Automatas as analogies of our Universe
Manuel de la Herrán GascónIndex:
- Discretización of the time in the automatas
- Secuencialización of the time in the automatas
- Automatas and Random
- The Problem of the simplification
In this document it is discuss several cases of utilization of analogies between our universe real and universes simulated in the creation of hypothesis about of the time, the space, the life and the random.
For opposition to the automatas used for the resolution of problems, we could to speak of automatas guided to the representation, for the contrast of hypothesis.
Exist several areas of study where to apply simulations guided to the representation. The objective of this type of automatas it is to accomplish analogies between the world real and the world simulated, of form that the automata serves for to know better the world real. The greater part of these applications belong to one of two groups, according to it is devote to to seek knowledge on some aspect concrete of:
· the life
· the universe
The topics they can to seem too vagrant, general or ambitious. It is would to introduce a third paragraph that it will be "... and all it other", paraphrasing to Douglas Adams in its famous series of humor "Guide of the Hitchhiker Galáctico", where it is try with great wisdom many of the topics fundamental in life artificial.
If well it is certain that concepts as "the life" or "the universe" not they are well defined, not they are in absolute negligible as phenomena susceptible of study. The lack of definition it is it must to the ignorance about of the nature of the own object of study, more that to a lack of concretud in how much to it that it is wants to study. It is to say, all we know of what it is try when we ask "¿what it is the life?", or "¿what it is the time?", though we continue without to know what it is the life or what it is the time. For other part, it is possible to study aspects very specific of the life and of the universe, as certain theories evolutionary, strategies cooperative, or interactions between particles.
The program "Ants and Plants" of the application "Ejemplos de Vida" not it is guided to the resolution of a problem determined, but to the representation of a universe imaginary that intend to gather of form very simplified several aspects of our universe real. The objective of this representation it is to extract conclusions, or to the less to contrast hypothesis of work, through the analogy with our universe real.
Though to the to work with analogies the validity of the conclusions it is with difficulty demonstrable, this not it is reason sufficient for restarles importance scientific. Such as comment Norbert Wiener in "God & Golem, S.TO., "If well it is dangerous to assert that exist a analogy with evidence insufficient, it is equally dangerous to reject a without tests of its lack of congruity". For the task scientific, that it is resemble to a great search in a huge space of states, puede be so important a fact demonstrated as a indicium promising.
The principal inconvenient of all simulation it is that require of a simplification, and not it is easy to obtain a criterion for to decide what level of detail it is accurate to simulate. Even in certain cases such time not it will be possible to decide neither even if exist or not said level. This problem it is try more forward in the paragraph "The Problem of the simplification."
Of all forms, the simulations they will be of usefulness. To describe some examples of applications of this type will help to show the form of to work with they. In physical theoretical abound the theories that intend to explain a behavior complex through the definition of the interactions between certain elements basic, with rules relatively simple. In these cases, result useful to accomplish simulations, already that in they it is possible to prove if in fact it is produces or not the behavior complex waited. In the case of to have success, these simulations not demonstrate the existence of said elements simple neither the veracidad of those rules relatively simple, but yes offer a explanation of the behavior complex known and object of study.
It is to say, it is offer a explanation, compatible with the beliefs outstanding, as alternative to not to possess no explanation of the phenomenon. A theory puede be, without to know it nobody, rigorously untruthful, or if it is prefer, "not thoroughly certain", and without embargo be useful.
This occur in physical, where many times new theories replace to other. In some cases the old theory not it is abandoned and puede be even more used that the new, to weigh of it be have proven that not it is absolutely certain. It is the case of the theory gravitational of Newton, that suppose a action instant of the gravity, front to the theory einsteniana, where the interaction gravitational it is propagate to the speed of the light. For supposed that the theory newtoniana, definida with its debidas restricciones it is thoroughly certain. Me i refer to that the theory newtoniana as theory general, without restricciones, it was untruthful, but it was useful. It was useful because was a explanation of a phenomenon, compatible with the beliefs outstanding (and this include the results of all the experiments accomplished for to prove it), and not import that more late it is discovered that was so alone a case particular. The theories physical current they are útiles, and not seem venturesome to suppose that it is try of cases particular, that they will be amplified in the future. Given the experience, it venturesome seem to suppose it opposite.
