Variations on Genetic Algorithms

Manuel de la Herrán Gascón

At work

Choosing mutations

For example, you have strings of 3 variables of 8-bit each one 10001000-10001000-10001000

It's a good idea to use binary mutations because in this way you can go from each value to all others (if you have lucky) only in 8 steps, and you can move from one to another in 2 ways:
-slowly, mutation the right bits, or
-faster, mutation the left ones.

If you choose to mutate ALL the variable (to choose randomly another complete 8-bit number) you can move faster to wherever you want (in one step) but not slow, and it is good to move slow when you are near the solution.

If you choose to mutate a variable for example, always "adding 1" circulary, you can move slowly, but not fast, and in some cases you have to make 255 mutations to get the correct value.

Ok. But when you have to recombinate two strings, i think is better to put the crossword point just between two complete variables. If you didn't, you are making a lot of mutations, and in this way you loose the good values that you have find before.

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Building self-adaptation Evolutionary Algorithms

To build self-adaptation Evolutionary Algorithms is a good idea to clasify all of the "variable aspects" or options that we want to give to the algorithm in 4 kinds:

Options of "gen" kind
Here are the things that, like the genes, are different inside only one agent.

Options of "agent" kind
Here are the things or options that are always the same (have the same value) in one agent, but are different from one agent to anothers. This is, we are thinking in give different values of something to each agent.

Options of "island" kind
Here are the things or options that are always the same in one subset of agents. There has to be more than one subset, this is, each subset has more than one agent and less than all.

Options of "all poblation" kind
Here are the things or options that are always the same for all agents.

For example, you have choose a unique mutation rate for all the poblation, but you're want to try a variable and autoconfigurable mutation rate ¿isn't it?

Options of "agent" kind (Note that i begin with the second one)
Ok, then, think that you can put a "mutation rate gen" in each agent, that is, you can add another gen to each agent with a mutation rate value. The agents inherits this gen like others, and the agents with good mutation rate will keep alive this mutation gen in their sons. This is thinking in mutation rate like an option of "agent" kind.

Options of "gen" kind
In other hand, you can choose a different mutation rate for each gen of each agent! This could be a very good idea:

If each gen of your agents represents a value of a variable of your problem, maybe there some "easy to find the value" variables that has values that are "always good values or bad values, with no dependencies with the rest of the string". In this case, i think could be better to has a low mutation rate for this variables.

This is the point of view of the mutation rate like an option of "agent" kind. The options of "gen" kind can be included inside each of the genes. You can think in a 2 dimension matrix of instead of an simple string of genes.

Options of "island" kind
To has a "different kinds of crossover type" is a option of "island" kind, because you have to try it with more than an agent but less than all.

To try Options of "island" kind you can add an special gen to each agent that only can take a little set of values ("A", "B", "C", or "D", for example). Then you establish that only the agents with the same value in that gen are able to reproduce within. An "A" agent only can be with an another "A" to has an "A" child. Then, you can make that the different letters are different methods or options of "island" kind, like crossover type. The islands can not combinate within them, and the islands with bad options (agents one value in this special gen) will have less population in the global soup.

To try more than one option, you can put two ore more letters, one for each option, and only the "AB" can be with another "AB" to has an "AB", and soon.

Options of "all poblation" kind
The "number of agents", the "selection scheme" and other general things are options of "all poblation" kind, and the more difficult to try.

If an option is "all poblation" kind, this is because we can not try it like an "island" kind. For example, to have more variation in the population, could be a good idea to make true this option:

"When a new agent borns, you compare it with all of the other agents. If there are one agent like it, you kill the new agent or give to the new agent a low fitness."

It has no sense to try it only with a subset of agents. You have to try with all.

To solve it, you have to made more than one paralell complete algorithms. For example, in the "first floor" there are some algoritms with different options. In the "second floor" there are the best agents of each algorithm of the "second floor"

[ Volver al Indice ]

[1] "J. Theo. Biology", vol. 17. Reed et al. 1967.

[2] Ph.D. dissertation, U.Mich. Rosenberg. 1967.

[3] "Handbook of Evolutionary Computation". Baeck, Thomas; Fogel, David B. (eds.) 1997. Oxford, NY.

[4] "Evolutionary Computation: Toward a New Philosophy of Machine Intelligence". Fogel, David B. 1995. IEEE Press.

