Reliable Computing 3: 325–333, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Concurrent Cooperating Solvers over Reals
e de Nice—Sophia Antipolis, I3S, Route des colles BP 145, 06903 Sophia Antipolis,
France, e-mail: email@example.com
e Lyon I, LISI, Bat. 710–43, bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France,
(Received: 29 November 1996; accepted: 28 February 1997)
Abstract. Systems combining an interval narrowing solver and a linear programming solver can
tackle constraints over the reals that none of these solvers can handle on their own. In this paper
we introduce a cooperating scheme where an interval narrowing solver and a linear programming
solver work concurrently. Information exchanged by the solvers is therefore handled as soon as it
becomes available. Moreover, to improve the pruning, the linear programming solver computes the
actual range of values of each variable with respect to the subset of linear constraints. To validate the
proposed architecture a prototype system—named CCC—has been developed. Several examples are
given to illustrate the gain in speed and precision we can expect with CCC.
Many industrial applications ranging from ﬁnancial applications and resource allo-
cation to thermal ﬂow problems and electro-mechanical engineering problems
involve solving arbitrary constraints over the reals. Such systems of constraints
are usually mixed: they are formed of linear equalities and inequalities, non-linear
equalities and often non-polynomial ones.
Interval arithmetic has been introduced in the constraint logic programming
(CLP) framework , , , , ,  because of its capabilities to narrow
the domains of the variables for any system of constraints over the reals. We
collectively call the local consistency algorithms used in these systems interval
narrowing (IN) algorithms
. Chiu and Lee ,  show that existing CLP languages
based upon IN algorithms are deﬁcient in handling linear systems of constraints
over reals and therefore fail to solve general constraints systems over reals. For
instance, IN fails to solve such trivial systems as
x + y =2
The motivation of cooperating approaches is to tackle such mixed systems of
constraints over the reals by combining solvers based on different algorithms. Many
systems combining a linear programming (LP) solver and an IN-solver have been
proposed during the last years –, , . The purpose of this paper is to
The overall scheme of the IN algorithm is given in Section 2.1.