has two fundamental and complementary aspects: its form and its funcfionalily. The system’s form refers to the system’s static properties, i.e. its composition and structure. The system’s functionality refers to the system’s dynamic properties, i.e. its behaviour. A specific system form gives rise to a specific system functionality, while a specific system functionality can be implemented by many different system forms.

Given a priori to a model in evolutionary compu- tation is a general form speclJication and a general functionality speczjkation. The evolutionary model should then, as efficient as possible, find a specific type of system form that have a specific type of sys- tem functionality, where the found system form and associated system functionality should be consistent with the given general form and functionality speci- fications. This is often a very difficult problem. The reason for this is that (1) the number of possible system forms that can be constructed from a given form specification often is very large and (2) the relation between system forms and system hnc - tionalities often is very complex. The number of systems forms often grows exponentially with the size of the structure, i.e. with the number of connec- tions used. The functionality of a system often can change very drastically when a small change is made in the system form This means, in search jargon, that the search space is very large and that the search topography is non monotonical

Currently there exist a wide variety of models in evolutionary computation. They share a common conceptual base of simulating evolution of artificial systems via the processes of construction

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F’roc. 19% Australian New Zealand Conf. on Intelligent Informanon Systems, 1620 November 1996, Adelaide, Australia. Editors, Narasimhan and Jain

(reproduction) and destruction (selection) acting on a population of artificial systems.

Construction of systems. The construction proc- ess use information from (1) the a priori given general form description (environment), and (2) from the population of system forms. Systems in the population that is participating in this con- struction process more frequently is said to have a higher reproduction capacity.

0 Destruction of systems. The destruction process use information from (1) the a priori given gen- eral fixnctionality description (environment), and (2). the population of system functionalities. Sys- tems in the population that is participating in this destruction process more frequently is said to have a lower survival capacity.

By repeatedly applying construction (reproduction) and destruction (selection) to a population of sys- tems, a model in evolutionary computation tries to get a population of systems where each system has a higher and higher survival Capacity (note: not fitness value which is determined by a system’s reproduc- tion capacity and survival capacity).

IniPopOfFor(ForSpe) Do (1 and 2 in parallel) in PopOfFor

1. ConFor(F0rSpe) 2. DesFor(FunSpe)

Loop Until Ready

Figure 1. General algorithm.

The general algorithm of models in evolutionary computation is shown in figure 1 . First a population of forms is initialised. In a loop that continues until some termination condition is fulfilled the following happens in parallel in the population of forms: (1) Construction of forms. The construction process operates on the form of systems. The construction process is accompanied by some alteration process. If the model use one single parent form then we call the alteration process ” mutation” and the actual con- struction of a child form ”asexual reproduction”. If the model use two unique parent forms then we call the alteration process ” recombination” and the ac- tual construction of a child form ”sexual reproduc- tion”. Often models use a combination of the two alteration processes. (2) Destruction of forms. The destruction process operates on the functionalities of systems. The destruction process is accompanied by some validation process. If the model uses a single system in the validation, then the validation process is similar to a fixed environmental background pres- sure. If the model uses more than one system in the validation? then the validation process is similar to a

coevolutionary arms race. Often models uses a com- bination of the two validation processes.

3 Traditional DARWIN Models Evolutionary computation consist currently of three, more or less different, sub fields called evolutionary progamming [F91] [F92a] [F92b] [FOW66], evolution strategies [R65] [R73] [S65] [SSl], and genetic algo- rithms [H62] [H75] [H86]. We follow the tradition and refer to all these approaches by the name evolution- ary computation (in our opinion artificial evolution would be a better name). The three traditional models in evolutionary computation are summarised in table 1.

Evolutionary Programming Evolution Strategies

Genetic Algo- rithms

Selection of Selection of Darents I ”deads -‘ All I Lowest survival

1 capacity Random 1 Version 1:

Parents Version 2: Lowest survival

I capacity Highest sur- I Parents viva1 capacity I

Table 1. Summary of models.