The problem of the variety

A AG try of to explore the regions more promising of a huge space of possibilities. To the to find a zone with weights high, this it is explore more to fund. But there is that to avoid that the algorithm it is reservoir in a determined zone, producing multitude of chains very similar. Of equal form, it is there is of to avoid it opposite, that the algorithm distribute so uniformly the space of search that almost it is let of to explore the better zones. It is try of econtrar a balance, the better of the options in each case.

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Solutions to the problem of the variety

It is try of to obtain a balance between exploration and development, or the speed optima of convergence of the algorithm, for that follows the addresses more promising but without to forget other possibilities that to the releases they can to produce better results; to arrive to the optimo quickly without but it be remained stagnant in a minimal local.

The algorithm genetic basic or canonical, in the that always it is seleccionan the better individual, tend for it general to the homogeneización of the population. It is try for so much of to increase the variety seleccionando some of the individual that not they are the better.

The increase of variety it is procures with the mutations, but this not tend to suffice, and not is advisable it be exceeded in the mutations. A possible solution would be seleccionar a great number of agents, for to avoid that a small group repeats its characteristic in a great population, but this not is sufficient to long term.

It that it is it must to make is seleccionar, furthermore of the population with greater weight, a fraction of the of smaller weight. If only seleccionaramos some few of the better, the very good, the algorithm probably never operate as we wait, already that it is will remain stagnant in a minimal local, (all a lesson for the life same, we go).

This it is puede to make of a form so radical as seleccionar the 40% better and the 10% worse, but exist other methods that seleccionan "of all a little" having each chain a probability of be seleccionada proportional to its weight, through the method of the roulette or of the tilts. In the method of the roulette, it is calculate the weight relative of each standard (equal to the weight own divided between the sumatorio of all the weights), with it that the sum of the weights relative is 1. Thereinafter it is assign to each entity a chunk of the values between 0 and 1 of the size of its weight relative, and it is generate numbers to the random between 0 and 1; the individual with those values they will be seleccionados.

The method of the tilts consist in to group individual in subsets created to the random and seleccionar to the better of each subset.

Also it is there is of to procure that, to the hour of the reproduction, the progenitors it is take to the random and not in order of weights, already that is more probable that agents of weights seemed possess genes seemed.

Other options they are to make that if it is reproduce two chains equal, the children they will be mutations in all its elements; or to travel of time in when all the population eliminating a of each two chains identical or very similar.

Also we can to generate blasts of mutations: in time of to establish a probability of mutation independent for each creation of a element of the chain, we can to make that if it is produces a mutation in a element, the following element has a probability of mutation greater. In the nature the mutations it is they can to produce, between other causes, for a excess of radiation. Is logical to think that if not there is excess for a gene, is less probable that it there is for the following, and conversely.

But, in definitive, it that it is seek is that the calculation of the weight not it will be independent for each chain, but relative: that it is to him assign a weight high when is evaluated positively for yes alone, and furthermore not exist chains that it is to him seem. For this we can to analyze the chains of greater weight that a given and to reduce the weight of this chain if exist a great seemed with the other chains.

In the life natural occur this same: ¿For what not we are all high, blond, strong and handsome? ¿For what there is tántas kinds in the nature? The beings seemed compete some with other, but the diversity and the specialization lives together peacefully.

¿What criterion to apply? There is that to have in account that puede have options that "logically" they will be better, already that reduce the number of cycles necessary for to arrive to the weight maximum, but that in the practical not they are profitable for the increase of the time of process had to to the complexity of the calculations (thing that would not to occur if we had a parallelism real).

To find the combination of parameters optimum result for so much, costly. But exist a form of to automate the election of these parameters, that is to take them of the result of the execution of other algorithm genetic, with it that we would would have algorithms genetic hierarchic.

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The problem of the reproduction

The characteristic positive of the individual they should power it be transmitted to the descendents.

In each cycle of a Algorithm Genetic it is selecciona a subset of the solutions or individual existing, eliminating the rest. The individual seleccionados it is will reproduce between yes, generating new solutions.

It is try of to choose the better solutions, for that to the it be reproduced, generate other new that combine the aspects positive of each progenitor. The problem is to find a mechanism of combination of information of two (or more) individual, of way that the (or the) individual resulting possess, to the less in the majority of the cases, tántas or more qualities positive that its progenitors, and for so much it is find tánto or more nearby to the objective that they.

In the case of not to exist reproduction, only mutations, also we should to find a type of mutations that in a number sufficient of cases permit to maintain and even to increase the kindness of the entity mutada.

