Evolution in a nutshell

Many people still believe that random developmental processes are hopelessly inefficient in solving difficult problems. I therefore want to point to a simple random process which may find good solutions to a very difficult problem; the travelling salesman problem.The salesman should visit a numer of towns, one at a time, and wants to know in what order the towns should be visited in order to make the tour as short as possible.Suppose that the number of towns is = 60. For a random process, this is like having a deck of cards numbered 1, 2, 3, ... 59, 60 where the number of permutations is of the same order of magnitude as the total number of atoms in the universe. If the home town os not counted the number of possible tours becomes 60*59*58*...*4*3 (about 10 raised to 80).The probability to find the shortest tour by random permutation of the cards is about one in 10 raised to 80 so, it will never happen.But the natural evolution uses an inversion operator which - in principle - is extremely well suited for finding good solutions to the problem. A part of the card deck - chosen at random - is taken out, turned in opposite direktion and put back in the deck again. If this inversion takes place where the tour happens to have a loop, then the loop is opened and the salesman is guaranteed a shorter tour. See figure below. In a population of one million card decks this might happen at least 200 times in every generation. I have simulated this with a population of 180 card decks, from which 60 decks are selected in every generation (using MATLAB, The language of technical computing). the figure below shows a random tour at start. After about 1500 generations all loops have been removed and the length of the random tour at start has been reduced to 1/5 of the original tour. In a special case when all towns are equidistantly placed along a circle, the optimal solution has been found when all loops have been removed.This means that this simple random process has been able to find one optimal tour out of as many as 10 raised to 80. This also means that random variation and selektion is a very important principle for creating a huge amount of information.So, there is no reason to distrust random developmental processes. Se also Goldberg in references.ReferencesBergstrm, M. Hjrnans resurser. Brain Books, ISBN 91-88410-07-2, Jnkping, 1992. (Swedish).Bergstrm, M. Neuropedagogik. En skola fr hela hjrnan. Wahlstrm & Widstrand, 1995. (Swedish).Cramér, H. Mathematical Methods of Statistics. Princeton, Princeton University Press, 1961.Dawkins, R. The Selfish Gene. Oxford University Press, 1976.Eigen, M. Steps towards life. Oxford University Press, 1992.Goldberg, D. E. Genetic Algorithms in Search Optimization & Machine Learning. Addison-Wesley, New York, 1989.Hartl, D. L. A Primer of Population Genetics. Sinauer, Sunderland, Massachusetts, 1981.Kandel, E. R., Schwartz, J. H., Jessel, T. M. Essentials of Neural Science and Behavior. Prentice Hall International, London, 1995.Kjellstrm, G. Network Optimization by Random Variation of component values. Ericsson Technics, vol. 25, no. 3, pp. 133-151, 1969.Kjellstrm, G. Optimization of electrical Networks with respect to Tolerance Costs. Ericsson Technics, no. 3, pp. 157-175, 1970.Kjellstrm, G. & Taxén, L. Stochastic Optimization in System Design. IEEE Trans. on Circ. and Syst., vol. CAS-28, no. 7, July 1981.Kjellstrm, G. On the Efficiency of Gaussian Adaptation. Journal of Optimization Theory and Applications, vol. 71, no. 3, Dec. 1991.Kjellstrm, G. & Taxén, L. Gaussian Adaptation, an evolution-based efficient global optimizer; Computational and Applied Mathematics, In, C. Brezinski & U. Kulish (Editors), Elsevier Science Publishers B. V., pp 267-276, 1992.Kjellstrm, G. Evolution as a statistical optimization algorithm. Evolutionary Theory 11:105-117 (January, 1996).Kjellstrm, G. The evolution in the brain. Applied Mathematics and Computation, 98(2-3):293-300, February, 1999.Levine, D. S. Introduction to Neural & Cognitive Modeling. Laurence Erlbaum Associates, Inc., Publishers, 1991.MacLean, P. D. A Triune Concept of the Brain and Behavior. Toronto, Univ. Toronto Press, 1973.Maynard Smith, J. Evolutionary Genetics. Oxford University Press, 1998.
Mayr, E. What Evolution is. Basic Books, New York, 2001.Middleton, D. An Introduction to Statistical Communication Theory. McGraw-Hill, 1960.Rechenberg, I. Evolutionsstrategie. Stuttgart: Fromann - Holzboog, 1973.Reif, F. Fundmentals of Statistical and Thermal Physics. McGraw-Hill, 1985.Ridley, M. Evolution. Blackwell Science, 1996.Zohar, D. Kvantjaget (The quantum self): En revolutionerande syn p mnniskans natur och medvetande med utgngspunkt i den nya fysiken. Forum AB, Stockholm, 1990. (Swedish).slund, N. The fundamental theorems of information theory (Swedish). Nordisk Matematisk Tidskrift, Band 9, Oslo 1961.This message has been edited by gregor, 04-11-2006 08:23 AMThis message has been edited by gregor, 04-11-2006 08:43 AMThis message has been edited by gregor, 04-12-2006 01:13 PMThis message has been edited by gregor, 05-01-2006 02:47 PMThis message has been edited by gregor, 05-01-2006 02:50 PM

