LOWER BOUNDS FOR GEOMETRICAL PROBLEMS 21 rr ? HHHY

The opposite direction, that for any given P, e, h and q, the language L satisfying (2) is Turing-computable, follows from the fact that we can enumerate all points of P that are reachable from a given point 2 e (the algorithm has to follow the structure of P inductively, simplex for simplex, by marking the already detected reachable areas). 5. Future work. We think that the ideas, which we developed in this paper, are interesting from both a complexity theoretic as well as from a physical point of view. Because of the intuitive power of geometrical reasoning, and also because of the close relationship to physical problems, it seems to be a challenging subject of research to develop a complexity theory based on the geometrical machine model introduced in theorem 4.4. Interesting questions in this context are, how known characteristics of Turing machines (number of tapes, nondeterminism etc.) are reeected in the geometrical model, and how characteristics of the geometrical model could help to classify problems in complexity theory. We state the following concrete problems: 1. The number of tapes of a Turing machine corresponds to the number of diierent structures on the simplices of P. Which restrictions on the possible structures lead to interesting subclasses of the class of Turing-computable languages ? 2. Can two structured polyhedra accepting the same language be transformed to each other by some reasonable set of given transformation rules ? 3. The dimension of the polyhedron that we get by taking the roundabout way via dynamic scenes depends on the Turing machine M. Is it possible to nd for each structured polyhedron anèquivalent' 3-dimensional polyhedron ? The latter of these problems is related to the probably most interesting question in practice, namely to detect the complexity of motion planning under constraints for lower-dimensional problem instances, especially instances involving only one movable object. On the complexity of motion planning for multiple independant objects: PSPACE-hardness of the warehouseman's problem, Int. a a a a a a a a a a a a a a a a a a a a a a a a Cf Ci horizontal structure diagonal structure horizontal structure structure vertical Fig. 17. A structured polyhedron: the polyhedron consists of 4 structured sides. E.g., the triangle on the right can only be crossed in diagonal direction.. Compared to the general model, the structured spaces that occured in this section constitute a very …