The evolution from a mathematical perspective

This time I'm going to share a fascinating proposal about a considered a cellular automaton, which is known as Game of Life, or simply Life among experts in Artificial Intelligence and mathematics, which was created by the English mathematician John Horton Conway in 1970.

The game is determined by its initial state, without need of further input.  The game begins with an initial configuration and it is seen how it evolves.

The game is due to the interest of Conway in a problem presented by the mathematician John von Neumann who was trying to find a machine that could build copies of itself. 

Neumann succeeded when he found a mathematician model for a machine of this type with rules applied in a rectangular grid, which is very similar to the so-called Universal Turing machine, which is described as an automaton artifact that was unveiled in 1936 in the Journal Proceedings of the London Mathematical Society and which is basically a device that manipulates symbols on a strip of tape according to a set of rules that can be adapted to simulate the logic of any algorithm which can help to understand the limits of mechanical calculations, which provides advances to complexity theory.

Returning to the Game of Life, Conway found a way to dramatically simplify the notion of von Neumann, using only 4 rules for the implementation of a cellular automaton.

The game was released in October 1970 in the magazine Scientific American column mathematical games of Martin Gardner and opened a line of mathematical research known as the field of cellular automatons because of the analogy of the emergence, transformation and fall of any society of living organisms belonging to well-known simulation games in which the patterns can evolve.

Life allows an example of emergence and self-organization of patterns by what has attracted the interest of multiple fields of science and giving way to studies of emerging complexity or self-organization systems.

Life  is played in an infinite orthogonal network cell square (I recommend widely to visit the link of Wikipedia that I am here referring), each cell is in one of two possible States, alive or dead.

Every cell interacts with its 8 neighbors, which are the horizontal or vertical, diagonal or adjacent cells, with the passage of time, it is possible to observe the following transitions which are the rules of the game:

The initial pattern is considered the seed of the system. The first generation is created through the application of the above rules simultaneously to every cells of seeds that includes births and deaths which can occur at the same time and discreetly called this a tick, which implies that each generation is a pure function of the precedent and the game continues until the last cell dies.

From these simple rules, life has become one of the examples of what is known as emergent complexity and self-organization systems.

Now, let me explain why this made me jump off my seat and go running the stairs to search Google in this respect:

These simple rules allow the understanding of such complex phenomena such as the arrangement of the petals of a rose, or patterns on the skin of a zebra, and it is that in life, science attempts to explain complex patterns, always applying the idea that simple is best, I remember clearly professor Colm Donaldson saying loudly: make it simple, simplicity is more beautiful in science.

So those simple patterns of interaction are the beginning a complex process of interaction that unlike the rest of the games, life part of the standards themselves to make patterns, while in conventional games, developers create multiple game situations that must be met to advance. 

Designed by Paul Rendell 02/April/00, disponible en http://rendell-attic.org/gol/tm.htm

Life has served even to analyze patterns of conduct testing different models and watching their interaction. For example the so-called R-Pentomino was the first pattern observed by Conway, which is very stable and thus easy to predict, although 1 103 steps are required for this purpose.

This is why some of the programs designed for Life at the beginning were limited to describe the fate of a pattern of small and specific, however with the development of computers, it is now possible to run more complex patterns.

One of the questions of Conway was to determine if the initial pattern of life could grow indefinitely, or if any system could, so it offered a prize of $50.00 to anyone who could respond to questions. In 1970 a group of MIT led by RW Gosper won the prize with a pattern known as Glider Gun, which emits a new agent every 30 generations indefinitely, by what the pattern grows forever.

As they were more and more patterns, other aspects of the game are defined, for example the speed of light is defined as maximum sustainable speed by any moving object, or the rate of spread given in one step either horizontally, diagonally or vertically by generation. In this regard for both processes are the maximum rate at which information can travel and thus determines the speed of the pattern.

Based on that, the mathematicians have played and created different theorems that are observing with developed patterns.

This is part of a set of ideas that are integrated into the Game Theory, and just one of them is known as Nash equilibrium, which is a concept of the solution of a game and decision-making, which analyzes the strategies employed by the players from the benefit that can be obtained by changing their strategies which creates a principle of stability in the solution exchange during the game and is known as the theory of Nash equilibrium.

 

Of course, one of the fields that it has enabled more development is known as artificial life that it is defined as a life made by a human mind and not by nature and relates also with the study of non-organic bodies, beyond of the creations of nature, possessing essential properties that allow you to understand it inside an artificial environment created specifically for its development within a programmable machine (usually you can think in) a computer) so that the artificial life (A Life) allows to understand three properties of the nature that are the reproduction, the emergent properties and evolution. 

I am sure you are wondering how does this explain students would ability to change patterns from simple action rules?, well, it seems that many researchers have tried to apply these principles on biology, personally I think that the program of Neuro-modulation environmental assisted which starting with micro-tasks capable of creating patterns of conduct specifies, can be an applied example, but undoubtedly there is much reading to do simple the complexity of learning.