Without embargo, in many problems is very difficult to give with a structure of data and a manner of reproduction that it is behave of this form. For the opposite, with frequency occur that in the combination of solutions not only not it is maintain the qualities positive of the progenitors, but that furthermore it is generate with frequency solutions not valid, whose approximation to the objective sought is void.

For example, if i am seeking the sequence of actions with the that it is procures a determined objective (programming automatic), the standard 234, 523, 12, 765, 256 (¿codes assembly?) would to represent said sequence of actions, supposing that it is there is assigned to each number a action or instruction.

Without embargo, the probability of that through the combination of two chains valid of this type, it is generate a chain valid is very decrease. Changing a line of a program to the random, it more probable is that the program let of to operate

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Solutions to the problem of the reproduction

The problem of the reproduction appear when the structure of data elected not is the adapted. In the majority of the cases, the structures of data more simple and intuitive not contain information about of why that solution there is been seleccionada, and for so much not is possible to transmit to the descendents characteristic positive of the individual progenitors already that these not they are represented in no place, and any method of reproduction elected not will produce the effects wished.

This seem a difficulty insalvable, but not it is. A time more, the life natural us gives the track: in reality, the genes, more that to determine directly the characteristic that define a new individual (color of the eyes, etc.), define a joint of rules of development that applied and combined will provoke said characteristic. For this, the endowment genetic, that exist in all the cells of each be live, control the accomplishment of certain reactions chemistries, of way that the genes they are the that determine, but not directly, the constitution and behavior of said individual.

We suppose that we go to to build a AG that plays to the tic-tac-toe. We should to decide what type of information is going to be stored in the chains protagonists of the algorithm. It more simple is that the standards contain each a of the positions of the chips put. For example, a series valid is: (1,1)-(1,2)-(1,3). This mean that the individual represented for this chain will put its first chip in the position (1,1), the second in (1,2) and the third in (1,3). Is possible that this individual there is cattle some departure with this sequence, and for so much it will be seleccionado. Without embargo, the information stored genetically not is representative of how and why earned, already that depend thoroughly of the movements accomplished for its opponent. The repetition of the sequence of movements (1,1)-(1,2)-(1,3) puede to originate with facility departures not valid, to the to attempt to put a chip in a cabin occupy, and evidently the combination of two chains with this structure rarely will produce a departure valid, much less a chain winning.

The solution to the problem of the reproduction is to find a structure of data capable of to contain the information critical necessary for to reach the objective proposed. Here we enter in the topic of the methods of representation of the knowledge ámpliamente studied in the systems expert. In many cases, is sufficient a system of rules.

In the tic-tac-toe, we could to have rules of the type "if i observe such situation, i accomplish such action". Each entity would be compound for a joint of rules of this type. To the to play a departure, the entity, observe the state of the checkerboard, and if possess some rule that to apply, the apply. In case opposite, plays to the random.

In this case, in the entities yes it is stored the knowledge key for to earn the game, that they are the rules, not the movements. Each joint of rules it will be a solution. The element minimal that it is combine or muta is a rule completes. And a possible function of evaluation in this case is to provoke a departure between two entities.

¿Then, the use of rules will solve always the problem of the reproduction? Is difficult to know when a structure of data is adapted; without embargo, yes exist a form of to know for anticipated when a structure of data not is adapted.

For that the algorithm operate as we wish, the reproduction and the mutations they would have to it be produced of such form that maintain, in the majority of the cases, the qualities positive of the individual. It is try of that the individual, in joint, they will be each time better. We think in the case of the mutations. Given a chain, a mutation will consist in to modify to the random one of the elements of that chain. For that all march well would be very interesting that the values of the genes they will be good or bad, independently of the values of other genes, or to the less, that they will be it more independent possible.

¿As to know when a structure of data not is adapted? A method is the following: if to the to modify a element anyone of a chain seleccionada (good) exist a probability high of that the chain resulting it will be pésima, then the structure of data elected not is correct, already that the components of the chain for yes onlys not contain the information that makes that that chain it will be seleccionada.

Is to say, "When the objective (weight) of a entity it is puede to calculate as a function in the that variations very small of the values of its variable (values of the genes) produce with a probability high, variations very large in the result of the function, the structure of data is wrong". A variation very small would be to change the value of a gene.

The example for excellence of this type of problems is the lottery. The change of one only of the digits of our bill puede to produce changes huge in the quantity that it is receive as premium. A AG that try of to predict the number rewarded of this form never will will have too success.