Sorry, my first message was only a test.

Thanks.
My intension was to use the references in the future, but it may perhaps be better to include them one at a time.

Mayr states (se references) that “it is sometimes claimed that evolution, by producing order, is in conflict with the ”law of entropy’ of physics, according to which evolutionary change should produce an increase of disorder. Actually there is no conflict, because the entropy law is valid for closed systems only, whereas the evolution of a species of organisms can reduce entropy at the expense of the environment and the sun supplies a continuing input of energy.”I have a different view of this. Creationists are right in the sense that random events do not produce order. But they have produced an enormous amount of disorder represented by millions of different species and billions of different individuals in certain species, in agreement with the entropy law. Because a more widespread gene pool is more disordered. The order in the biologic sphere was biggest when the first living organism ruled the roost. Disorder/entropy may also be called biological diversity because - as I see it - there is no reason to distinguish between disorder and diversity because it is the same random evolution, giving rise to both.The illusion of order in the biologic sphere is due to the fact that only a very tiny little fraction of all possible DNA-messages may manifest themselves as living organisms. Thus, the disorder becomes restricted, and this restricted disorder is interpreted as order by both creationists and biologists. Intuitively, this may be understood, if we observe that the duality order-disorder is like cold-warmth. Actually there is no cold, only limited warmth. Likewise, there is no order, only limited disorder.So, for our purposes, evolution may be seen as a random process climbing a genetic landscape, ruled by the restrictions, and which will completely determine the shape of living organisms. The landscape is completely dependent of the almighty laws of nature (in the sense that they are valid throughout the whole universe), properties of DNA molecules, proteins etcetera, whose origin is not known.If it were possible to prove that the electro-magnetism, for instance, is a product of some random process, then we have perhaps proved that there is no god. But, to my knowledge, there is no such proof. So, it may be possible to believe that god created the laws of nature inclusive the entropy law and the evolution. In this way creationists may hopefully accept that evolution is a part of the creation.I hope, later on, that I will be able show that evolution gives rise to intelligent design.This message has been edited by gregor, 04-13-2006 02:15 AM

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gkm

In the middle of the 60-ties, I worked at a Swedish telephone company with analysis and optimisations of signal processing systems. Formerly such systems consisted of interconnected components such as resistors, inductors and capacitors.In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.If we have only two components - each having a parameter value - the problem is very simple. Let the first parameter value be the shortest distance to the left edge of a picture while the second value is the distance to the bottom edge. Then, if the interconnection is given, a point in the picture represents the system unambiguously.Suppose now that all points inside a certain triangle (region of acceptability) will meet all requirements according to the specification of the system, while all other points does not, and that the spread of parameter values is uniformly distributed over a circle. Then, if the circle touches the three sides of the triangle, the centre of the circle would be a perfect solution to the problem.But if we have 10 or 100 parameters, then the number of possible parameter combinations becomes super-astronomical and the region of acceptability will not possibly be surveyed. I begun to think that the man was not all there.The problem was almost forgotten until a system designer entered my room. He wanted to maximize the manufacturing yield of his system that was able to meet all requirements according to the specification, but with a very poor yield. Oh, dear! I would not like to get fired immediately. So, we wrote a computer program in a hurry, using a random number generator giving normally Gaussian distributed numbers. The system functions of each randomly chosen system were calculated and compared with the requirements. In this way we got a population (generation) of about 1000 systems from which a certain fraction of approved systems was selected. For the next generation the centre of gravity of the normal distribution was moved to the centre of gravity of the approved systems and this process was repeated for many generations.After about 100 generations the centres of gravity reached a state of equilibrium. Then the designer said “but this looks very god”. And we were both astonished, because we had only put some things together by chance. A closer look revealed that there is a mathematical theorem (the theorem of normal or Gaussian adaptation) valid for normal distributions only stating: “If the centre of gravity of the approved systems coincides with the centre of gravity of the normal distribution in a state of selective equilibrium, then the yield is maximal.” For the proof see references Kjellstrm (1970) & Taxén, 1981 in message 1.This gave an almost religious experience. Here a mathematical theorem solved our problem without our knowledge and independently of the structure of the region of acceptability. Our very simple process was similar to the evolution in the sense that it worked with random variation and selection. Later, it turned out that evolution might as well use the theorem and much more than that.Today I would not hesitate to regard this an example of “intelligent design” effectuated by a mathematical theorem and a process using random variation and selection.This message has been edited by gregor, 04-13-2006 05:41 AM

