The algebra image: a framework for qualitative fuzzy temporal reasoning Original Research Article

The aim of this work is to integrate the ideas of flexibility and uncertainty into Allenʹs interval-based temporal framework, defining a new formalism, called image, which extends classical Interval Algebra (IA), in order to express qualitative fuzzy constraints between intervals. We generalize the classical operations between IA-relations to image-relations, as well as the concepts of minimality and local consistency, referring to the framework of Fuzzy Constraint Satisfaction Problem. We analyze the most interesting reasoning tasks in our framework, which generalize the classical problems of checking consistency, finding a solution and computing the minimal network in the context of IA. In order to solve these tasks, we devise two constraint propagation algorithms and a Branch & Bound algorithm. Since these tasks are NP-difficult, we address the problem of finding tractable sub-algebras of image, by extending to our fuzzy framework the classical pointizable sub-algebras image and SA, as well as the maximal tractable subalgebra image introduced by Nebel. In particular, we prove that the fuzzy extension of the latter, called image, shares with its classical counterpart a maximality property, in that it is the unique maximal subalgebra of image which contains the fuzzy extensions of Allenʹs atomic relations.