Every model use different methods of alteration in the creation of a child form from the selected parent form@). Evolutionary programming uses only one parent and mutation. Evolution strategies uses two parents, mutation as the main alteration operator and recombination as an operator to adapt the mutation degree. Genetic algorithms use two parents, recom- bination as the main alteration operator and mutation as an operator to keep the population diversified. It is a remarkable fact that each model emphasises differ- ent features as being most important to a successful search process. However, since they all select an individual randomly or with the highest survival capacity in the population to be a parent, they are, in their most general outline, based on the following version of the principle of natural selection:

”The differential survival of classes of entities that differ in one or more hereditary characteris- tics: the difference in survival is not due to chance, and it must have the potential conse- quence of altering the proportions of the different entities, to constitute natural selection.”

This view of natural selection was first introduced by Darwin himself [C93] [R89] and has since then been redefined (see below). We will call models based on Darwin’s original definition of the principle of natu- ral selection for traditional, or DARWIN models.

hoc 1996 Australian New Zealand Conf on Intelligent Information Systems, 18-20 November 1996, Adelaide, Australia. Editors, Namstmhan and Jain

These models focuses accordingly solely on the con- cept of differential survival of classes of entities. For example, in their view, the fitness of two individuals A and B is not equal if the following two facts holds.

1. The individual A has twice the survival capacity than the individual B.

2 . The individual A has half the reproduction ca- pacity than the individual B.

In their view, if an individual A has a higher survival capacity than another individual B, then it is regarded as more fit independent of what their respective re- production capacities are.

high F------- rep. cap.

low 1 , low sur cap. high

Figure 2. Survival and reproduction capacity.

Figure 3. Survival capacity and fitness.

Figure 2 shows the relation between the survival capacity and the reproduction capacity in this case when the reproduction capacity is considered to be equal between individuals of different survival ca- pacities. Figure 3 shows the, from this assumption, resulting relation between the survival capacity and the fitness. In this case, the individuals with the high- est survival capacities will be associated to the high- est fitness.

4 The FISHER Model In this section we present an altemative to the DAR- WIN models called FISHER. The representation that F I S H E R use is the general concept of a system. F I S H E R can be used as an optimiser, although its main purpose is to construct artificial systems of high complexity.

PopOfFor = IniFor(ForSpe) PopOfSC = EvalSurCap(PopOfFor, FunSpe) Do ParFor = SelPar(PopOfFor, PopOfSurCap) ParSC = EvaSC(ParFor, FunSpe) ChiFor = Mut(ParFor, ForSpe) ChiSC = EvaSC (ChiFor, FunSpe) DeaFor = SelDea (PopOf For, PopOf S C )

DeaSC = EvaSC(DeaFor, FunSpe) PopOfFor = PopOfFor + ChiFor - DeaFor PopOfSC = PopOfFor + ChiSC - DeaSC Loop Until Ready

Figure 5. General algorithm of FISHER.

The general algorithm for FISHER is shown in fig- ure 5. First a population of forms is initialised. Then the survival capacity of the individuals in this popu- lation of forms is evaluated. In a loop that continues until some termination condition is fulfilled the fol- lowing happens: (1) One parent form is selected ac- cording to a probabilistic function based on relative survival capacity. Individuals with lower relative survival capacities are more likely to be selected as parents. (2) One child form is created from the parent form by a fixed low degree of mutation. The form of mutation is based on the representation used, and is not adaptive. (3) The survival capacity of the child form is evaluated. (4) The individual with the lowest survival capacity is removed from the population.

Since FISHER select an individual with lower survival capacity in the population to be a parent with higher probability, F I S H E R is, in its most gen- eral outline, based on the following version of the principle of natural selection:

”The differential survival and/or reproduction of classes of entities that differ in one or more he- reditary characteristics: the difference in survrval andor reproduction is not due to chance, and it must have the potential consequence of altering the proportions of the different entities, to consti- tute natural selection.” [F86]

This view of the principle of natural selection, that now is the standard in the field of evolutionary biol- ogy, is a bold redefinition of Darwin’s original prin- ciple of natural selection made by the British statis- tician and geneticist R. A. Fisher IC931 [R89]. F I S H E R considers accordingly both differential survival and differential reproduction capacities. For example, in this more complete interpretation, the fitness of two individuals A and B is equal if the following two facts holds:

