Complete sets of initial vectors for pattern growth with elementary cellular automata

Computer simulations of complex spatio-temporal patterns using cellular automata may be performed in two alternative ways, the better choice depending on the relative size between the spatial width W of the expected patterns and their corresponding temporal period T. While the traditional timewise   updating algorithm is very efficient when W≪TW≪T, the complementary spacewise   algorithm wins whenever T≪WT≪W. Independently of the algorithm used, the key to obtaining exhaustive answers, not just statistical estimates, is to have explicit knowledge of the complete sets of initial conditions that need to be individually tested as sizes grow. This paper reports an efficient algorithm for generating complete sets (without redundancy) of k-vectors of initial conditions allowing one to perform definitive classifications of patterns in systems with a minimal characteristic length k, either spatial or temporal.