Autonomous selection and indefinite goals: A system using Bezier curves as dynamically re-defined transition rules

At the beginning, we point out a serious problem which arises when we describe the process of the living system as the computational process of a formal system. The problem is the indefiniteness in terms of correspondences of input and output states. Such indefiniteness can be grasped as the contradiction in a formal system. That is why we can say that the process of the living system is executed properly in spite of the existence of such a serious problem. The process proceeding in spite of a contradiction is represented by a perpetual change of partial function as a transition rule depending on the state. To achieve this purpose, we introduce the Bezier curve as a partial function. Given some control points in a plane, the Bezier curve that roughly connects all control points is calculated, and then all control points are moved horizontally and vertically at the same time as far as such moves can cross the Bezier curve. The new configuration of control points yields for the new Bezier curve. This sequential process is iterated, and that a transition rule of the Bezier curve is recursively re-defined. As a result, we have found that the system can generate particular output states even though the system is constructed without any explicit mechanism ensuring such events. Finally, we show that our system does work as a conceptual tool to embody ourselves who observe the process of the living system. A framework of the argument presented here is so-called internal measurement [1,2].