We see other example more concrete, this time taken of the biology. It is knows that between males and female of several kinds of mamíferos exist a difference between the time necessary for to reach the crest of pleasure sexual in the copulación. We can to defend the theory of that this difference has a origin evolutionary. It is to say, that the duration and intensity of the pleasure in the relationships sexual it is engraving genetically, and that a trend to a smaller duration and greater intensity for the males front to the female it is sees seleccionada, it is to say, it is extend in the population. Seem logical, already that thus there is a greater probability of that the male deposit its sperm, and for so much, of that arrive to it be fertilized the ovum. For to prove this we could to simulate a joint of entities of two sexes with different "delays orgasmic" individual, engravings in the genes, that transmit its information genetic to its falling, of form that the culmination sexual of one of the two sexes it will be requirement for a fecundación probabilística of the ovum virtual. In various executions we could to prove if in fact it is produces or not this desfase between delays orgasmic. In no case we will demonstrate the theory, but in the case of not it be verified in the simulation we will will have serious indicia of that it is untruthful, and in the case of that it is produce the results wished, we will be able to follow optimistic this line of investigation, without have had that to accomplish no costly study of field.
The following plan reflect the location of several examples of this type of applications of the life artificial
TO continuation i will describe briefly some of the examples of applications proposed, justifying its definitions. More forward i will develop several of they. As i have commented, the utilization of life artificial as tool of simulation not permit to arrive to conclusions definitive, but yes provide a new point of sight, "possible explanations" or "roads promising".
- Artificial Intelligence
- Approach Symbolic
- Systems Expert
- Approach Subsimbólico
- Simulation of systems complex
- Resolution of problems: search, classification, forecast and optimization.
- Nets Neuronales
- Computation Evolutionary
- Simulation of universes: contrast of hypothesis
- Artificial Life
- ¿What it is the life?
- ¿What it is the individuality?
- ¿It is possible the cooperation?
- ¿Puede to last a revolt?
- ¿How operate the universe?
- ¿What they are the time and the space?
- ¿What it is the random?
Two examples carácterísticos within of "¿What it is the life?" they are "¿What it is the individuality?" and "¿It is possible the cooperation?". Both questions, for the form in that the i have outlined, require of a justification.
I have preferred the interrogation "¿What it is the individuality?" to "¿What it is the conscience?" to weigh of that the same topic it is more known for this second definition. The reason it is that the term "conscious" would it be confused with the idea of a entity autorreferente, and without embargo in the language common it is continues supposing that the be conscious it is something more.
The autorreferencia there is been treaty profusely for Douglas R. Hofstadter in "Gödel, Escher, Bach. A eternal and graceful curl". Some examples of autorreferencia they are two mirrors faced, or the joint formed for a chamber of vídeo that it is recording it that it is sample in a monitor that it is precisely the exit of that same chamber of vídeo, or it that in programming it is know as a function recursiva, that it is flame to yes same.
If we reject the mirrors and the rest of examples cited as beings conscious, and in change we accept that the be human it is the be more conscious known, being the rest of the animal also conscious, though more mildly, we arrive to that the characteristic fundamental of the individual conscious not it is the be autorreferente, but the to feel that autorreferencia, and the to feel not seem be complex for that that it is sit, that puede be described better or worse, but for the existence of who it sit, it is to say, in the individuality receiving.
Have elected as second example of study within of "¿What it is the life?", the question "¿It is possible the cooperation?" also require of a explanation. It is try of the study of the viability of different strategies that include interaction between individual, as they are the strategies egotistic, cooperative or rencorosas. I make hincapié in the search of the viability of the cooperation because implicitly it is it that it is it is trying of to find in the majority of the studies. Not it is of to miss, already that the cooperation, if it is fulfil certain requirements, offer the greater benefits potential so much to individual as to groups.
If well seem obvious that all the investigating intend to maintain a objectivity that you permit to predict the evolution of the strategies in a group without that in this affect its preferences personal, also it is evident that the great majority try of to make it with the objective of to increase in it possible the benefit of future or hypothetical communities, and me seem just to emphasize this reality.
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There is a aspect of the definition classic of automata that the program "Ants and Plants" of the application "Ejemplos de Vida" not fulfil. Typically, a cellular automata it is a machine of states finite that consist in a cuadrícula of cells in the which the evolution of each cell depend of its state current and of the states of its neighboring immediate. It is says that in a cellular automata all the automatas simple or cells happen to the following generation to the same time and according to a same algorithm of change that puede to make to vary its state within of a joint limited of states.
