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Overview
The Stochastic Game of Life or more formally (and hence will be used interchangeably), Stochastic Cellular Automation, is an important extension to the well-known mathematical zero-player game, “Conway’s Game of Life”, which is a Cellular Automaton devised by the British mathematician John Horton Conway in 1970. It is a game governed by several deterministic simple rules and with initial condition as given that could generate fantastic complex patterns as the evolution goes on. Loosely speaking, the Stochastic Game of Life is adding randomness element to make the evolution process “more fluctuated” but more approaching to the real world system. Consequently, by incorporating the randomness into the original model, the Stochastic Cellular Automation makes “Conway’s Game of Life” a special case of it. The impetus for the interest in Stochastic Cellular Automation is that although deterministic Cellular Automation could exhibit complexity, the stochastic one may be more accurate in modeling physical systems because they allow for random fluctuations which are ubiquitous as experimental noise.

1.1 Game of life
The universe of the “Game of Life” is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, alive or dead. Every cell interacts with its eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur: The initial pattern constitutes the seed of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed—births and deaths occur simultaneously, and the discrete moment at which this happens is sometimes called a tick (in other words, each generation is a pure function of the preceding one). The rules continue to be applied repeatedly to create further generations.
 * 1) Any live cell with fewer than two live neighbors dies, as if caused by under-population.
 * 2) Any live cell with two or three live neighbors lives on to the next generation.
 * 3) Any live cell with more than three live neighbors dies, as if by overcrowding.
 * 4) Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

1.2 Incorporating the Randomness
As the evolution process of “Game of Life” is uniquely determined by the initial configuration and the transition rules, it is naturally to compare “Game of Life” with Markov Process, which is uniquely determined by the initial probability distribution and transition probability matrix. Researchers later on has introduced the randomness to this model by assigning an initial probability distribution and make the deterministic transition rules stochastically, i.e. the probability distribution of state at next period rather than the state of the next period depends on its current state (the state of some or all of the eight neighbor cells) of a cell. Nowadays, the Stochastic Cellular Automation has been widely used in many different areas to study the evolution behavior of the dynamic system. Smith III (1972) and Mahajan (1992) has done research with Stochastic Cellular Automation in language theory; Nishio and Kobuchi (1975) has introduced this tool to reliable computation area; K. C. Clarke, S Hoppen (1997) analyze the urbanization phenomenon with this model, etc. Unlike the Deterministic Cellular Automation, which has been studied for decades and has a relatively well-established framework such as Wolfram’s qualitative classification and Gutowitz’s quantitative measure, the work done on Stochastic Cellular Automation, however, has dealt primarily with applications aforementioned. Few works have addressed the important issues of classification and characterization. In the next section, I will provide a characterization of the Stochastic Cellular Automation which I consider as comprehended.

General setting of Stochastic Cellular Automation in 1-Dimension
A 1-dimensional cellular automaton is an ordered, linear array of N cells. The state of a cell at position $$ x $$ and time step $$ t$$is given by $$ v_x^t $$. Each cell has two possible states, 0 or 1. For Stochastic Cellular Automation, the evolution of a cell $$ v_x^t $$ depends on a probabilistic rule, $$\boldsymbol{\Phi}(v_x^t,v_{x+1}^t)$$,which is a function of the state of a neighborhood consisting of the cell $$ v_x^t $$ itself and its neighbor to the right,$$ v_{x+1}^t $$. The state $$ v_x^t $$ of at the next time step is then given by:

$$ v_x^{t+1}=\boldsymbol{\Theta}(\boldsymbol{\Phi}(v_x^t,v_{x+1}^t)-X) $$ (1)

where $$\boldsymbol{\Theta}$$ (•)is the Heaviside step function (a discontinuous function whose value is zero for negative argument and one for positive argument) and X$$ \in $$[0,1] is an evenly-distributed, continuous random variable. The rule $$\boldsymbol{\Phi}(i,j)$$ is simply a look-up table of probabilities $$\phi_{ij}$$:

$$\boldsymbol{\Phi}(i,j)$$ = {$$\phi_{ij}$$} = {$$\phi_{11},\phi_{10},\phi_{01},\phi_{00}$$}.(2)

We will refer to $$\phi_{ij}$$ as unit probabilities where $$\phi_{ij}\in$$ [0,1]. $$\phi_{ij}$$ gives the probability that $$ v_x^{t+1} $$ has the value 1, given $$(v_x^t, v_{x+1}^t)=(i,j)$$:

$$\phi_{ij}=P(v_x^{t+1}=1|v_x^t=i,v_{x+1}^t=j)$$.(3)

The rule space of $$\boldsymbol{\Phi}$$ is then a 4-dimensional unit cube where the vertices are the deterministic rules (see Figure 1 for an example). These four unit probabilities ($$\phi_{ij}$$) along with initial conditions and boundary conditions determine the dynamics of a stochastic Cellular Automation. Initial conditions are given by randomly assigning each cell a value of 0 or 1 with probability ½. One could use circular boundary conditions where the neighbor to the right of the right-most cell,$$ v_N^t $$, is the left-most cell,$$ v_1^t $$.