In the tic-tac-toe with rules in geenral occur that each rule, or is good, or is wrong, independently of the demas. If a entity seleccionada possess 40 or 50 rules, probably we could to change a without that it is convert in a pésimo player.

But in almost all the problems exist this inconvenient of form unavoidable. For example, in games complicated as the chess, they can it be produced for random rules very interesting, but that not they are seleccionadas because need of other for to demonstrate its usefulness and to produce a weight high. Is to say, exist chunks of chain whose usefulness depend of the values of other segments, and that only they are útiles when all they contain determined data, disappearing the usefulness to the to fail one of they.

How much more occur this, more slowly convergerá our AG. This is thoroughly unavoidable, already that not is possible to find something if not we know what is it that we are seeking. ¡Without embargo, the nature seem that yes it procures! ¿How they have been able it be generated gradually organs complex, as the eyes or the pens for the flight, that not they can to depend of a only mutation, if sólamente the organ i complete is useful to the individual? Not always seem easy to explain all the features that present a organization with a mechanism adaptativo.

In my opinion, this position is wrong. The beings live draw benefit of any organo to middle to form. A duck puede to flee of a depredador land raising the flight so only a meter. The system visual of the frog it is "tuned in" only to four classes of stimulus, these they are: the contrast of light and darkness; a edge of light or darkness; a sudden decrease of the luminosity; and the constant movement of a small object black. ¿Of truth solely the organ i complete (vision in colors) is useful to the individual?

Is customary to consider our intelligence or the vision of a hawk as perfect, in time of simple stadiums intermediate, that will arrive to levels much more advanced thanks to the evolution. The nature probably it will be much more progressive and constant of it that seem, and we we can be "the frog".

The method of reproduction

Well, we are resolving a problem with a AG and for fín we have obtained that appear together the values wished in the segments dependent. ¿We go to let that this information so valuable it is lose in the following generation? If we combine segments of the chain of any form, is very probable that we divide the chunk that us interest, it being lost all its properties.

¿Exist some form in the that the nature it would be capable of to avoid this?

We suppose a population with high rate of birthrate and mortality in the that a individual acquires a rare and very interesting combination of genes. If exist reproduction sexual, to the to combine its genes with the of its couple is possible that these qualities it is lose, but if the number of children the sufficiently large, is possible that it is could to repeat in the falling the same combination.

If we take a segment anyone of genes of one of the falling live of this mutant, is but probable (though certainly, not much but) to find exactly this segment interesting that any other, simply because if the falling it is live, is but probable that it will be one of the heirs of said characteristic.

We can to make that segments with equal values in chains that it is reproduce see reinforced its union, of form that in all the descendents successive increase the probability of that those segments it is inherit complete of a only progenitor, without it be combined with the of the other.

This would be very interesting in the last phases of the evolution (in the solution of a problem). It normal is that in the joint of the population it is they may have formed several "types" seemed of individual (for example, in a population of 200 individual, of one to five "types") each one of the which there is found a maximum local of the function that represent the problem, and attempt to maximize the weight in its zone. In each "type" not is interesting to combine the chains of any form, but maintaining the characteristic common of the "type".

This criterion would be untruthful when it is reprodujeran two chains sisters, already that though would would have segments equal, this not would be a indicative of features positive, but that solely it is try of the part that both they have heir of the same progenitor. This option presuppose that exist a number very large of chains and that the probability of reproduction of a chain with its sister it will be equal that with any other, thing that not exist in the nature, unless maybe for the taboo of the incest.

If the repetition of a segment in the two progenitors not it would be sign sufficient of that it is try of a feature valuable, it is would to make that each entity had a endowment genetic double, (dominant and recesivo) and to compel to a coincidence in the four segments.

In definitive, it is try of to try the segments independent of form independent. We could to travel all the population each certain time, detecting them. A time located, it is puede to provoke a microbúsqueda within of the own segment independent, maintaining constant the rest, that for be it, not will affect to the weight resulting, until to find some combination valuable.

We go to to see this topic with more depth. A chain of a AG has several elements, that correspond with the variable of the problem to to solve. When it is muta, it is muta a element, and in the reproduction, it is combine subcadenas of elements complete. Is to say, a element (gene) is the unit minima that it is recombina or muta.

A element puede be or not "linked" (for to call it of some form) with other. Of equal form, a joint of elements puede be or not "linked" with other joint.