Discretización of the time in the automatas
In the program of "Ants and Plants" not occur exactly thus. We go to to see for what, already that the reason permit to draw to to glow the matter of the nature of the time in the automatas. It is says that in a automata "all the cells happen to the following generation to the same time". For that it would be possible to make this of form literal, it is to say, for that two events it is produce, "to the time" of form literal, the time it must to have a nature discreet. It is to say, it is necessary that the time it be formed for certain units indivisible. If the precision with the that it is measures a interval of time puede be infinite, not seem possible to control that two actions it is give to a same time, with infinite precision.
A discussion very similar it is the that proposed the philosopher greek Zenón in its known paradoxes (or aporías), in wheel to the movement, as the of Aquiles that never reach to the tortoise, or the of the arrow immobile. Zenón denied the possibility of the movement of a arrow in flight, already that in each instant the arrow appear "congealed". In fact, if the arrow i had that to travel a number infinite of positions intermediate, the arrow never would it be moved. "If puede to make it", - would rebut a mathematical-. "A arrow puede to travel infinite positions intermediate, with the condition of that solely it is stops a time infinitesimal in each a of the positions". Certain, but ¿they are the infinite and the infinitesimal concepts real of our universe, or so alone abstractions non-existent?
Not it is try of to answer still to these questions, already that they are formulated in terms of our own universe, that not we control. The discussion of Zenón was referred to the universe of the that we form part and of the which not we are owners. Without embargo, here it is try of to endow of movement to beings virtual within of a universe virtual. Now we yes we are the owners, yes we are the constructing of the universe virtual, and as such, us we find exactly with the problem that Zenón anticipated long ago: "not we can to create movement in a universe if this not possess units discreet of time". TO this affirmation there is that to add the puntualización of the mathematical: "to not be that we have of infinite and infinitesimales".
Thus occur, in effect: not we have of infinite neither of infinitesimales. For so much, in the accomplishment of automatas, it is discretiza the time, and it is define it that they will be "units indivisible of time" within of the universe of the automatas.
These "units indivisible of time" not have why be it for we, but yes it they will be for the inhabitants of that world virtual. It is to say, if a automata could it be asked and to investigate on the nature of its own time, would arrive to the conclusion of that this it is discreet.
All it previous it is puede to summarize in the fact of that the expression "all the cells happen to the following generation to the same time" not it is refer to a appraisal strict, but to other of nature practical, that comes to to mean "all the cells it is behave as if happened to the following generation to the same time", it that imply to say "all the cells it is behave as if existed in a universe whose time i had a nature discreet (compound for elements indivisible)".
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Concluding, dices n automatas or cells
Secuencialización of the time in the automatas
TO1, TO2, ..., An
that it is find in a "instant" (unit minimal of time) T=k in the following states
AND1(T=k), AND2(T=k), ..., In(T=k)
it is try of that the new states of each one of the automatas in the following "instant"
AND1(T=k+1), AND2(T=k+1), ..., In(T=k+1)
it is calculate in function of the states of the automatas in the instant previous T=k
In the practical, customarily the automata it is implement in a computer monoprocesador, or it is solved, with paper and pen for a only person. In these cases, the method consist in to calculate the new states of the cells, storing them in a report temporary, but without be used still. A time that it is there is ended the calculation of the new states for all the automatas simple, then it is modify the states of all the automatas, simply copying them of said report temporary. Of this form it is procures that all the cells change of state "to the time".
Here comes the problem. Not always it is possible to make this. There is cases in the that the states future of two automatas they are incompatible. If both it is calculate for separated, and enter in conflict ¿what state to choose?. This occur with the space, it is to say, when one of the components of the state of a automata it is the position. We go to to see how, if the positions of the automatas they can to vary in the time, a of the solutions it is that all the cells not happen to the following generation to the same time.
We imagine two automatas simple mobile, each one occupying its corresponding cell, separated for a cell empty. In certain generation both automatas "see" the cell empty, thus that they can to decide it be moved to the same position, with it that in the following generation will exist the conflict of to decide which of the two has right to to occupy the cell.
In reality, not it is necessary that the automatas they will be actually "mobile". We can to have a grid of automatas immobile and to call "position spatial" to the state of each automata, including the restriction of that two automatas not they can it be found in the same state.
The problem exist, so much if the automata it is solved for a or hundred persons, for one or hundred computadores. It is they can to take all type of decisions for to solve the conflict:
1. To permit both actions, making possible the living together of two automatas in a same cell.
2. To permit both actions, making to disappear to the first automata to the be occupied for the second.
3. To prevent both actions: to the to detect the second action, to disqualify the first, rescinding it.
4. To prevent the movement solely to the second automata, it is to say, to break the norm according to the which "all the cells it is behave as if happened to the following generation to the same time" and to make that the second automata detect that the first already it is there is moved, and to forbid its movement.