My intension was to discuss my contributions with interested biologists ad creationists.My intension also was to expound an alternative theory of biology based on the theorem of Gaussian adaptation (centering) concerning the maximization of mean fitness and genetic disorder/diversity. Here the mean fitness is calculated as a mean over the set of individuals in a large population, and the law of entropy has, so to say, been married with the mean fitness. A certain theorem of efficiency rules the balance between perfect order and total chaos.This is in contrast to the fundamental theorem of biology due to Fisher (1930) concerning the increase of mean fitness as calculated over the set of genes under the assumption that a gene may have a fitness of its own. According to Maynard Smith the theorem states that; “the rate of increase of mean fitness of any organism at any time is equal to its genetic variance at that time”.But a population may reach a state of selective equilibrium, in which case the increase of mean fitness is equal to zero, but not necessarily the genetic variance. So the fundamental theorem can hardly be a fundamental truth. Of course, the calculations of Fisher are certainly correct. But the result is wrong, because a necessary condition for evolution to be able to select a gene is that it has a fitness of its own, and this is not possible for all genes.The same way of thinking appears in the definition of fitness according to Maynard Smith: “Fitness is a property, not of an individual, but of a class of individuals - for example homozygous for allele A at a particular locus.” This definition is certainly useful in breeding programs. But unfortunately, a theory based on this is completely useless as a basis of a model of an evolution selecting individuals. In addition, Mayr (see my references) states that selection takes place at the individual level.Dawkins metaphor concerning “the selfish gene” is an example where selfish genes are supposed to be units of selection. These may even cause individuals to become selfish. With respect to the observation that not even the “fundamental theorem” is correct, it seems dangerous to draw such far reaching conclusions from this way of thinking.An additional example is the evolution of helper behavior, which is explained in terms of egoism as kin-selection (Hamilton, Ridley). But there is no need for any egoism to explain the phenomenon. If the individuals of some primitive species do not help their offspring to survive, then mean fitness may increase if a certain helper behavior evolves and - vice versa - an increase of mean fitness may cause a helper behavior to evolve. Further, if this behavior is extended to include relatives or even any individual independent of “race” or religion, then the mean fitness of mankind may increase even more.Because the way of thinking in the fundamental theorem forms a basis of very many fields in biologic theory and socio-biology, my contributions will become very fragmented, and it will be difficult to see the wood for the trees. I therefore prefer to have a thread of my own, where my contributions can be discussed. But if this is impossible, then I have to look for some other forum or medium.This message has been edited by gregor, 04-14-2006 05:47 AMThis message has been edited by gregor, 04-14-2006 05:54 AM

