Project : coprin

Section: New Results

Keywords : modal intervals , universally quantified constraints .

Modal Intervals and Universally Quantified Constraints

Modal intervals and their arithmetic are a generalization of classical intervals which let one change the quantifiers usually appearing in classical interval arithmetic. Hence, they have a great potential applicability to universally quantified constraints. Within this latter class of problems, we begin to study the reliable projection of the solution set of a parametric system of equations:

f : E \times X \subset ℝ^{p} \times ℝ^{n} \to ℝ^{n}

Reliable projection can be expressed in the following quantified formulation: $B \in ???? ℝ^{p}$ is inside the projection of the solution set $f = 0$ iff,

\forall t \in B, \exists x \in X, f (p, x) = 0

We first proposed a classical method – not using modal intervals – to solve this problem relying on the parametric Miranda theorem [21] and we plan to build a solver using this method.

In a next step, we will use modal intervals in order to overcome the main restriction of the latter method, i.e. dealing with under-constrained parametric systems.

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