Discrete System Theory and Large-Scale Optimization Methods Based on Binary Decision Diagrams and

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In this research, we aimed at proposing a unified approach to discrete system theory based on binary decision diagrams, and developing a prototype system for the unified system. By our results, we can now represent the whole moderate-scale discrete structure in a implicit and compact manner, which could not be done by the existing methods. We applied our approach to network reliability computation and also computing the Jones polynomial of a knot. These computation problems are known to be #P-hard, but, with our methods, moderate-size problems can be solved rigorously in practice. For example, the network reliability function of a grid of 14x14 can be computed. We also explore fundamental theory of binary decision diagrams, by investigating the difference in size when a monotone Boolean function is represented directly and when it is represented by their prime implicants.New approaches have also been demonstrated, one is based on algebraic approach, and the other is based on quantum approach. Concerning the algebraic approach, Grobner bases are fully investigated for network flow problems. Concerning the quantum approach, we investigate a quantum analog of binary decision diagrams in order to investigate the computational power of quantum computing. These new results will be published soon.