On Automata II

"It bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what a tiny piece of space-time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple, like the checker board with all its apparent complexities." (Richard P Feynman, "The Character of Physical Law", 1965)

"Perhaps the most exciting implication [of CA representation of biological phenomena] is the possibility that life had its origin in the vicinity of a phase transition and that evolution reflects the process by which life has gained local control over a successively greater number of environmental parameters affecting its ability to maintain itself at a critical balance point between order and chaos." ( Chris G Langton, "Computation at the Edge of Chaos: Phase Transitions and Emergent Computation", Physica D (42), 1990)

"Cellular automata are now being used to model varied physical phenomena normally modelled by wave equations, fluid dynamics, Ising models, etc. We hypothesize that there will be found a single cellular automaton rule that models all of microscopic physics; and models it exactly." (Edward Fredkin, "Nonlinear Phenomena", Physica D (45), 1990)

"Over and over again we will see the same kind of thing: that even though the underlying rules for a system are simple, and even though the system is started from simple initial conditions, the behavior that the system shows can nevertheless be highly complex." (Stephen Wolfram, "A New Kind of Science", 2002)

"Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell." (Ramon Alonso-Sanz, "Cellular Automata with Memory", 2009) 

"Cellular automata are dynamical systems that are discrete in space, time, and value. A state of a cellular automaton is a spatial array of discrete cells, each containing a value chosen from a finite alphabet. The state space for a cellular automaton is the set of all such configurations." (Burton Voorhees, "Additive Cellular Automata", 2009) 

"Cellular automata (CA) are idealizations of physical systems in which both space and time are assumed to be discrete and each of the interacting units can have only a finite number of discrete states." (Andreas Schadschneider et al, "Vehicular Traffic II: The Nagel–Schreckenberg Model" , 2011)

"Discrete dynamic systems that evolve in space and time. A cellular automaton is composed of a set of discrete elements – the cells – connected with other cells of the automaton, and in each time unit each cell receives information about the current state of the cells to which it is connected. The cellular automaton evolve according a transition rule that specifies the current possible states of each cell as a function of the preceding state of the cell and the states of the connected cells." (Francesc S. Beltran et al, "A Language Shift Simulation Based on Cellular Automata", 2011)

"Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogenous, simple units, the atoms or cells. [...] Secondly, CA are abstract: they can be specified in purely mathematical terms and physical structures can implement them. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems." (Francesco Berto & Jacopo Tagliabue, "Cellular Automata", Stanford Encyclopedia of Philosophy, 2012) [source]

"One of the unique features of typical CA [ cellular automata] models is that time, space, and states of cells are all discrete. Because of such discreteness, the number of all possible state-transition functions is finite, i.e., there are only a finite number of “universes” possible in a given CA setting. Moreover, if the space is finite, all possible configurations of the entire system are also enumerable. This means that, for reasonably small CA settings, one can conduct an exhaustive search of the entire rule space or phase space to study the properties of all the 'parallel universes'." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

Stephen Wolfram - Collected Quotes

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata may be considered as discrete dynamical systems. In almost all cases, cellular automaton evolution is irreversible. Trajectories in the configuration space for cellular automata therefore merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto 'attractors'. These attractors typically contain only a very small fraction of possible states. Evolution to attractors from arbitrary initial states allows for 'self-organizing' behaviour, in which structure may evolve at large times from structureless initial states. The nature of the attractors determines the form and extent of such structures." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica D (10), 1984)

"Computation offers a new means of describing and investigating scientific and mathematical systems. Simulation by computer may be the only way to predict how certain complicated systems evolve." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"Scientific laws give algorithms, or procedures, for determining how systems behave. The computer program is a medium in which the algorithms can be expressed and applied. Physical objects and mathematical structures can be represented as numbers and symbols in a computer, and a program can be written to manipulate them according to the algorithms. When the computer program is executed, it causes the numbers and symbols to be modified in the way specified by the scientific laws. It thereby allows the consequences of the laws to be deduced." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"Scientific laws have conventionally been constructed in terms of a particular set of mathematical functions and constructs, and they have often been developed as much for their mathematical simplicity as for their capacity to model the salient features of a phenomenon." (Stephen Wolfram, "Computer Software in Science and Mathematics", 1984)

