From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability

This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in tree

Structural theory of trees I. Branching and condensations of trees

Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of branching in trees and of condensed trees, which are trees in which every node is a branching node. We then introduce and investigate two different constructions of tree condensations-one shrinking, and the other expanding, the tree to a condensed tree

Structural theory of trees II. Completeness and completions of trees

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of tree completions that extend any tree to a minimal one satisfying the respective completeness property

Evolutionary leap in large-scale flood risk assessment needed

Current approaches for assessing large-scale flood risks contravene the fundamental principles of the flood risk system functioning because they largely ignore basic interactions and feedbacks between atmosphere, catchments, river-floodplain systems and socio-economic processes. As a consequence, risk analyses are uncertain and might be biased. However, reliable risk estimates are required for prioritizing national investments in flood risk mitigation or for appraisal and management of insurance portfolios. We review several examples of process interactions and highlight their importance in shaping spatio-temporal risk patterns. We call for a fundamental redesign of the approaches used for large-scale flood risk assessment. They need to be capable to form a basis for large-scale flood risk management and insurance policies worldwide facing the challenge of increasing risks due to climate and global change. In particular, implementation of the European Flood Directive needs to be adjusted for the next round of flood risk mapping and development of flood risk management plans focussing on methods accounting for more process interactions in flood risk systems

Branching-Time as a relative closeness relation among histories

In the traditional approaches to branching-time, histories are dened as linearly ordered and maximal sets of moments. The `geometrical' approach considers both moments and histories as primitive entities with no set-theoretical and ontological dependency of the latter on the former. In the a topological approach the original perspective is inverted: only histories are primitive entities and moments are dened as sets of histories. Moreover, these particular sets of histories can be dened also by means of a relative closeness relation among histories

Quantification over Sets of Possible Worlds in Branching-Time Semantics

Temporal logic is one of the many areas in which a possible world semantics is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart from the genuine Kripke semantics in that they involve a quanti\uafcation over histories, which is a second-order quanti\uafcation over sets of possible worlds. In the paper, variants of the original Prior's semantics will be considered and it will be shown that all of them can be viewed as \uafrst-order counterparts of the original semantics