Abstract For every natural numberm, the existentially closed models of the theory of fields withmcommuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

Abstract Fields of characteristic zero with several commuting derivations can be treated as fields equipped with aspaceof derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

A learning method by which a mobile robot may analyze an initially uninterpreted sensorimotor apparatus and produce a useful characterization of its set of actions is demonstrated. By initially uninterpreted it is meant that the robot is given no knowledge of the structure of its sensory system nor of the effects of its actions. It merely sees and produces vectors of real numbers. The method is applied to the case of a simulated robot with an array of 16 range finders, and a motor apparatus with which it can make combinations of turning and advancing actions. The robot learns a set of primitive actions allowing it to make pure turns (both clockwise and counterclockwise) and pure travels. It is believed that this approach is robust and will apply to sensory systems used for motion detection, such as arrays of photoreceptors or range finders.>