Open Synchronous Cellular Learning Automata Hamid Beigy Computer Engineering Department, Sharif University of Technology, Tehran, Iran Institute for Studies in Theoretical Physics and Mathematics (IPM), School of Computer Science, Tehran, Iran beigy@ce.sharif.edu M. R. Meybodi Computer Engineering Department, Amirkabir University of Technology, Tehran, Iran Institute for Studies in Theoretical Physics and Mathematics (IPM), School of Computer Science, Tehran, Iran mmeybodi@aut.ac.ir Abstract. Cellular learning automata is a combination of learning automata and cellular automata. This model is superior to cellular learning automata because of its ability to learn and also is superior to single learning automaton because it is a collection of learning automata which can interact together. In some applications such as image processing, a type of cellular learning automata in which the action of each cell in next stage of its evolution not only depends on the local environment (actions of its neighbors) but it also depends on the external environments. We call such a CLA as open cellular learning automata. In this paper, we introduce open cellular learning automata and then study its steady state behavior. It is shown that for class of rules called commutative rules, the open cellular learning automata in stationary external environments converges to a stable and compatible configuration. Then the application of this new model to image segmentation has been presented. Keywords. Cellular Automata, Learning Automata, Cellular Learning Automata, Dynamical Sys- tems. 1 Introduction In recent years cellular automata (CA) have frequently been used to model the dynamics of spatially extended physical systems. Examples include a wide range of topics, such as periotic evolution [1], the development of pigment patterns in mollusks [2], and growth of clonal plants [3], to mention a few. Cellular automata is a collection of cells that each adapts one of a finite number of states. CA updates the state of each cell by employing a local rule that depends on the environment of the cell. The environment of a cell usually taken to be a small number of neighboring cells. The dynamics of cellular automata is generated by repeatedly applying the local rule to all cells in the cellular automata. The cellular automata evolves in discrete steps, changing the states of all its cells according to the local rule, homogenously applied at each step. Cellular automata perform complex computation with high degree of efficiency and robustness. In other hand, learning automata (LA) are simple agents for doing simple things. The learning au- tomata have finite set of actions and at each stage choose one of them. The choice of an action depends on the state of automaton which is usually represented by an action probability vector. For each action chosen by the automaton, the environment gives a reinforcement signal with fixed unknown probability distribution, which specified the goodness of the applied action. Then upon receiving the reinforcement signal, the learning automaton updates its action probability vector by employing a learning algorithm. The interaction of the learning automaton and its environment is shown in figure 1. The learning algorithm is a recurrence relation and is used to modify action probability vector p . Various learning algorithms have been reported in the literature. Below, a learning algorithm, called L R−I , for updating the action probability vector is given. Let α i be the action chosen at time k as a sample realization from probability distribution p (k). In L R−I algorithm, the action probability vector is updated according to the following rule. p j (k + 1) =  p j (k)+ b × [1 − p j (k)] if i = j p j (k)(1 − b) if i = j (1)