Applications of fuzzy logic in heuristic control have been highly successful, but which aspects of fuzzy logic are essential to its practical usefulness? This paper shows that an apparently reasonable version of fuzzy logic collapses mathematically to two-valued logic. Moreover, there are few if any published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In the future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems. 1

In this paper, we present an image understanding system using fuzzy sets and fuzzy measures. This system is based on a symbolic object-oriented image interpretation system. We apply a simple, powerful three-dimensional (3-D) recursive filter to tracking moving objects in a dynamic image sequence. This filter has a time-varying 3-D frequency-planar passband that is adapted in a feedback system to automatically track moving objects. However, as objects in the image sequence are not well-defined and are engaged in dynamic activities, their shapes and trajectories in most cases can be described only vaguely. In order to handle these uncertainties, we use fuzzy measures to capture subtle variations and manage the uncertainties involved. This enables us to develop an image understanding system that produces a very natural output. We demonstrate the effectiveness of our system with complex real traffic scenes.

Qualitative counterparts of expected utility can be expressed by means of Sugeno integrals. Both von Neumann and Morgensternlike and Savage-like axiomatic justifications are provided. In this framework, pessimistic (i.e., risk-averse) as well as optimistic attitudes can be captured. 1

Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We present complete descriptions of the function classes axiomatized by each of these properties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasipolynomial function which appears as a natural extension of the well-established notion of polynomial function. We present further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and present characterizations accordingly.

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems. Key words: weighted lattice polynomial, lattice polynomial, order statistic, cumulative distribution function, moment, reliability. 1

Aggregation functions for decision

Representations and characterizations of polynomial

Abstract. In [6] the authors introduced the notion of quasi-polynomial function as being a mapping f: X n → X defined and valued on a bounded chain X and which can be factorized as f(x1,..., xn) = p(ϕ(x1),..., ϕ(xn)), where p is a polynomial function (i.e., a combination of variables and constants using the chain operations ∧ and ∨) and ϕ is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions. 1.

Axiomatizations of quasi-polynomial functions on

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of “indicator ” variables. A connection is studied between Y and order statistics of the set of arguments.