1. 1 Cellular Automata and Applications Ajith Abraham Telephone Number: (918) 5948188 E-mail: aa@cs.okstate.edu WWW: http://ajith.softcomputing.net

2. 2 Automata Theory An automaton is a system which obtains, transforms, transmits and uses information to perform its functions without direct human participation. It is self- operational (Lerner, 1972). The most general form of deterministic automata is the Turing machine. The notions of message, amount of disturbance or 'noise', quantity of information, coding technique, etc. in the study of automata are related to cybernetics.

3. 3 Machine Intelligence refers back to 1936, when Alan M Turing proposed, the idea of universal mathematics machine, a theoretical concept in the mathematical theory of computability. Turing and Emil Post independently proved that determining the decidability of mathematical propositions is equivalent to asking what sorts of sequences of a finite number of symbols can be recognized by an abstract machine with a finite set of instructions. Such a mechanism is now known as a Turing machine Turing Machine: http://www.turing.org.uk/turing/http://www.turing.org.uk/turing/ Turing Machine

4. 4 A Turing machine is an abstract representation of a computing device. It consists of a read/write head that scans a (possibly infinite) one-dimensional (bi-directional) tape divided into squares, each of which is inscribed with a 0 or 1. Computation begins with the machine, in a given "state", scanning a square. It erases what it finds there, prints a 0 or 1, moves to an adjacent square, and goes into a new state. This behavior is completely determined by three parameters: (1) the state the machine is in, (2) the number on the square it is scanning, and (3) a table of instructions. Turing Machine

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6. 6 The table of instructions specifies, for each state and binary input, what the machine should write, which direction it should move in, and which state it should go into. (E.g., "If in State 1 scanning a 0: print 1, move left, and go into State 3".) The table can list only finitely many states, each of which becomes implicitly defined by the role it plays in the table of instructions. These states are often referred to as the "functional states" of the machine. A Turing machine, therefore, is more like a computer program (software) than a computer (hardware). Any given Turing machine can be realized or implemented on an infinite number of different physical computing devices. Turing Machine

7. 7 Automata: key concepts and principles: Deterministic automata Statistical automata Memoryless automata Finite memory automata Infinite memory automata

8. 8 Deterministic automata The output is uniquely determined by the input sequences; that is, you get a definite output given any input. The behavior of such automata can be accurately predicted if the transfer operator is known and given in the form of a table of a logical function, and if you also know the initial state and the input sequence. http://www.article19.com/shockwave/lifegame.htm

9. 9 Statistical Automata Random output sequences are generated given any fixed input. The amount of randomness can be set by applying a probability of any output given the systems current state and input sequence.

10. 10 Memoryless automata Recognizes only one input at a time and produces output based on that input. The output is not influenced by any additional inputs which arrived before that one. The reaction time of the automaton is constant for all input signals. The internal state of such an automaton is independent from any external action.

11. 11 Finite Memory Automata It refers to the type of automata where the group of output signals generated at a given quantized time depends not only on the signals applied at the same moment, but also on those which arrived earlier. These preceding external actions (or fragments of them) are recorded in the automaton by a variation of its internal state. The reaction of such an automaton is uniquely determined by the group of input signals which has arrived and by its internal state at a given time. These factors also determine the state into which the automaton goes.

12. 12 Infinite memory automata It refers to an abstract circuit of a logical automaton which in principle is suitable for realizing any information processing algorithm. Turing machine belongs to this type of automata.

13. 13 Cellular Automata (CA) Cellular Automata were introduced in the late 1940´s by John von Neumann (von Neumann, 1966; Toffoli, 1987) and Stanislaw Ulam. From the more practical point of view it was introduced in the late 1960´s when John Horton Conway developed the Game of Life (Gardner, 1970; Dewdney, 1989; Dewdney, 1990). CA´s are discrete dynamical systems and are often described as a counterpart to partial differential equations, which have the capability to describe continuous dynamical systems.

14. 14 The meaning of discrete is, that space, time and properties of the automaton can have only a finite, countable number of states. The basic idea is not to try to describe a complex system from "above" - to describe it using difficult equations Not to describe a complex system with complex equations, but let the complexity emerge by interaction of simple individuals following simple rules.

15. 15 Hence the essential properties of a CA are a regular n-dimensional lattice (n is in most cases of one or two dimensions), where each cell of this lattice has a discrete state a dynamical behavior, described by so called rules. These rules describe the state of a cell for the next time step, depending on the states of the cells in the neighborhood of the cell.

16. 16 Building Cellular Automata The Cell The basic element of a CA is the cell. A cell is a kind of a memory element and stores - to say it with easy words - states. In the simplest case, each cell can have the binary states 1 or 0. In more complex simulation the cells can have more different states.

17. 17 The Lattice These cells are arranged in a spatial web - a lattice. The simplest one is the one dimensional "lattice", meaning that all cells are arranged in the form of a line (string?). The most common CA´s are built in one or two dimensions. Whereas the one dimensional CA has the big advantage, that it is very easy to visualize. The states of one time step are plotted in one dimension, and the dynamic development can be shown in the second dimension.