"Computational reducibility may well be the exception rather than the rule: Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be formally undecidable." (Stephen Wolfram, Undecidability and intractability in theoretical physics", Physical Review Letters 54 (8), 1985)

"In the existing sciences much of the emphasis over the past century or so has been on breaking systems down to find their underlying parts, then trying to analyze these parts in as much detail as possible. [...] But just how these components act together to produce even some of the most obvious features of the overall behavior we see has in the past remained an almost complete mystery." (Stephen Wolfram, "A New Kind of Science", 2002)

"Over and over again we will see the same kind of thing: that even though the underlying rules for a system are simple, and even though the system is started from simple initial conditions, the behavior that the system shows can nevertheless be highly complex." (Stephen Wolfram, "A New Kind of Science", 2002)

"The idea of describing behavior in terms of mathematical equations works well where the behavior is fairly simple. It almost inevitably fails whenever the behavior is more complex. Indeed, there are many common phenomena about which theoretical science has had remarkably very little to say. Degree of difficulty encountered in mathematical representation of a phenomenon increases exponentially by the degree of its complexity." (Stephen Wolfram, "New Kind of Science", 2002)

"Our best way of summarizing and communicating knowledge tends to be through language. And when mathematics became formalized, it did so essentially by emulating the symbolic structure of traditional human natural language." (Stephen Wolfram, Some Modern Perspectives on the Quest for Ultimate Knowledge, 2007)

"Could it be that some place out there in the computational universe, we might find our physical universe?" (Stephen Wolfram, "Computing a Theory of Everything", 2010)

On Automata I

"Ninety-nine [students] out of a hundred are automata, careful to walk in prescribed paths, careful to follow the prescribed custom. This is not an accident but the result of substantial education, which, scientifically defined, is the subsumption of the individual." (William T Harris, "The Philosophy of Education", 1889)

"We are automata entirely controlled by the forces of the medium being tossed about like corks on the surface of the water, but mistaking the resultant of the impulses from the outside for free will. The movements and other actions we perform are always life preservative and tho seemingly quite independent from one another, we are connected by invisible links." (Nikola Tesla, "My Inventions", 1919)

"Besides electrical engineering theory of the transmission of messages, there is a larger field [cybernetics] which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method." (Norbert Wiener, "Cybernetics", 1948)

"Automata have begun to invade certain parts of mathematics too, particularly but not exclusively mathematical physics or applied mathematics. The natural systems (e.g., central nervous system) are of enormous complexity and it is clearly necessary first to subdivide what they represent into several parts that to a certain extent are independent, elementary units. The problem then consists of understanding how these elements are organized as a whole. It is the latter problem which is likely to attract those who have the background and tastes of the mathematician or a logician. With this attitude, he will be inclined to forget the origins and then, after the process of axiomatization is complete, concentrate on the mathematical aspects." (John Von Neumann, "The General and Logical Theory of Automata", 1951)

"A world of automata – of creatures that worked like machines – would hardly be worth creating." (Clive S Lewis, "Mere Christianity", 1952)

"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Cellular automata may be considered as discrete dynamical systems. In almost all cases, cellular automaton evolution is irreversible. Trajectories in the configuration space for cellular automata therefore merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto 'attractors'. These attractors typically contain only a very small fraction of possible states. Evolution to attractors from arbitrary initial states allows for 'self-organizing' behaviour, in which structure may evolve at large times from structureless initial states. The nature of the attractors determines the form and extent of such structures." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite; that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. [...] We take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata." (Edward Fredkin, "A New Cosmogony", PhysComp ’92: Proceedings of the Workshop on Physics and Computation, 1993)

