Combining Local Consistency, Symbolic Rewriting and Interval Methods

. This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three different methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we define a generic two-step constraint processing technique based on an extension of the Constraint Satisfaction Problem, called Extended Constraint Satisfaction Problem (ECSP). The first step is a rewriting step, in which the initial ECSP is symbolically transformed. The second step, called approximation step, is based on a local consistency notion, called weak arc-consistency, defined over ECSPs in terms of fixed point of contractant monotone operators. This notion is shown to generalize previous local consistency concepts defined over finite domains (arc-consistency) or infinite subsets of the Reals (arc B-consistency and interval, hull and box-consistency). A filtering algorithm, derived from AC-3, is given and is shown to be correct, conflue..

Constraint reasoning for differential models

The basic motivation of this work was the integration of biophysical models within the interval constraints framework for decision support. Comparing the major features of biophysical models with the expressive power of the existing interval constraints framework, it was clear that the most important inadequacy was related with the representation of differential equations. System dynamics is often modelled through differential equations but there was no way of expressing a differential equation as a constraint and integrate it within the constraints framework. Consequently, the goal of this work is focussed on the integration of ordinary differential equations within the interval constraints framework, which for this purpose is extended with the new formalism of Constraint Satisfaction Differential Problems. Such framework allows the specification of ordinary differential equations, together with related information, by means of constraints, and provides efficient propagation techniques for pruning the domains of their variables. This enabled the integration of all such information in a single constraint whose variables may subsequently be used in other constraints of the model. The specific method used for pruning its variable domains can then be combined with the pruning methods associated with the other constraints in an overall propagation algorithm for reducing the bounds of all model variables. The application of the constraint propagation algorithm for pruning the variable domains, that is, the enforcement of local-consistency, turned out to be insufficient to support decision in practical problems that include differential equations. The domain pruning achieved is not, in general, sufficient to allow safe decisions and the main reason derives from the non-linearity of the differential equations. Consequently, a complementary goal of this work proposes a new strong consistency criterion, Global Hull-consistency, particularly suited to decision support with differential models, by presenting an adequate trade-of between domain pruning and computational effort. Several alternative algorithms are proposed for enforcing Global Hull-consistency and, due to their complexity, an effort was made to provide implementations able to supply any-time pruning results. Since the consistency criterion is dependent on the existence of canonical solutions, it is proposed a local search approach that can be integrated with constraint propagation in continuous domains and, in particular, with the enforcing algorithms for anticipating the finding of canonical solutions. The last goal of this work is the validation of the approach as an important contribution for the integration of biophysical models within decision support. Consequently, a prototype application that integrated all the proposed extensions to the interval constraints framework is developed and used for solving problems in different biophysical domains