Learning by Exploration Including Risk

In this paper we describe how methods of concept (classi cation) learning can be used to solve constraint satisfaction problems in the spatial domain, for example if a set of numerical constraints de nes a complicated region in the physical or feature (parameter) space. In this case it is in general not easy or even not possible to nd explicit boundary descriptions by some kind of formula manipulation system. Formally, the task is learning a decision function A(x1; : : : ; xn) which decides whether a vector x = (x1; : : : ; xn) belongs to a region where the predicate A is true, i.e. the corresponding constraints are satis ed. In the examples given below, the constraints consist of equations and inequations containing trigonometric functions which lead to computational di culties well known in robotics (cf. Ambler and Popplestone (1975) and Taylor (1976)). Seen from the cognitive point of view, our claim is that at least parts of human knowledge (facts and general rules) are learned by experience and directed exploration using the type of inductive inference introduced in this paper. Learning rules (logical implications) of the form 8x(A(x1; : : : ; xn) ! B(x1; : : : ; xn)) is the more general case which can be handled by our method, too (see Sect. 2.1). In the following we identify a region where a constraint A is satis ed with a class (concept) A. Now algorithms of classi cation learning (e.g., decision tree learning or Neuronal Nets) by means of a training set construct classi ers which | by inductive generalization | decide the class membership of an arbitrarily chosen x (not necessarily contained in the training set). Usually, there is a generalization error due to the unavoidable approximation of the boundaries between the A-region and the not A-region. The generalization error can be measured using a test set of classi ed example vectors di erent from the training set. Decision tree learners approximate the class boundaries piecewise linearly by axis-parallel hyperplanes, generalized Perceptrons by hyperplanes in general position. In the case of rule learning, the implication 8x(A(x) ! B(x)) can be \decided" within the limits of approximation error by testing whether every x taken from the test set and classi ed as belonging to the A-region also belongs to the B-region. This procedure will be demonstrated by means of an example problem taken from the area of spatial reasoning using the decision tree learner CAL5 (M uller and Wysotzki 1994) and the neuronal net DIPOL (Schulmeister and Wysotzki 1997) to learn the semantics of spatial relations like right or