3.1 Heavy Traffic System
Consider a one dimension Stochastic Cellular Automation to model the traffic system, where the road is represented by a string of cells, which are either occupied by exactly one car (with speed $$ v\in[0,v_{max}]\cap N_0, v_{max}\in N$$) or empty. A conﬁguration is characterized by the positions of the cars and their respective velocities. Usually periodic boundary conditions are employed (circular lane), and one uses the following stochastic update rules:

	Acceleration: If the velocity $$ v $$ is lower than $$ v_{max} $$and if the distance to the next car (number of empty sites ahead) is larger than $$ v+1 $$, the speed is increased by one ($$v\rightarrow v+1$$)

	Slowing down: Let $$ d $$ be the distance to the next car ahead. If $$d < v$$,, then the speed is reduced to $$d-1$$ ($$v\rightarrow d-1$$). (There are no accidents in this model.)

	Stochastic braking: With probability $$ p_{slow} $$the velocity of a moving vehicle is reduced by one($$v\rightarrow v-1$$).

	Propagation: Each car proceeds by the value of its velocity $$ v $$.

The dynamics of the traﬃc ﬂow is determined by the initial conﬁguration, the car density $$ \rho\triangleq \frac{n_{cars}}{n_{sites}} $$, the maximum velocity $$ v_{max} $$ and the braking probability $$ p_{slow} $$. For very small densities (”non-interacting cars”), one has an average speed of $$ \left \langle v \right \rangle = v_{max}-p_{slow} $$. If the density of cars is increased, the traﬃc ﬂow is still laminar for $$ p_{slow} $$=0, but the average speed is reduced. In fact, for high densities, $$ v_{max} $$ has little inﬂuence on$$ \left \langle v \right \rangle$$. When stochastic braking is introduced by choosing $$ p_{slow}>0 $$, laminar ﬂow may convert to more erratic one and a further increase of car density leads to stop-and-go waves (See Figure 2).

Figure 2: CA traﬃc model: trajectories of cars: a) only one car on the road, b) laminar traﬃc ﬂow, c) laminar ﬂow with same average speed $$ \left \langle v \right \rangle$$as in (b) although the maximum speed $$ v_{max} $$ has been dramatically increased, d) erratic traﬃc ﬂow due to stochastic braking, e) stop-and-go waves due to stochastic braking and an increased car density

3.2 Wildland Fire Spread Dynamics
The wildland fire spread dynamics could also be model as a Stochastic Cellular Automation. Consider a two-dimensional lattice L$$\times$$L, each cell is defined by: i. PD: probability of the initial distribution of the burnable cell and hence for each cell, initially it has PD chance to be burnable and 1- PD to be unburnable.
 * Its discrete position (i, j) in the lattice, where i = 1, 2,…,L and j = 1, 2,…,L;
 * Each cell has a state $$ S_{(i,j)}^n\in$${E,V,F,O} where E denotes the unburnable cells, V denotes cells with potential to burn, F denotes the burning cell and O is the burned cell.
 * The set of finite neighbor cells N(i, j) consisting eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent.
 * Transition function: $$\boldsymbol{f}$$ : $$ S_{(i,j)}^n\times S_{N(i,j)}^n\rightarrow S_{(i,j)}^{n+1}$$ that calculate the future cell state. The time evolution of the model is driven by the interaction between the cell states and the cell neighborhood states. Starting from a given configuration of cells initial states, the cellular automaton self-replicates the sequent cell states. The cellular automata model is stochastic because the state transition function is performed according to probabilities values.

''ii. PB: probability of a burning cell becomes a burnt cell at each step n''.

''iii. PI'': probability of a burning cell making its neighbor burnable cell become combusted. Figure 3 shows the transition pattern. Figure 3: the transition is one direction. The burnable cell could transit to a burning cell and a burning cell could transit to a burnt cell but the other way around is not allowed. And the unburnable cell it isolated with the other three kinds of cells.

The Figure 4 characterizes some fire patterns using different values of PD, PB and PI, time step n = 100 for a lattice of size 201×201 with the fire starting in the middle of the lattice. Each cell state along the lattice is represented by colors that which are, empty cell (black), vegetation cell (green), burning cell (red) and burnt cell (gray). Figures 4: Comparing the a), b) and c). Higher values of PI are related with fire fronts which spread most quickly.

Reference
[1]	Smith Iii, A. 1972. Real-time language recognition by one-dimensional cellular automata. J. Comput. Syst. Sci. 6, 233–253.

[2]	Mahajan, M. 1992. Studies in language classes defined by different types of time-varying cellular automata. Ph.D. Dissertation.

[3]	Nishio, H. And Kobuchi, Y. 1975. Fault tolerant cellular spaces. J. Comput. Syst. Sci. 11, 150–170.

[4]	Clarke, K. C. And Hoppen, S. 1997. A self-modifying cellular automaton model of historical urbanization in the San Francisco Bay area. Environment and Planning B: Planning and Design, Vol. 24, 247–261

[5]	Almeida, R. M. And Macau, E. E. N. 2010. Stochastic cellular automata model for wildland fire spread dynamics. 9th Brazilian Conference on Dynamics, Control and their Applicarions, June 07-11, 2010.

[6]	Wolfgang von der Linden And Ewald Schachinger. 2005. Computer simulation, lecture notes.

[7]	Klaus Lichtenegger. 2005. Stochastic cellular automation models in disease spreading and ecology

[8]	Benny Brown. 2003. Phase transition of two-neighbor stochastic cellular automata.

[9]	http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life