¿That is that be "linked" or "not linked"? The values for a element "not linked" they are always intrinsecamente "good" or "bad" independently of the values of the rest of the chain. In change, in a element "linked" occur it opposite: certain value of the segment linked it will be good or bad (is to say, will produce a weight high or under) in function of the values of other segments. A segment i could be "relatively linked" in function of the degree in that it is fulfil this property.

The property of be or not linked puede it be applied to a joint of elements, if it is have in account all they together, as a unico segment. A joint of elements (not have for that it is contiguous) it will be "not linked" if any series of values for those elements is always intrinsecamente "good" or "wrong" independently of the values of the rest of the chain.

Us interest to seek segments and groups of segments "not linked" with the rest of the chain, already that if we can to divide the chain in several segments "not linked", we will be able to apply a AG independent for each segment (in the that the rest of the chain is always fixed, with a value anyone, and in reality is as if not existed, unless for the evaluation) and to obtain the solucion much but rapido.

There is several problems pending:

1.- How to detect segments not linked
2.- How to try a AG in the that it is they have detected segments not linked
3.- How to try segments "little linked"

1.- How to detect segments not linked

To times it is knows for intuicion if a segment is not linked according to this definition intuitive: those that they are good or bad independently of the rest they will be not linked. But it more interesting is to find a method automatic that permit to detect it.

For to know if a element (or a joint of elements) this "not linked" with the rest of the chain we can to make it following:

-Selecciónamos the joint of elements to to analyze
-We fix values whatever to the random for the rest of the chain
-We evaluate the chains that result of all the possible combinations of values for the joint of elements elected, letting the rest fixed such as it is there is indicated
-We order the chains segun its weight (fitness) of greater to smaller

Now we repeated the process using other values whatever to the random in the rest of the chain, different to the already used. If always that we make this, we obtain the chains in the same order (having in account the case particular in the that the order not import if two chains have exactly the same value of weight), then we are before a segment "not linked".

Thus it is detect segments not linked in a Algorithm Genetic.

We suppose a chain of 4 genes having each one of they 8 bits, is to say, 256 possibilities in each gene.

G1 G2 G3 G4

We go to to analyze if G1 is "not linked". For this, we let fixes the part G2 G3 G4 with any value and we prove all the combinations of G1

c1 = 0 5 98 66
c2 = 1 5 98 66
c3 = 2 5 98 66
c4 = 3 5 98 66
c5 = 4 5 98 66
c6 = 5 5 98 66
c7 = 6 5 98 66
c256 = 255 5 98 66

despues we evaluate and we order these chains for weights of greater to smaller. Podriamos to obtain for example the chains in this order with the following weights:

c37 = 140
c2 = 70
c98 = 70
c44 = 30
c23 = 20

Now it we make with other values of G2 G3 G4: in time of 5 98 66, this time we prove with 81 29 120. After of to create the chains, to evaluate them and to order them, we could to obtain:

c37 = 200
c2 = 120
c98 = 120
c44 = 30
c23 = 10
This already would be a good indicium of that it is try of a segment not linked. For to obtain a greater safety, we can to repeat the process for some how much combinations of values of G2 G3 G4. If the classification of the chains always appear in the same order, is to say, if each value of the segment is better or worse that other values, independently of the values of the rest of the chain, the segment is not linked and puede it be tried independently. In the example there is that to fix the value of this segment in 37, already that is the that greater weight produces in all the cases.

In reality, yes puede to exist a "dependency" between the values of the rest of the chain and the values of the segment that we are analyzing, is more, this dependency exist practically always; for that i speak of "vinculacion" (for to designate it of some form) and not of "dependency". For example in the second case the weights that obtain they are much but high; but it that import not is if it is obtain weights more or less high, in reality not import the value of the weight: only the order of the chains classified in order of weights. It is try of something qualitative and not quantitative.

Is evident that we can to find cases in that it is fulfil the rule of the "not link", but not of form absolute. If the order of the chains not change, unless eu some few cases, it is will try of a segment "little linked".

2.- How to try a AG in the that it is they have detected segments not linked

If we have detected segments or groups of segments not linked pure, it more appropriate it will be to divide the chain in said groups and to try them independently, as it is there is explained before, as if each one it would be a problem to to solve with a AG of form totally independent, joining finally the solutions in a only chain.

3.- How to try segments "little linked"

As normally not we will will have the safety of that a joint of elements it will be not linked, or such time we know that not it is but that in certain degree, it is they could to use algorithms genetic to two levels. In the level under it is would apply a AG to each segment. In the level high it is would take segments complete as if they would be genes, and the unit minima to recombinar or mutar to level high serious a segment "not linked" i complete.

