Reliable Computing 8: 21–42, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Approximate Quantiﬁed Constraint Solving by
Cylindrical Box Decomposition
Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria,
(Received: 1 September 2000; accepted: 21 March 2001)
Abstract. This paper applies interval methods to a classical problem in computer algebra. Let a
quantiﬁed constraint be a ﬁrst-order formula over the real numbers. As shown by A. Tarski in the
1930’s, such constraints, when restricted to the predicate symbols <, = and function symbols +,
in general solvable. However, the problem becomes undecidable, when we add function symbols like
sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide
partial information before computing the total result, cannot satisfactorily deal with interval constants
in the input, and often generate huge output. As a remedy we propose an approximation method
based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition—as
introduced by G. Collins. We describe an implementation of the method and demonstrate that, for
quantiﬁed constraints without equalities, it can efﬁciently give approximate information on problems
that are too hard for current exact methods.
This paper deals with solving quantiﬁed constraints (i.e. ﬁrst-order formulae over
the real numbers). This problem has various applications, for example in control
theory and stability analysis of differential equations , , .
A. Tarski  showed that the theory of real-closed ﬁelds allows quantiﬁer
elimination, which means that one can solve quantiﬁed constraints that contain
only the predicate symbols < and =, and the function symbols + and
. But as soon
as one adds function symbols like sin, the theory becomes undecidable , 
(although, by imposing certain restrictions on the constraint structure one can come
up with solvers for special cases , ).
Up to now, Collins’ method  is the only general quantiﬁer elimination method
in the theory of real-closed ﬁelds for which a practically useful implementation 
exists. This implementation is quite usable for many small and a few middle-size
examples, however it usually cannot ﬁnish big examples. More efﬁcient solvers
exist only for special cases , , .
Further problems that occur when trying to solve quantiﬁed constraints exact-
ly include: Current methods cannot provide partial information when interrupted
before computing the total result , , , ; they cannot deal with input
that contains constants that come from measurements and are thus not exact but