Reliable Computing 8: 139–174, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Efﬁcient Interval Linear Equality Solving in
Constraint Logic Programming
CHONG-KAN CHIU and JIMMY HO-MAN LEE
Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin,
N.T., Hong Kong SAR, P.R. China, e-mail: firstname.lastname@example.org
(Received: 10 August 1996; accepted: 12 July 2001)
Abstract. Existing interval constraint logic programming languages, such as BNR Prolog, work under
the framework of interval narrowing and are deﬁcient in solving systems of linear constraints over
real numbers, which constitute an important class of problems in engineering and other applications.
In this paper, we suggest to separate linear equality constraint solving from inequality and non-linear
constraint solving. The implementation of an efﬁcient interval linear constraint solver, which is
based on the preconditioned interval Gauss-Seidel method, is proposed. We show how the solver can
be adapted to incremental execution and incorporated into a constraint logic programming language
already equipped with a non-linear solver based on interval narrowing. The two solvers share common
interval variables, interact and cooperate in a round-robin fashion during computation, resulting in
an efﬁcient interval constraint arithmetic language CIAL. The CIAL prototypes, based on CLP(R),
are constructed and compared favorably against several major interval constraint logic programming
The current status of Prolog arithmetic suffers from two deﬁciencies. First, the
system predicate “is”  is functional in nature. It is incompatible with the rela-
tional paradigm of logic programming. Second, real numbers are approximated
by ﬂoating-point numbers. Roundoff errors induced by ﬂoating-point arithmetic
destroy the soundness  of computation. The advent of constraint logic pro-
gramming  presents a solution to the ﬁrst problem but the implementation of
CLP languages, such as CLP(R) , are mostly based on ﬂoating-point arithmetic.
The second problem remains.
The languages CAL  and RISC-CLP(R)  use symbolic algebraic meth-
ods to refrain from ﬂoating-point operations. Algebraic methods guarantee the
soundness of numerical computation but they are time-consuming.
Previous efforts in the sub-symbolic camp, such as BNR Prolog , employ
interval methods  and belong to the family of consistency techniques .
The main idea is to narrow the set of possible values of the variables of arbitrary
real constraints using approximations of arc-consistency . We collectively call
these techniques interval narrowing. Interval narrowing has been shown to be
applicable to critical path scheduling , X-ray diffraction crystallography ,
boolean constraint solving , and disjunctive constraint solving , . However,