In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all filled squares above it drop by one row. We prove that in the offline version of Tetris, it is [Formula: see text]-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p1-ε, when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2-ε, for any ε>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.

Cellular automata are used in many fields to generate a global behavior with local rules. Finding the rules that display a desired behavior can be a hard task especially in real world problems. This paper proposes an improved approach to generate these transition rules for multi dimensional cellular automata using a genetic algorithm, thus giving a generic way to evolve global behavior with local rules, thereby mimicking nature. Three different problems are solved using multi dimensional topologies of cellular automata to show robustness, flexibility and potential. The results suggest that using multiple dimensions makes it easier to evolve desired behavior and that combining genetic algorithms with multi dimensional cellular automata is a very powerful way to evolve very diverse behavior and has great potential for real world problems.

Abstract. Genetic Algorithms have been used before to evolve transition rules for one dimensional Cellular Automata (CA) to solve e.g. the majority problem and investigate communication processes within such CA [3]. In this paper, the principle is extended to multi dimensional CA, and it is demonstrated how the approach evolves transition rules for the two dimensional case with a von Neumann neighborhood. In particular, the method is applied to the binary AND and XOR problems by using the GA to optimize the corresponding rules. Moreover, it is shown how the approach can also be used for more general patterns, and therefore how it can serve as a method for calibrating and designing CA for real-world applications.

Abstract.It is an unconventional computation approach to evolve solutions instead of calculating them. Although using evolutionary computation in computer science dates back to the 1960s, using an evolutionary approach to program other algorithms is not that well known. In this paper a genetic algorithm is used to evolve behavior in cellular automata. It shows how this approach works for different topologies and neighborhood shapes. Some different one dimensional neighborhood shapes are investigated with the genetic algorithm and yield surprisingly good results.

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