One of the main problems of interval computations is, given a function f(x 1 ; :::; x n ) and n intervals x 1 ; :::; x n , to compute the range y = f(x 1 ; :::; x n ). This problem is feasible for linear functions f , but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute the enclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute the optimal enclosure, i.e., the range itself.

This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (non-fuzzy) case.

We consider the problem of finding interval enclosures of all zeros of a nonlinear system of polynomial equations. We present a method which combines the method of Grobner bases (used as a preprocessing step), some techniques from interval analysis, and a special version of the algorithm of E. Hansen for solving nonlinear equations in one variable. The latter is applied to a triangular form of the system of equations, which is generated by the preprocessing step. Our method is able to check if the given system has a finite number of zeros and to compute verified enclosures for all these zeros. Several test results demonstrate that our method is much faster than the application of Hansen's multidimensional algorithm (or similar methods) to the original nonlinear systems of polynomial equations. 1 Introduction The general problem we address is: Find, with certainty, all solutions in IR n of the nonlinear system f k (x 1 ; x 2 ; : : : ; x n ) = 0 for k = 1; : : : ; m of m ...

This chapter provides a general introduction to interval computations, especially to interval computations as an important part of granular computing.

Traditionally, most planning research in AI was concentrated on systems whose state can be characterized by discrete-valued fluents. In many practical applications, however, we want to control systems (like robots) whose state can only be described if we used continuous variables (like coordinates). Planning for such systems corresponds, crudely speaking, to Level 2 of the planing language PDDL2.1. In this paper, we analyze the computational complexity of such planning problems.

We prove that no efficient necessary and sufficient conditions are possible for checking whether straightforward interval computations lead to the exact result.

We prove that no "efficient" (easy-to-check) necessary and sufficient conditions are possible for checking whether straightforward interval computations lead to the exact result.

This paper deals with representation and reasoning on information concerning the evolution of a physical parameter by means of a model based on Fuzzy Constraint Satisfaction Problem formalism, and with which it is possible to define what we call Fuzzy Temporal Profiles (FTP). Based on fundamentally linguistic information, this model allows the integration of knowledge on the evolution of a set of parameters into a knowledge representation scheme in which time plays a fundamental role.

i . In this case, the goal is to find all real numbers that can be the values of this quantity y (for the given measurement results). The goal of data processing is, however, often more complicated. For example, we may want to control a certain system; in this case, we must find a control value that, e.g., guarantees stability of the system for all possible values of the parameters x i (i.e., for all x i 2 x i ). In this case, we want to find all real numbers y for which stability must occur. Similarly, many real-life problems of design, control, and optimization lead to complicated mathematical formulations. 1.3 Shary's Approach: Successes and Limitations Shary's approach. For the case when the relationship between different variables is described by a system of (linear or non-linear) equations, different possible problems have been described by Shary (see, e.g., [12]). Shary distinguishes between different possible formulations by using different quantifiers for different variabl

A new methodology 1 is presented for computing a minimal envelope for the reachable space of a mechanism, i.e. the space that contains a given mechanism in all its admissible configurations. The research is motivated by the packaging process in Digital Mock-Up applied in automotive industry. An important task in the concept phase, the automated determination of the space requirement for all parts, is still an unsolved problem in the case of mechanisms. The particular benefit of the method presented is its generality and robustness: It is able to deal with both open- and closed-loop mechanisms. The reachable space is computed with regard to the geometric description of each part. The approximation is enclosing and always converges in a uniform way and the tolerance can be pre-defined by the user. The method combines the use of bounding object hierarchies and the application of interval analysis. It is also able to approximate the swept volume of an object following a parameterized trajectory. We describe how the efficiency can be improved by lazy evaluation and by a deeper problem analysis. The presented algorithms are implemented and tested to a large extent 2. 1