Analytic machines

In this paper we present some results about it analytic machines regarding th power of computations over sf bf Q and sf bf R, solutions of differential equations and the stability problem of dynamical systems. We first explain the machine model, wich is a kind of sc Blum-Shub-Smale machine enhanced by infinite convergent computiations. Next, we compare the computional power of such machinesofer the fields sf bf Q and sf bf R showing that finite computations with real numbers can be simulated by infinite converging computations on rational numbers, but the precision of the approximation is not known during the process. Our attention is then shifted to it ordinary differential equations (ODEs), dynamical systems described by ODEs and the undecidability of a class of stability problems for dynamical syste

Analytische Maschinen

In dieser Arbeit präsentieren wir einige Resultate über analytische Maschinen hinsichtlich des Berechenbarkeitsbegriffs über Q und R, der Lösungen von Differentialgleichungen und des Stabilitätsproblems dynamischer Systeme. Wir erläutern zuerst das Maschinenmodell, das eine Art von BLUM-SHUB-SMALE Maschine darstellt, erweitert um unendliche, konvergente Berechnungen. Danach vergleichen wir die Mächtigkeit dieses Berechnungsmodells über den Körpern Q und R und zeigen z.B., daß endliche Berechnungen mit reellen Zahlen durch unendliche, konvergente Berechnungen mit rationalen Zahlen simuliert werden können, wobei die Genauigkeit der Approximation während des Prozesses nicht bekannt ist. Analytische Berechnungen über R sind echt mächtiger als über Q. Unsere Aufmerksamkeit wendet sich dann gewöhnlichen Differentialgleichungen (DGl) zu, bei denen wir hinreichende Kriterien für die Berechenbarkeit von Lösungen innerhalb unseres Modells angeben. Schließlich untersuchen wir dynamische Systeme, die durch DGl beschrieben werden, und zeigen die Unentscheidbarkeit einer Klasse von Stabilitätsproblemen für dynamische Systeme.In this thesis we present some results about analytic machines regarding computability over Q and R, solutions of differential equations, and the stability problem of dynamical systems. We first explain the machine model, which is a kind of BLUM-SHUB-SMALE machine enhanced by infinite convergent computations. Next, we compare the computational power of such machines over the fields Q and R showing e.g. that finite computations with real numbers can be simulated by infinite converging computations on rational numbers, but the precision of the approximation is not known during the process. Analytic computations over R are strictly more powerful than over Q. Our attention is then shifted to ordinary differential equations (ODEs) where we establish sufficient criteria for the computability of their solutions within our model. We investigate dynamical systems described by ODEs and show the undecidability of a class of stability problems for dynamical systems

Revising Type-2 Computation and Degrees of Discontinuity

By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0

A topological view on algebraic computation models

We investigate the topological aspects of some algebraic computation models, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable analysis can be. The framework for this is Weihrauch reducibility. As a consequence of our characterizations, we establish that the solvability complexity index is (mostly) independent of the computational model, and that there thus is common ground in the study of non-computability between the BSS and TTE setting

Heuristic motion planning with many degrees of freedom

We present a general heuristic approach to the geometric motion planning problem with the aim to quickly solve intuitively simple problems. It is based on a divide-and-conquer path search strategy which makes inquiries about feasible paths; to answer these, we develop an efficient collision detection scheme that handles translations and rotations of polyhedra to compute all times of collision. The whole algorithm can be easily implemented and universally applied and has been successfully tested in a program for assembly planning

Heuristic motion planning with many degrees of freedom

We present a general heuristic approach to the geometric motion planning problem with the aim to quickly solve intuitively simple problems. It is based on a divide-and-conquer path search strategy which makes inquiries about feasible paths; to answer these, we develop an efficient collision detection scheme that handles translations and rotations of polyhedra to compute all times of collision. The whole algorithm can be easily implemented and universally applied and has been successfully tested in a program for assembly planning

Heuristic Motion Planning with Movable Obstacles

We present a heuristic approach to geometric path planning with movable obstacles. Treating movable obstacles as mobile robots leads to path planing problems with many degrees of freedom which are intractable. Our strategy avoids this computational complexity by decoupling the whole motion planning problem into a series of tractable problems, which are solved using known path planning algorithms. The individually computed solutions are then coordinated to a path plan. This method results in a powerful and practicable strategy for path planning with movable obstacles, which can be applied using a wide variety of known motion planning algorithms. 1 Introduction Objects in geometric path planning problems are usually divided into moving objects and fixed ones called obstacles. In a problem description the moving objects and the obstacles are given - usually as polygons or polyhedra - together with their positions and orientations. For the moving objects goal configurations are specified ..