Kripke‐style Semantics of Orthomodular Logics

We present here a Kripke‐style semantics for propositional orthomodular logics that is based on the representation theorem for orthomodular lattices by D.J. Foulis ([2]), in which a sort of semigroups is employed. This semantics can characterize the logics above the orthomodular logic by some elementary conditions.     

A bipolar model of assertability and belief

Valuation pairs are introduced as a bipolar model of the assertability of propositions. These correspond to a pair of dual valuation functions, respectively, representing the strong property of definite assertability and the dual weaker property of acceptable assertability. In the case where there is uncertainty about the correct valuation pair for a language then a probability distribution is defined on possible valuation pairs. This results in two measures, μ⁺ giving the probability that a sentence is definitely assertable, and μ⁻ giving the probability that a sentence is acceptable to assert. It is shown that μ⁺ and μ⁻ can be determined directly from a two dimensional mass function m defined on pairs of sets of propositional variables. Certain natural properties of μ⁺ and μ⁻ are easily expressed in terms of m, and in particular we introduce certain consonance or nestedness assumptions. These capture qualitative information in the form of assertability orderings for both the propositional variables and the negated propositional variables. On the basis of these consonance assumptions we show that label semantics, intuitionistic fuzzy logic and max-min fuzzy logic can all be viewed as special cases of this bipolar model. We also show that bipolar belief measures can be interpreted within an interval-set model.     

A simulated annealing heuristic for the team orienteering problem with time windows

This paper presents a simulated annealing based heuristic approach for the team orienteering problem with time windows (TOPTW). Given a set of known locations, each with a score, a service time, and a time window, the TOPTW finds a set of vehicle tours that maximizes the total collected scores. Each tour is limited in length and a visit to a location must start within the location’s service time window. The proposed heuristic is applied to benchmark instances. Computational results indicate that the proposed heuristic is competitive with other solution approaches in the literature.     

An exact algorithm for team orienteering problems

Optimising routing of vehicles constitutes a major logistic stake in many industrial contexts. We are interested here in the optimal resolution of special cases of vehicle routing problems, known as team orienteering problems. In these problems, vehicles are guided by a reward that can be collected from customers, while the length of routes is limited. The main difference with classical vehicle routing problems is that not all customers have to be visited. The solution method we propose here is based on a Branch & Price algorithm. It is, as far as we know, the first exact method proposed for such problems, except for a preliminary work from Gueguen (Methodes de résolution exacte pour problémes de tournées de véhicules. Thése de doctorat, école Centrale Paris 1999) and a work from Butt and Ryan (Comput Oper Res 26(4):427–441 1999). It permits to solve instances with up to 100 customers.      

A balanced team generating model

This paper introduces a general team balancing model. It first summarizes existing balancing methods. It is shown that for these methods it is difficult to meet all the conditions posed by Belbin on balanced teams. This mainly is caused by the complexity of the balancing problem. A mathematical model is then introduced that gives one the opportunity to take into account these conditions in mutual coherence and to generate a balanced team in terms of Belbin. A practical application is given.     

A spatial modal logic with a location interpretation

A spatial modal logic (SML) is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space (or location) interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut‐elimination theorem for a modified subsystem of SML are also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On semilattice relevant logics

The semilattice relevant logics ^∪ R , ^∪ T , ^∪ RW , and ^∪ TW (slightly different from the orthodox relevant logics R , T , RW , and TW ) are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut‐free labelled sequent calculus with plural succedents (like LK ). We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents (like LJ ). Moreover, using this equivalence, we give a new Hilbert‐style axiomatizations for ^∪ R and ^∪ T and prove equivalence between two semantics (commutative monoid and distributive semilattice) for the contractionless logics ^∪ RW and ^∪ TW .