References: 

Ashwani, K.  (2013) Cellular automation: A discrete approach for modeling and simulation of artificial life systems. International Journal of Scientific and Research Publications.  3 (10) Available at: http://www.ijsrp.org/research-paper-1013/ijsrp-p2213.pdf

Gardner, M (1970) Mathemathical Games: The fantastic combinations of John Conway’s new solitarie game “life”. Available at: http://web.archive.org/web/20090603015231/http://ddi.cs.uni-potsdam.de/HyFISCH/Produzieren/lis_projekt/proj_gamelife/ConwayScientificAmerican.htm

Gymerek, M. (2010) Conway’s game of life. Available at: http://web.mit.edu/sp.268/www/2010/lifeSlides.pdf

Wikipedia. Conways’s Game of Life. Available at: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

 If you want to see very cool patterns, you can visit here: http://www.frank-buss.de/automaton/golautomaton.html

Adaptability of plants: dandelion

Usually when researchers speaks about adaptive systems is intended of course, this apply in animal and artificial intelligence, since we can believe is needed a cognitive process, however, the plants have given us too much to learn in this area too.

In order to understand the process in other species more than animals, some plants have been object of study thanks to its ability to adapt to the environment and even annihilate other nearby plants, a plant in particular is peculiar due to its yellow flower and which is popularly known as dandelion but its scientific name is Taraxacum, which is posible to found  practically on all continents both in free areas in large cities, though they are native to Eurasia and America.

At least 3100 kind of taraxacums have been recognized of which only 500 are accepted species and varieties described as harmful plants for natural areas and gardens.

Its name comes from from French dent-de-lion, meaning lion's tooth. There are varieties of which reproduce sexually and others asexually, but it has been shown that those with more effective parthenogenesis are the asexual species, one of the most common explanations is that they have greater phenotypic plasticity due to the efficient selection of genotypes of general purpose.

However, some studies have showed that plants sexually reproduction have an advantage with respect to pests control and pathogens because sexual patterns produces genetically variables to allow  offspring  reduces the risk of infection.

The way in which some of these plants manage to cover extensive grounds is quiet simple and witty since its seeds float helped by the wind, wearing its form by way of parachute, so the strength of the wind can push them by wide sections.

In a study conducted by Honek, Martinkova and Saska (2005) it was found that seed dispersal takes place 10 days after flowering produced massively, however, only between 11 and 13% will land in the right place to grow a plant, since there are several predators around them, especially birds and beetles produce a need to develop a sustaintable way to survive.

Given the difficulty of plantation, a study conducted in 2000 found that the ability of this plant to populate large areas even with such low rate of sucessful plantation is because they have developed processes both cloning and recombination, which makes them a species capable of survive anywhere in the world.

It has been shown that dandelions and some other plants has two strategies of survival, which makes them peculiar to adaptive studies to such strategies called K & R that defines the ability of disturbance of the response of a population. Both terms are derived from the logistic growth equation, where K is equal to the environmental carrying capacity and R the intrinsic rate of population increase.

By what they called strategic play is characterized by the production of a large amount of seeds (or reproductive units) and high dispersion of the same, which potentiates the growth rate of a population given under environmental conditions that are capable of promoting growth, which makes them difficult to control plants.

The infestation level depends on the space the plant can occupy, in some places in the United States they can cover huge areas, while if chemical methods are applied consistently, they reproduce more controlled, but never disappear. 

In this context, I began to observe them in my own garden, where two different varieties can be found, one long and one short plant. 

In both cases they can be planted between the grass than on fertile land. Some ways to prevent them are chemical options, through liquids that affect only the roots, which is used by the city to eradicatethe amount of plants, or by mechanical methods which consists of boot plant from the root. However, this latter procedure is difficult because of the depth of the roots.

A third (not recommended) method, is pouring water boiling on the affected areas, but it should be taken into account that there is no selectivity in the process, whereupon all the roots are killed around, including grass.

The first year of observation implied recognition of the spaces that plants occupied and cleaning by mechanical methods. Something that catch my attention is the tendency to find them in the middle of the rose-bushes, which create an ideal space for its growth, however it doesn’t mean that the plant has the ability to choose the best place, but that I'm not so stupid to put my hands among the roses.

Its presence extends from the first weeks of spring, until the first weeks of autumn.  When temperature are  low, plants  disappear like the rest of perennials. 

During the summer, it is easy to see the flight of seeds, which are eaten by birds, beetles and some slugs around. The pace of  growing lowest between 75 and 85% in the third year of analysis but still be can’t be eradicated, since while there are plants in nearby areas and means of transportation, these will continue to grow.

I have seen other means of transportation as well as the air like the fur of pets, squirrels and other creatures or the feathers of birds. 

Surely, if you  has reached this article to the  end begins to ask your self: what is the relationship of the incredible ability of the dandelion to survive with learning?. Well, if you have not noticed it, because the species require biological and technological innovation to maintain its rate of growth and reproduction, as well as learning. The same answers cannot be used when the culture is not static.

In the next entry I will share a fascinating mathematical theory able to explain evolution.

References: 

Honek, A., Martinkova, Z., & Saska, P. ( 2005) Post-dispersal predation of Taraxacum officinale (dandelion) seeds. Journal of Ecology. 93 (2) 345-352.

Van Der Hulst, RGM., Mes, THM., Den Nijs, JCM., & Bachmann, K. (2000) Amplified fragment lenght polymorphism (AFLP) markers reveal that population structure of tripoid dandelions  (Taraxacum officinale) exhibits both clonality and recombination. Molecular Ecology. 9 (1) 1-8.

Verhoeven, KJF., and Biere, A. (2013) Geographic parthenogenesis and plant-enemy interactions in common dandelion. BMC Evolutionary Biology. doi:10.1186/1471-2148-13-23. Disponible en red: http://www.biomedcentral.com/1471-2148/13/23

Zimdahl, RL. (2007) Fundamental of Weed Science. Academic Press.  UK.