18. 18 A flat plot of a one dimensional CA hence shows the states from time step 0 to time step n. A two dimensional plot can evidently show only the state of one time step. So visualizing the dynamics of a 2D CA is by that reason more difficult. By that reasons and because 1D CA´s are generally more easy to handle first of all; the one dimensional CA´s have been exploited be theoreticians.

19. 19 Neighborhoods To introduce dynamic into the system, we have to add rules. The "job" of these rules is to define the state of the cells for the next time step. In cellular automata a rule defines the state of a cell in dependence of the neighborhood of the cell.

20. 20 von Neumann Neighborhood four cells, the cell above and below, right and left from each cell are called the von Neumann neighborhood of this cell. The radius of this definition is 1, as only the next layer is considered. Moore Neighborhood the Moore neighborhood is an enlargement of the von Neumann neighborhood containing the diagonal cells too. In this case, the radius r =1 Extended Moore Neighborhood equivalent to description of Moore neighborhood above, but neighborhood reaches over the distance of the next adjacent cells. Hence the r=2 (or larger). Margolus Neighborhood a completely different approach: considers 2x2 cells of a lattice at once.

21. 21 von Neumann Neighborhood Moore Neighborhood Extended Moore Neighborhood

22. 22 What to do with cells at borders? The influence depends on the size of the lattice. Example: In a 10x10 lattice about 40% of the cells are border cells, in a 100x100 lattice only about 4% of the cells are of that kind. Anyway, this problem must be solved. Two solutions of this problem are common: 1. Opposite borders of the lattice are "sticked together". A one dimensional "line" becomes following that way a circle, a two dimensional lattice becomes a torus. 2. the border cells are mirrored: the consequence are symmetric border properties. The more usual method is the possibility 1.

23. 23 Applying Rules An example of "macroscopic" dynamic resulting from local interaction is "the wave" in a - say soccer-stadium. Each person reacts only on the "state" of his neighbor(s). If they stand up, he will stand up too, and after a short while, he sits down again. Local interaction leads to global dynamic.

24. 24 Every group of states of the neighborhood cells is related to a state of the core cell. E.g. consider a one-dimensional CA: a rule could be "011 -> x0x", what means that the core cell becomes a 0 in the next time step (generation) if the left cell is 0, the right cell is 1 and the core cell is 1. Every possible state has to be described. Totalistic Rules: the state of the next state core cell is only dependent upon the sum of the states of the neighborhood cells. E.g. if the sum of the adjacent cells is 4 the state of the core cell is 1, in all other cases the state of the core cell is 0. Legal Rules: a special kind of totalistic rules are the legal rules. As it is not of advantageous in most cases to use rules that produce a pattern from total zero-state lattices (all cells in the automaton are 0), Wolfram defined the so called legal rules. These rules are a subset of all possible rules, a selection of rules that produce no one´s from zero-state lattices.

25. 25 Important properties CA´s develop in space and time CA is a discrete simulation method. Hence Space and Time are defined in discrete steps. CA is built up from cells, that are lined up in a string for one-dimensional automata arranged in a two or higher dimensional lattice for two- or higher dimensional automata the number of states of each cell is finite the states of each cell are discrete all cells are identical the future state of each cell depends only of the current state of the cell and the states of the cells in the neighborhood the development of each cell is defined by so called rules

26. 26 The Behavior of CA´s Universality and the Turing-Stuff A system that is capable to do universal computation is able to perform any finite algorithm. Only a CA calculating for an infinite period of time can be universal.

27. 27 Class 1 cellular automata After a finite number of time-steps, class one automata tend to achieve an unique state from (nearly) all possible starting conditions. Class 2 cellular automata This type of automata usually creates patterns that repeat periodically (typical with small periods) or are stable. Class 3 cellular automata From nearly all starting conditions, this type of CA´s lead to non- periodic - chaotic patterns. The statistical properties of these patterns and the statistical properties of the starting patterns are almost identical (after a sufficient period of time). Class 4 cellular automata After finite steps of time, this type of CA´s usually "dies" - the state of all cells becomes zero. Nevertheless a few stable (periodic) figures are possible. One popular example of an automaton of this type is the Game of Life.

28. 28 CA Applications The Game of Life (GOL) was one of the first "applications" showing that cellular automata are capable of producing dynamic patterns and structures. The GOL is "plays" on a two dimensional lattice with binary cell states, Moore neighborhood and arbitrary border conditions. To be vivid: a 1 can be interpreted that the cell is "living", a 0 that the cell is "dead".

29. 29 a cell that is dead at the time step t, becomes alive at time t+1 if exactly three of the eight neighboring cells at time t were alive. a cell that is alive at time t dies at time t+1 if at time t less than two or more than three cells are alive.

30. 30 Gas Models Rules base on 2x2 parts of the lattice "o" stands for a particle (or a billiard ball) and a "." stands for an empty cell After applying the rule and proceeding to the next time step, the 2x2 raster is moved for one block diagonally. Hence the movement of a particle needs two time-steps.

31. 31 Gas simulation of collision of two particles

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