In this paper, we introduce open Cellular learning Automata and then study its convergence behavior. It is shown that for a class of rules called commutative rules, the open Cellular learning Automata in stationary external environments converges to a stable and compatible configuration. The numerical results also confirm the theory.  

Rapid, irregular and unbalanced growth of the world population during past years, particularly in developing countries, have been caused many problems in the areas of environmental, economic, social and cultural for urban planners. These situations request new methods and tools to model and predict urban future changes. Due to simple and dynamic structure and utilizing spatial characteristics, Cellular Automation (CA) model widely uses in spatial-temporal modeling problems such as urban extension. This paper develops a Fuzzy-CA method to model urban extension. In conventional CA method, state and position of pixels and transition rules are defined certainly. The definitive expression of components of the complex processes needs a large amount of data. However in most cases accurate data are not available. As a result, integration of CA method and fuzzy theory would be useful to model urban extension. In this paper a Fuzzy-CA method is developed and tested in Shiraz city between the years 2004 to 2009. The results of the proposed method show 80% accuracy in comparison to real data have been captured from satellite images. However, an accuracy of 75% has been reported for this case study with utilizing conventional Cellular automation.