It more interesting is that it is they could to make but levels: within of a segment "not viculado", that it is try as a AG totally independent, it is would to follow the same philosophy and to seek within of the "segments not linked" applying to each one of they a AG.

To weigh of the relative beauty of the plan that it is it is explaining, seem obvious that puede to suppose a overcharge of calculation very large, that such time not deserve the penalty. Evidently, if the problem to to solve is such that all the elements they are linked, we will be losing the time.

Serious interesting to have of a metodo that not it would be so strict and that not hiciese so many checkings, levels and subniveles. It interesting would be to make more or less it same, or to use of some form this knowledge, but in a alone AG.

With this we return to the idea initial: that the segments related they should traterse together. Podriamos to compel to that existed a kind of "cohesion" between the genes "linked" between if, of form that if two genes not they are neither good neither bad independently, but taken as a group of two genes, that group if it will be always or good or bad.

The "cohesion" haria it following:

-Is but probable that those genes between the that exist cohesion it is inherit together, is to say, the blocks with cohesion (not have for that be contiguous) it is inherit all (or none) with a greater probability. For example, we have 10 genes with cohesion: if the first of they it is inherit sure, increase the probability of that it is inherit each one of the remaining. and to the opposite: if the first not it is inherit, the probability of that the rest it is inherit disminnuye much

-Is but probable that those genes between the that exist cohesion it is muten together, is to say, For example, we have 10 genes with cohesion: if the first of they it is muta sure, increase the probability of that it is mute each one of the remaining. and to the opposite: if the first not it is muta, the probability of that the rest it is mute reduce much

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The problem of the selection

In each cycle of a Algorithm Genetic it is selecciona a subset of the solutions or individual existing, eliminating the rest. The individual seleccionados it is will reproduce between yes, generating new solutions.

The function that decide what entities they will be seleccionadas, the call function of evaluation (FAITH), puede be very difficult or impossible of to obtain. ¿How we can to know which they are the better solutions? ¿Which they are the that more it is approach to the objective sought? ¿which will produce a convergence toward the resolution of the problem?

The problem it is aggravate when the algorithm genetic it is directed for a objective that not possess nature more or less continuous. When the objective puede be achieved in various degrees, the algorithm puede it be strengthened in to obtain that the objective it is fulfil in the maximum degree possible, detecting small variations. But if the state sought it would be all-nothing, is to say, if occur that: a of two, or it is procures the objective thoroughly or not it is procures in absolute, not is possible that the evolution produces gradually entities each time more nearby to the objective.

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Solutions to the problem of the selection

The function of evaluation more current consist in to assign to each solution a weight or value in function of the degree in that it is about to the objective. A time fact this, it is order all the solutions according to this criterion. Finally, it is selecciona a number determined of solutions, beginning for the first of the ready.

In the problem of the travel you, that it must to travel several cities until to return to the point of departure incurring in the minimal cost, the FAITH will calculate the sum of the distances between the cities traveled in the order specified and will return the inverse of this value of form that the tours cut possess a weight greater that the long.

Without embargo, as it is try of seleccionar some solutions and to eliminate other, not is necessary to know quantitatively (with a number) the degree in the that each solution it is about to the solution sought. Enough with to know cualitativamente what solutions it is approximate more that other.

Of this form, we can to group the solutions for criteria of proximity physical or random and to make that "play", "fight" or demonstrate of some form which they are the better preparations in function of the objective sought.

In the tic-tac-toe each joint of rules is a solution (agent), and we have fact that the entities play departures between they. The entity winning will receive a weight high and the losing one under, that would to vary in function of it that there is delayed in to lose.

The case of the programming automatic, each entity would be a program or joint of instructions. But, ¿how we defined what is it that it must to make a program, without to have that to write that program? It is they could to offer several pars of joint of data of entry and exit, and to suppose that the program is correct when generate the data of exit waited in all the examples.

In general, the problems in the that the goal sought possess a character continuous (in the that the objective it is could to fulfil in different degrees) they will be more easy of to solve that those in the that the goal sought is all-nothing, is to say, in the that or it is procures the objective thoroughly or not it is procures in absolute.

Finally, the problems more prickly of all they are those in the that, to the less for now, the FAITH not is automatizable, as the case of a program capable of to create music, already that ¿where we will find to someone disposed to to listen and to hold about of endless melodies initially absurd?

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