1. The individual A has twice the survival capacity than the individual B.

2. The individual A has half the reproduction ca- pacity than the individual B.

Roc. 1996 Australian New Zealand Conf. on Intelligent Information Systems, 18-20 November 1996, Adelaide, Australia. Editors, Nmimhan and Jain

In this more complete interpretation, if an individual A has a higher survival capacity than another indi- vidual B, then it is not regarded as more fit inde- pendent of what their respective reproduction ca- pacities are.

c

Figure 6 . Survival and reproduction capacity.

high

fit

low\ low sur. cap. high

Figure 7. Survival capacity and fitness.

Figure 6 shows the relation between the survival capacity and the reproduction capacity in this case when the reproduction capacity is considered to be different between individuals of different survival ca- pacities. Figure 6 shows a more realistic situation than figure 2 where individuals with lower survival capacity has a higher reproduction capacity. A high reproduction capacity can compensate for a low survival capacity and a high survival capacity can compensate for a low reproduction capacity. Figure 7 shows the, from this assumption, resulting relation between the survival capacity and the fitness. Here will not, in contrast to figure 3, the individuals with the highest survival capacities be associated to the highest fitness. The fitness of different individuals will be the same independent of the to each individ- ual associated survival capacity.

5 Difference Between DARWIN and FISHER

Given a general form description, we can order the systems in the population so that systems more simi- lar to each other, in relation to the alteration method, is closer to each other into a multidimensional space. Associated to every system in this space is a value reflecting the systems survival capacity. Assume that we plot the survival capacity function in this space. We will call the resulting topography the survival landscape (note: we will not call it the fitness land- scape, which is flat). In this new terminology, the goal for the models in evolutionary computation is to

find, in an efficient manner (in time and space re- sources), a high peak in the survival landscape. A local optima in this survival landscape is a point P where the following holds:

1. There exist higher points than point P in the sur- vival landscape.

2. There exist no path (by a series of minimal al- terations) from this point P to a higher point Q in the survival landscape that never goes to a point lower than the point P.

If all the systems in the population are associated to local optima points in the survival landscape then there is two ways to deal with this problem:

1.

2.

In

Make a larger degree of change. With a larger degree of change, applied to some parent system form, a larger number of systems in the survival landscape can be reached. The number of systems that can be reached grows very fast (expo- nentially) when the degree of change increases.

Make a change to the same system form repeat- edly, evaluating the system between changes. With a repeated change, to the same system form, a larger number of systems in the survival land- scape can be reached. The number of systems that can be reached grows very fast (exponentially) when the number of repetitions increases.

method one, the search quickly becomes more random as the degree of change increases. In method two, the search quickly becomes more random as the number of repetitions increases And random search is a synonym for inefficient search. The difference between method one and method two above is that method two repeatedly explores the more close points next to the point A before exploring the points that are further away from the point A, while method one explores any of the reachable points point indis- criminantly without consideration of closeness or order. This means that the search does not become random in the same way for method two as with method one. The resulting search process is in method two, in other words, more efficient.

A method similar to method one is to use a larger degree of change that is biased towards a small de- gree of change. This method is more similar to method two. There is however still a difference be- tween the methods: method two evaluates the system being changed between the changes while method one does not. In method one there is also the question of where the bias originates from.

At best DARWIN models, using the random parent selection procedure, the probability for using the same system repeatedly r times in a population of size n is equal to n-r. The probability for using the

F’rw 1996 Australian New Zealand Conf. on Intelligent Information Systems, 18-20 November 1996, Adelaide, Australia. mtors, Naraslmhan and Jam

same system repeatedly is accordingly decreasing exponentially with the number of repetitions. This means that DARWIN models not can use method two to resolve the problem of local optima. They must instead use the, less efficient, method one.

