初等幾何学の補助線問題におけるフラストレーションに基づく学習

抄録

We have developed a learning system, AUXIL, which has an ability to solve auxiliary-line problems in geometry in an intelligent way. First, we show that a basic mechanism for producing auxiliary-lines is to associate a certain condition or subgoal in the problem with an appropriate figure-pattern and that AUXIL can produce a right auxiliary-line by making use of associative knowledge, which we call figure-pattern strategies. Secondly, we proposed a new method, frustration-based learning, which can learn associative knowledge from experiences of solving a variety of auxiliary-line problems. AUXIL simulates the following expert behavior. When an expert tries to solve such a problem, he feels frustration because enough information is not given in a problem space for him to proceed an inference and to find a correct path from given conditions to the goal. Here, he concentrates himself on the conditions or subgoals which have caused frustration. After he has produced an auxiliary-line and made a complete proof-tree, he would learn several pieces of associative knowledge. Each frustration-causing condition or subgoal will constitute the if-part of each knowledge. He will then recognize several lumps of figure-patterns in the proof-tree, each of which has contributed to resolving each frustration. All pieces of geometrical information of each figure-pattern will constitute the then-part of each knowledge. A figure-pattern strategy has two characteristics as an associative knowledge. One is that its application to problems does not necessarily contribute to successful paths of the problems because it is a mere successful instance in the past experiences. The other is that it can enjoy flexible application to problems under no constraint of their goal-structures because its if-part can be unified, if unifiable, to any partial element of the problems. The second characteristic enables AUXIL to produce an appropriate auxiliary-line by a multiple use of figure-pattern strategies in response to several frustrations occurring in a problem, which is sufficient for making up for the first undesirable characteristic.