The first the i discard for to ignore precisely the restriction that it is intend to implement and that cause the conflict. The second seem a little arbitrational, already that produces a disappearance of automatas difficult of to compensate with creations, but would be interesting of to investigate.
The third seem to have sense, it is would to say that the automatas "crash and bounce", but compared with the fourth it is a complication unnecessary, though also puede be of great interest in other projects.
In the program "Ants and Plants" the decision taken there is been this last, for to seem the more consistent with the objective of the simulation and for its simplicity in programming, already that for to apply this method enough with to use, in time of two reports, a alone, calculating always the new states in function of the information content in this report. To the to use a criterion of simplicity in programming not i am applying solely criteria of economy of time of investigation, but also the criterion general of that "the explanations more simple have greater probability of be certain that the complicated", simply for the fact of that in a theory complicated it is more easy let some "end loosened". De igual forma:a) 1 + 1 = 2 b) 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 = 2tanto a) como b) son ciertas, pero en b) es más fácil cometer un error, esto es, es más fácil creer que b) es cierto y que no lo sea realmente.
The decision for so much it is to break the norm according to the which "all the cells it is behave as if happened to the following generation to the same time". It interesting it is that with this option, if a automata could it be asked and to investigate on the nature of its own time, would follow arriving to the conclusion of that this it is discreet, divided in "instants", and that between one and other "instant" it is happen "simultaneously" multitude of events in all its universe (all this, we recall, from the point of sight of the automata). This it is had to to that the time of the processing it is distribute sequentially between all the "matter" existing in the universe virtual, and neither the automatas neither the machines built for they they will be able never to make observations during the time in that not it is they are "executing", with it that all the environment, from the point of sight of the automata, it will be able to change "of blow", to intervals discreet, though in reality it there is fact sequentially. In fact, for the automatas exist "instants" but the beginning and end of each "instant" it is different for each automata.
It is more, we can to imagine us to the automatas attempting to discover what norm it is there is used for to solve the problem of two states future incompatible. They could to accomplish a experiment seemed to this. The automata TO and the automata B (blue), separated for a unit minimal of space, it is coordinan for to attempt it be moved to the same position that exist between both in the same unit minimal of time. For this, it is send a sign (automata orange mobile) that it is divide, rebota, and arrive to to both simultaneously.
Would be necessary to have of the following properties:
Exist a movement in 8 direcciones The cells negras they are obstacles, automatas immobile The automatas orange bounce in the obstacles The automatas orange they can "empujar" to the blue When a automata orange passes between two red, it is descompone in two automatas that modify its address in a octavo of return, each one in a sense.
The analogy of all this with our universe it is makes patent in the phrase neither the automatas neither the machines built for they they will be able never to make observations during the time in that not it is they are "executing". Transladado to our reality, it is to say, observing our Universe as a automata, diríamos that neither the men neither the machines built for they they will be able never to make observations during the time in that not it is they are "executing". Such time, these reflections serve for to venture hypothesis that describe our own time and space, of agreement with the beliefs current.
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To develop a work on automatas probabilísticos it is a good excuse for to introduce the topic of the random. Customarily it is use terms as "random", "stochastic", "random" and "chaotic" with a sense a so much confusing, for it that i go to to begin with several definitions, of form that remain clear the sense of the terms that i go to to use thereinafter.
Automatas and Random
System chaotic: that so sensitive to the conditions initial that small changes to the principle it is translate in large changes to the final
Random: without cause, and for so much, theoretically impredecible, except in form of probability
Pseudoaleatorio: with cause, but apparently impredecible, or impredecible in the practical had to to the spread of mistakes initial own of a system chaotic, but predictable in form of probability
Determinista: with a cause, and for so much, theoretically predictable, if it is knew the cause
Process random: process whose result it is impredecible, except in form of probability
Process pseudoaleatorio: process whose result it is apparently impredecible, except in form of probability
Process determinista: process whose result it is predictable
Chain random: chain for the that not exist no form of to compress its description
Chain pseudoaleatoria: chain for the that apparently not exist no form of to compress its description
Chain determinista: chain for the that if exist a form of to compress its description
It is there is of to note that a process random not always generate chains random, and puede to generate chains deterministas. Also occur that a process pseudoaleatorio, as a function of random of ordering, puede to generate chains deterministas.