On Information Theory I

"[…] information theory is characterised essentially by its dealing always with a set of possibilities; both its primary data and its final statements are almost always about the set as such, and not about some individual element in the set." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Just as in applied statistics the crux of a problem is often the devising of some method of sampling that avoids bias, our problem is that of finding a probability assignment which avoids bias, while agreeing with whatever information is given. The great advance provided by information theory lies in the discovery that there is a unique, unambiguous criterion for the 'amount of uncertainty' represented by a discrete probability distribution, which agrees with our intuitive notions that a broad distribution represents more uncertainty than does a sharply peaked one, and satisfies all other conditions which make it reasonable." (Edwin T Jaynes, "Information Theory and Statistical Mechanics" I, 1956

"Cybernetics is concerned primarily with the construction of theories and models in science, without making a hard and fast distinction between the physical and the biological sciences. The theories and models occur both in symbols and in hardware, and by 'hardware’ we shall mean a machine or computer built in terms of physical or chemical, or indeed any handleable parts. Most usually we shall think of hardware as meaning electronic parts such as valves and relays. Cybernetics insists, also, on a further and rather special condition that distinguishes it from ordinary scientific theorizing: it demands a certain standard of effectiveness. In this respect it has acquired some of the same motive power that has driven research on modern logic, and this is especially true in the construction and application of artificial languages and the use of operational definitions. Always the search is for precision and effectiveness, and we must now discuss the question of effectiveness in some detail. It should be noted that when we talk in these terms we are giving pride of place to the theory of automata at the expense, at least to some extent, of feedback and information theory." (Frank H George, "The Brain As A Computer", 1962)

"Now we are looking for another basic outlook on the world - the world as organization. Such a conception - if it can be substantiated - would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with ‘systems’, ‘wholes’ or ‘organizations’; and in their totality, they herald a new approach." (Ludwig von Bertalanffy, "General System Theory", 1968)

"The general notion in communication theory is that of information. In many cases, the flow of information corresponds to a flow of energy, e. g. if light waves emitted by some objects reach the eye or a photoelectric cell, elicit some reaction of the organism or some machinery, and thus convey information." (Ludwig von Bertalanffy, "General System Theory", 1968) 

"My analysis of living systems uses concepts of thermodynamics, information theory, cybernetics, and systems engineering, as well as the classical concepts appropriate to each level. The purpose is to produce a description of living structure and process in terms of input and output, flows through systems, steady states, and feedbacks, which will clarify and unify the facts of life." (James G Miller, "Living Systems: Basic Concepts", 1969)

"The 'flow of information' through human communication channels is enormous. So far no theory exists, to our knowledge, which attributes any sort of unambiguous measure to this 'flow'." (Anatol Rapoport, "Modern Systems Research for the Behavioral Scientist", 1969)

"Probability plays a central role in many fields, from quantum mechanics to information theory, and even older fields use probability now that the presence of "noise" is officially admitted. The newer aspects of many fields start with the admission of uncertainty." (Richard Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"The field of 'information theory' began by using the old hardware paradigm of transportation of data from point to point." (Marshall McLuhan & Eric McLuhan, Laws of Media: The New Science, 1988)

"Without an understanding of causality there can be no theory of communication. What passes as information theory today is not communication at all, but merely transportation." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence: the use of mathematical logic to understand the operation of the nervous system; the invention of information processing machines (as digital computers), thus laying the basis for artificial intelligence; the establishment of the metadiscipline of system theory, which has had an imprint in many branches of science, such as engineering (systems analysis, control theory), biology (regulatory physiology, ecology), social sciences (family therapy, structural anthropology, management, urban studies), and economics (game theory); information theory as a statistical theory of signal and communication channels; the first examples of self-organizing systems. This list is impressive: we tend to consider many of these notions and tools an integrative part of our life […]" (Francisco J Varela, "The Embodied Mind", 1991)

"[...] the mean information of a message is defined as the amount of chance (or randomness) present in a set of possible messages. To see that this is a natural definition, note that by choosing a message, one destroys the randomness present in the variety of possible messages. Information theory is thus concerned, as is statistical mechanics, with measuring amounts of randomness. The two theories are therefore closely related." (David Ruelle, "Chance and Chaos", 1991)