In the extreme case, when FISHER models se- lects the system with the lowest survival capacity as parent and selects the system with the lowest survival capacity as ”dead”, the probability for using the same system repeatedly, until a system with higher survival capacity is found, is equal to 1. This means that FISHER models can use method two in order to resolve the problem of local optima. FISHER models will, in other words, be more efficient than the DAR- W I N models.

6 An Illustrating Example In this section we present an illustrating example which shows that F ISHER can solve the problem of local optima in a novel way, without increasing the degree of random change. In this example we test FISHER on a highly non monotonic search topogra- phy defined by the function where f(x) = -~Sinjxl”~ where the task is to find the global minima in the interval (-500, 500) (see figure 8).

-400 -200 0 200 400 Figure 8. Graph of the function f(x) = -xSinjxl”2.

We have repeated the simulations 25 times and each simulation have been run for 10 000 generations. Figure 9 show the results of these simulations. From left to right and from top to bottom FISHER models with increasing bias towards selecting an individual with low survival capacity as parent are specified. Figure 9 shows the generation on the x-axis and the average survival capacity in the population on the y- axis. We see that the following holds: The larger the bias is towards selecting an individual with low sur- vival capacity as parent, the more capable is the model to handle the problem of local optima.

I tm “;y I i

Figure 9. Results from illustrating example.

These simulations show that F I S H E R models can handle non monotonic search topographies if the parent selection bias is set in an appropriate way.

7 The Travelling Salesman Problem The travelling salesman problem is hard to solve but simple to describe:

Given a set of n nodes and distances for each pair of nodes, find a round-trip of minimal total length visiting each node exactly once. The distance from node i to node j is the same as from node j to node i.

Given a set of n nodes, the number of possible per- mutations is n!. An exhaustive search for the optimal round-trip is accordingly impossible, even for mod- erately sized values of n. The problem is extremely multi-peaked and at many levels. Such topography arises because there are many different approaches to visiting the nodes, both within any cluster and also for travels between the clusters. Once a fairly good round-trip has been devised all simple ”cut and paste” operations of segments of round-trips are most likely to be harmful even though they may suggest, to an intelligent viewer, dramatically im- proved round-trips that could follow. The difficulty of the travelling salesman problem provides a chal- lenging test bed for the models in evolutionary com- putation. A concise review of the application of models in evolutionary computation to the travelling salesman problem can be found in Michalewicz [M92].

We present an example that compares a number of DARWIN models with a number of F I S H E R models on the TSP problem. In this comparison we consider models with one parent. We have repeated the simulations 25 times and each simulation have been run for 10 000 generations. We have used 20 nodes, which means that the number of different round-trips is equal to 20!. Figure 10 show the result of this example. On the top row traditional models of the type that select the individual with the highest survival capacity as parent are specified. These five

Roc. 1996 Australian New Zealand Conf. on Intelligent Information Systems, 18-20 November 1996, Adelaide, Australia. Editors, Narasimhan and Jain

models have a degree of random change (mutation) that increases from left to right (from one to five mutations). On the middle row traditional models of the type that select the individual with a random survival capacity as parent are specified. These five models have a degree of random change (mutation) that increases from left to right (from one to five mutations). On the bottom row, left to right, FISHER models with increasing bias towards selecting an individual with low survival capacity as parent are specified. Figure 6.10 shows the generation on the x- axis and the average survival capacity in the popula- tion on the y-axis. We see that the following holds: FISHER models performs better than some DARWIN models when FISHER is using a minimal (linear) bias towards selecting an individual with lower sur- vival capacity as parent and DARWIN models use a minimal degree of random change. In the other cases a comparison is difficult to make.

Figure 10. Results from TSP problem.

These simulations show that, when using one parent, FISHER models can be better than some DARWIN models in evolutionary computation on a difficult problem.

8 Conclusion Our work should be viewed as a first step in a new direction. We have not intended to show that FISHER models in some way is superior to any of the traditional models of evolutionary computation: we have only showed that there exist an alternative route at a very fundamental level. We do not yet know where this route will lead and if it is fruitful. We hope that the results of this paper has taken us one step closer towards the twofold goal of under- standing the design principles of nature and the abil- ity to use these principles in the design of artificial systems.

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