Henri Poincaré tried the topic of the determinismo in 1903 in its book Science and Method: "If we knew with precision infinite the laws of the nature and the situation of the universe, we could to predict exactly the situation of this same universe in a instant subsequent". A forecast it is a classification in the that intervene the time, and puede to have a form declaratory: "Today is going to to rain" or conditional: "If arrive Juan, María it is will march". If the Universe not it would be more that particles in movement, then any event (movement of particles) would it be predicted of form conditional in function of other events previous (more movements of particles).
Apparently this it is certain. For any event seem to exist a cause, but it is they can citar three exceptions to this rule:
- The Big Bang
- The processes cuánticos
- The depresiones
The first of they it is the Big Bang. The Big Bang it is know as the origin of our Universe, of the time and of the space, the first cause or the cause of all it other. In the chain of associations, and of agreement with the theories current, we would arrive to the first instant of the Universe. But then, ¿which it is the cause of the Big Bang? Continuing a reasoning similar to this, San Agustín identified to God as the cause first, as the only event without cause. It is to say, the cause of the Big Bang would be God. For not to enter to discutir the concept of God, enough with reescribir the affirmation as "The cause of the Big Bang it is it would be of our Universe", already it will be this cause God or a fluctuación in the contínuo of vete to to know that thing that inhabit it would be of our universe.
TO end of accounts, the Big Bang such time had a cause. It that it is sure it is that that cause not could it be found in our Universe, already that still not was existing.
The definition that it is there is given of random, thus as the rest of definitions, make reference implicit to our universe. The definition it is would reescribir as:
Random: without cause in our universe, and for so much, theoretically impredecible in our universe, except in form of probability
The second exception they are the processes cuánticos, for the that it is there is demonstrated, not without polemics, that they are impredecibles, and for so much authentic processes random.
In how much to the depresiones, me i refer to the phenomenon psicosomático, not to the accidentes geográficos. It is discuss about of if the depresiones have or not a cause. In any case, many authors coincide in that the lack of a cause it is a element indispensable of a depression.
The principal analogies between the simulations of life artificial and our universe in how much to the random they would be:
1. The Big Bang correspond with the initialization "to the random" of the variable of the model. The cause of the eleción of some or other parameters it is matter personal of the programmer. TO times it is fix conscientiously, in other it is use a function pseudoaleatoria. In any case the decision it is it would be of the universe of the simulation. The programmer it is sample in fact as "god" and cause of the "big bang" of the simulation.
2. The processes cuánticos correspond with those details that it is consider that not it is necessary to simulate, sustituyéndose for a function random. Of the same form that the processes cuánticos they are impredecibles for we, the aspects pseudoaleatorios of the simulation they would have to be impredecibles for the agents that inhabit in that simulation. In the analogy, the processes cuánticos that the men we consider impredecibles correspond with the aspects of the simulation that depend directly of the function pseudoaleatoria. The function pseudoaleatoria of a simulation alone it will be good if it is sample as random for the entities simulated, of form that not they could predecirla neither explicita neither implicitamente. For example, if we are studying phenomena of cooperation between entities simulated compete for certain food that emerge "to the random" in a space two-dimensional, the entities not they would have to be capable of to predict where is going to to appear the next element of food (to not be that precisely it is busque this behavior, as it is obvious). Now well ¿It is possible be seguros of that, implicitamente, the entities not accomplish certain type of forecast, not to level individual, but as group? It is to say, ¿we can be seguros of that the behaviors cooperativos that we are studying not they are influidos for the fact of to use a function pseudoaleatoria in time of a actually random?
3. I wait that the agents that inhabit in the simulation not sufran depresiones. That it would be certain and to know it would to affect seriamente to the ánimo of the programmers.
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A problem that have the simulations of life artificial it is that not we know if they are or not models excessively simplified of the great complexity of the world real. This it is had to to that not it is puede to determine when a function pseudoaleatoria it is it sufficiently good as for to accept the results obtained in a simulation. For now it only that we know it is that if we use a function pseudoaleatoria wrong, easily predictable, not we will be able to rely in the results. In general, it that it is makes it is precisely that: not to rely in the results.
The Problem of the simplification
The functions pseudoaleatorias it is use in Artificial Life for to avoid the modelización of those aspects that result too complex to represent. The functions pseudoaleatorias they are necessary, because in general always there is some aspect that result too complex to represent.
If we see the model as a box black that us offer a result, as for example a forecast, it is possible to obtain this without to use functions pseudoaleatorias, but this not would be a simulation. In change, if we want that the model internally it is asemeje it but possible to the reality, if we want to represent the aspects internal, we will have to to use functions pseudoaleatorias that sustituyan precisely the behaviors that for reasons practical not they can be represented of other form in the model.
