Haar's Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control constraints of practical interest are naturally expressed as the inclusion of two systems of linear (w.r.t. idempotent semiring or semifield operations) inequalities. Thus, we establish tropical version of Haar's Lemma to obtain the algebraic characterization of such inclusion. As in the linear case this Lemma exhibits the links between two apparently different problems: comparison of DES and control via positive invariance. Our approach to the control differs from the ones based on formal series and is a kind of dual approach of the geometric one recently developed. Control oriented applications of the main results of the paper are given. One of these applications concerns the study of transportation networks which evolve according to a time table. Although complexity of calculus is discussed the algorithmic implementation needs further work and is beyond the scope of this paper.
In two subsequent parts, Part I and II, monotonicity and comparison results will be studied, as generalization of the pure stochastic case, for arbitrary dynamic systems governed by nonnegative matrices. Part I covers the discrete-time and Part II the continuous-time case. The research has initially been motivated by a reliability application contained in Part II. In the present Part I it is shown that monotonicity and comparison results, as known for Markov chains, do carry over rather smoothly to the general nonnegative case for marginal, total and average reward structures. These results, though straightforward, are not only of theoretical interest by themselves, but also essential for the more practical continuous-time case in Part II (see \cite{DijkSl2}). An instructive discrete-time random walk example is included.
This second Part II, which follows a first Part I for the discrete-time case (see \cite{DijkSl1}), deals with monotonicity and comparison results, as generalization of the pure stochastic case, for stochastic dynamic systems with arbitrary nonnegative generators in the continuous-time case. In contrast with the discrete-time case the generalization is no longer straightforward. A discrete-time transformation will therefore be developed first. Next, results from Part I can be adopted. The conditions, the technicalities and the results will be studied in detail for a reliability application that initiated the research. This concerns a reliability network with dependent components that can breakdown. A secure analytic performance bound is obtained.
Lukowski has argued that, if it is the case that there are actual non-monotonic inferences, they are very hard to find. In this paper, a representative kind of inference that is often considered to be non-monotonic is addressed. Likewise, certain arguments provided by Lukowski to demonstrate that that type of inference is not really non-monotonic are reviewed too. Finally, I propose an explanation of why, despite the fact that the arguments given by him seem to be convincing, it is usually thought that those inferences are not monotonic. In this way, I also try to account for the role that disjunction has in this issue and argue in favor of the idea that we can continue to suppose that the human mind does not ignore the essential requirements of classical logic. and Lukowski argumentoval, že pokud je tomu tak, že existují skutečné nemontotonické závěry, je velmi těžké je najít. V tomto příspěvku je řešen typický závěr, který je často považován za nemontonický. Stejně tak některé argumenty, které poskytl Lukowski k prokázání toho, že tento typ závěru není skutečně nemonotonický, jsou také přezkoumány. Nakonec navrhuji vysvětlení toho, proč navzdory skutečnosti, že jeho argumenty se zdají být přesvědčivé, je obvykle myšleno, že tyto závěry nejsou monotonické. Tímto způsobem se také snažím vysvětlit úlohu, kterou má disjunkce v této otázce, a argumentovat ve prospěch myšlenky, že můžeme i nadále předpokládat, že lidská mysl ignoruje základní požadavky klasické logiky.