The algebra IA(fuz): a framework for qualitative fuzzy temporal reasoning

The aim of this work is to integrate the ideas of flexibility and uncertainty into Allen's interval-based temporal framework, defining a new formalism, called IAfuz, which extends classical Interval Algebra (IA), in that qualitative fuzzy constraints can be expressed between intervals. We generalize the classical operations between IA-relations to IAfuz-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-complete, we address the problem of finding tractable sub-algebras of IAfuz, by extending to our fuzzy framework the classical pointizable sub-algebras SAc and SA, as well as the maximal tractable subalgebra H introduced by Nebel. In particular, we prove that the fuzzy extension of the latter, called Hfuz, shares with its classical counterpart a maximality property, in that it is the unique maximal subalgebra of IAfuz which contains the fuzzy extensions of Allen's atomic relations.