For this we will have to to analyze the aspects complex of form estadistica, implementandolos through different functions of probability.
Modelizar situations of form statistics it is a process mental that all we make in our cabeza to daily. For example, to the to say that there is a 50% of probabilities of to have a child in time of a girl. When not we can to know the complicated mechanisms internal of a process, as it is the determination (it is understand, natural) of the sex, we attended to a calculation of probabilities, it is to say, we assign a confidence to a event in function of the number of times that it is there is produced previously. For example, the directing of the maternidad obtain of the observacion the knowledge of that the 50% of the births tend be children. With 100 partos inminentes, and supposing that the attention to the recien nacidos it would be different according to the sex, the directing puede elucubrar if the center possess the resources necessary for to attend to "50 children and 50 niñas"; probably "48 children and 52 niñas" or something asi as "32 children and 68 niñas", but almost sure that not "97 children and 3 niñas". For to make these reflections the directing it is using mentalmente a generating of random. For 100 births, the function of probability of "i number of children" it is a distribution normal of mean 50. The generating of pseudoazar it will be a function that will return values between 0 and 100, being the 50 the more probable and the 0 and the 100 the less probable.
In simulation, given that a precision infinite not it is possible, it is to say, already that the model not it is a copy exact, but a simplification, we have that to work with a margin of mistake. If the system to to study has nature chaotic, it is to say, if occur that small differences in the conditions initial they can to provoke large mistakes in the result final, then not it is possible to rely in the model.
Resumiendo, the problems they are:
1.- To determine if it is acceptable to establish a level for below of the which it is sustituye the simulation of the processes real that it is they are studying for a "box black" or function random or pseudoaleatoria that produces a behavior estadísticamente similar to the that it is gives in the reality toward the rest of the model. It is to say, it is try of to decide if it is acceptable or not to use a function random or pseudoaleatoria. Such time not it will be acceptable, or such time not it will be possible to decide if it is acceptable or not.
2.- In the case of not be acceptable or to suppose that not it is acceptable, the model would have to be a copy identica of the original, until the level of granularidad maximum (particles elementales, elements constructing of all it other). In the case of be acceptable or to suppose that it it is, the problem it is to define where it is that level, it is to say, in that cases we go to to use the function random or pseudoaleatoria.
3.- A time defined in that cases we go to to use the function pseudoaleatoria, the third problem it is to define the caracerísticas of that function for power to use a it sufficiently good. In the case of not power decidirlo, it but sure would be, or well to use the better function of that it is disponga, or to use a in the that intervene processes cuánticos, that de acuerdo with the theories current they are authentic generating of random.
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DIAZ, JosuKa. Departamento de Lenguajes y Sistemas Informáticos. 1996. "Teoría de Autómatas y Lenguajes Formales". Universidad de Deusto. http://www.redcientifica.com/gaia/auto_i.htm
Cellular Autómatas Aranguren, Ricardo
Gods, Men, Deamons and Machines. Rosa Aguilar, Luciano de la
Prata, Stephen. 1993. "Vida Artificial". Ed. Anaya. Referencia básica en castellano y muy buen libro para iniciarse en el tema. Hofstadter, Douglas R. 1987. "Gödel, Escher, Bach. Un eterno y gracil bucle". Tusquets Editores & CONACYT. Obra maestra, trata acerca de la recursividad, vista desde diversos y apasionantes puntos de vista. http://www.uv.es/~buso
Página web de Vicent Castellar-Busó Vicent.Castellar@uv.es. Departament de Matemática Aplicada, Universidad de Valencia, con unos trabajos muy interesantes.
Wiener, Norbert. 1988. "Dios & Golem, S.A." Siglo XXI editores. México, D. F. 1988. http://rev.upv.es/conjetur.htm
Morey, Miguel. Conjeturas sobre Zenon de Elea.
Chalmers, David J. "El problema de la consciencia". Revista Investigación y ciencia, Febrero 1996. Gell-Mann, Murray. 1994. "El Quark y el Jaguar. Aventuras en lo simple y lo complejo". Tusquets editores. Serie metatemas. Adams, Douglas. 1983. "Guía del Autoestopista Galáctico". Editorial Anagrama. Ciencia Ficción, novela, relato de humor... no está claro de que se trata pero me parto de risa cada vez que lo vuelvo a leer. Presenta el planeta Tierra como un super-ordenador biológico.
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