In short, there is a pocketful of mathematical theorems ruling the evolution, at least at a fairly good statistical second order approximation. This means that the gene pool is approximated by a normal distribution. Then by using the rules of genetic variation ( crossing over, inversion etcetera) as a random number generator, evolution may effectuate a simultaneous maximization of mean fitness and genetic disorder/diversity.This means that evolution strives to secure our survival with largest possible margins to spare, while the disorder stands for imagination and creativity.1 The central limit theorem: The sum of a large number of random steps tend to become normally Gaussian distributed.Since the development from fertilized egg to adult individual may be seen as a stepwise modified repetition of the evolution of a particular individual, morphological characters (parameters) tend to become normally distributed. As examples of such parameters we may mention the length of a bone or the distance between the pupils. Even mental parameters such as IQ may also be normally distributed. See Cramér in references.2 The normal distribution is the most disordered distribution among all statistical distributions having the same variance. See Middleton.3 The theorem of normal (Gaussian) adaptation or normal (Gaussian) centering (we have many names for the things we love): If the centre of gravity (m*) of the gene pool of the parents to offspring in the next generation coincides with the centre (m) of the normally distributed gene pool in the next generation - in a state of selective equilibrium (m* = m) - then the mean fitness is maximal. See Kjellstrom (1970) & Taxén, 1981.This theorem may be proved in two different ways. Firstly, one may maximize mean fitness while keeping the disorder of the normal distribution constant. Secondly, one may maximize the disorder of the normal distribution keeping the mean fitness constant. In both cases the condition of optimality will be the same, m* = m. This means that evolution effectuates a simultaneous maximization of mean fitness and genetic disorder/diversity.A more general formulation of the theorem includes the mean value of information and the moment matrix M of the normal distribution allowing he disorder to increase even more, still keeping the mean fitness constant. The condition of optimality becomes M* proportional to M, where M* is the moment matrix for the parental distribution. This will make normal adaptation a second order approximation of evolution.Note that mean fitness is calculated as a mean over the set of individuals, in contrast to the fundamental theorem of biology (Fisher, 1930) where mean fitness is calculated over the set of genes leading to a dubious teorem.4 The theorem for the choice of a breeding partner (Hardy-Weinberg): If mating takes place at random, then the allele frequencies in the next generation are exactly the same as they were for the parents. See Hartl, 1981.Since the centers of gravity (m* and m) will behave similarly, evolution will strive to fulfill the condition of optimality of the theorem of normal adaptation according to point 3, and mean fitness will be maximized.5 The second law of thermodynamics (the entropy law): The disorder will always increase in all isolated systems. But in order to avoid considering isolated systems I prefer an alternative formulation: A system attains its possible states in proportion to their probability of occurrence. See Reif, 1985, and Brooks, 1986.Thus, the system will attain its most probable disordered states of occurrence even if some force from outside influences it. If the mutation rate is sufficiently high,evolution will also be able to maximize the genetic disorder/diversity in accordance with the theorem of normal adaptation, point 3.6 The theorem of efficiency. The most important difference between the natural and the simulated evolution in my PC is that the natural one is able to test millions of individuals in parallel, while my PC has to test one at a time.But the efficiency of evolution also depends on the mutation rate and this is plain from the theorem of efficiency. It is based on the theory of information (Shannon 1948, see Middleton). So, if P is the probability that an individual in a large population will be able to survive, then the negative logarithm of P, -log(P), is the information in the art of survival gained when a survivor has been found. Since the inverse of P is proportional to the work or time needed to find a survivor, then -P*log(P) becomes a measure of efficiency.A simplified version of the theorem states that all measures of efficiency, that satisfy certain postulates, are asymptotically proportional to -P*log(P) when the number of statistically independent parameters tend towards infinity. See Kjellstrm, 1991.Maximum efficiency is attained when P = 1/e = 0.3679, where e is the base of the natural logarithmic system.As an example of a measure (not based on the theory of information) of efficiency I may mention the average speed of a random walk in simplex region. The average speed will asymptotically tend to -P*log(P) when the number of dimensions tend towards infinity. See Kjellstrm, 1969.ReferencesBrooks, D. R. & Wiley, E. O. Evolution as Entropy, Towards a Unified Theory of Biology. The University of Chicago Press, 1986Cramér, H. Mathematical Methods of Statistics. Princeton, Princeton University Press, 1961.Hartl, D. L. A Primer of Population Genetics. Sinauer, Sunderland, Massachusetts, 1981.Kjellstrm, G. Network Optimization by Random Variation of component values. Ericsson Technics, vol. 25, no. 3, pp. 133-151, 1969.Kjellstrm, G. Optimization of electrical Networks with respect to Tolerance Costs. Ericsson Technics, no. 3, pp. 157-175, 1970.Kjellstrm, G. & Taxén, L. Stochastic Optimization in System Design. IEEE Trans. on Circ. and Syst., vol. CAS-28, no. 7, July 1981.Kjellstrm, G. On the Efficiency of Gaussian Adaptation. Journal of Optimization Theory and Applications, vol. 71, no. 3, Dec. 1991.Middleton, D. An Introduction to Statistical Communication Theory. McGraw-Hill, 1960.Reif, F. Fundmentals of Statistical and Thermal Physics. McGraw-Hill, 1985.