Given modal logics 1, 2, their lexicographic product 1 ⇤ 2 is a new logic whose frames are the Cartesian products of a 1-frame and a 2-frame, but with the new accessibility relations reminiscent of a lexicographic ordering. This article considers the lexicographic products of several modal logics with linear temporal logic (LTL) based on “next” and “always in the future”. We provide axiomatizat...

In this paper fully fuzzy linear programming (FFLP) problem with both equality and inequality constraints is considered where all the parameters and decision variables are represented by non-negative trapezoidal fuzzy numbers. According to the current approach, the FFLP problem with equality constraints first is converted into a multi–objective linear programming (MOLP) problem with crisp const...

We consider context-free languages equipped with the lexico-graphic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than ω ω. As a corollary of this result we obtain that an ordinal is the order type of a well-ordered context-free language iff it is less than ω ω ω .

It is shown how sorting -numbers and lexicographic can be done completely within TEX. Lexicographic sorting allows words with ligatures and diacritical marks. As applications I selected sorting of address labels, and sorting and compressing i ndex . tex, Knuth's index reminders file. It is claimed that a set can be sorted within TEX once the ordering of the set is defined and encoded in a compa...

Résumé We give an axiomatization of Recursive Path Orders in the Calculus of Inductive Constructions. Then, we show that they are monotonic strict partial orders, and that they are well-founded. The proof of the well-foundedness is particularly short and elementary. Finally, we produce three relations that are proved to model the axiomatization: the Multiset Path Ordering, the Lexicographic Pat...

In this paper we give the deenition of a solution concept in multicri-teria combinatorial optimization. We show how Pareto, max-ordering and lexicographically optimal solutions can be incorporated in this framework. Furthermore we state some properties of lexicographic max-ordering solutions , which combine features of these three kinds of optimal solutions. Two of these properties, which are d...

The Knuth-Bendix ordering is usually preferred over the lexicographic path ordering in successful implementations of resolution and superposition calculi. However, it is incompatible with certain requirements of hierarchic superposition calculi, and it also does not allow non-linear definition equations to be oriented in a natural way. We present two extensions of the Knuth-Bendix ordering that...

Optimization problems in IMRT inverse planning are inherently multicriterial since they involve multiple planning goals for targets and their neighbouring critical tissue structures. Clinical decisions are generally required, based on tradeoffs among these goals. Since the tradeoffs cannot be quantitatively determined prior to optimization, the decision-making process is usually indirect and it...

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are ...