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Quotient Space Based Problem Solving: A Theoretical Foundation of Granular Computing

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The term problem solving is used in many disciplines, sometimes with different perspectives. As one of the important topics in artificial intelligence (AI) research, it is a computerized process of human problem-solving behaviors. So the aim of problem solving is to develop techniques that program computers to find solutions to problems that can properly be described.

In the early stage of AI, symbolists play a dominant role. They believe that all human cognitive behaviors, including problem solving, can be modeled by symbolic representation and reasoning and do not advocate the use of strict mathematical models. The most general approach to tackle problem-solving processes is “generation and test”. Applying an action to an initial state, a new state is generated. Whether the state is the goal state is tested; if it is not, repeat the procedure, otherwise stop and the goal is reached. This principle imitates human trial-and-error behaviors in problem solving sufficiently. The principle has widely been used to build AI systems such as planning, scheduling, diagnosis, etc. and to solve a certain kind of real problems. Therefore, the heuristic and scratch method is misunderstood as a unique one in AI for many people. We believe that more and more modern sciences such as mathematics, economics, operational research, game theory and cybernetics would infiltrate into AI when it becomes mature gradually. Over the years, we devoted ourselves to introducing mathematics to AI. Since 1979 we have introduced statistical inference methods to heuristic search, topological dimension reduction approach to motion planning, and relational matrix to temporal planning. Due to the introduction of these mathematical tools, the efficiency and performance of AI algorithms have been improved significantly. There are two main trends in AI research recently. One is attaching importance to the usage of modern scientific methods, especially mathematics; the other is paying attention to real-world problem solving. Fortunately, our efforts above are consistent with these new trends.
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