Positivity 8: 215–228, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Convergence ofa Reactive Planning Algorithm
and BERNARD BEAUZAMY
Kergueven, 29470 Plougastel Daoulas, France;
Société de Calcul Mathématique SA,
111 Faubourg Saint Honoré, 75008 Paris, France. (E-mail: Bernard.Beauzamy@wanadoo.fr)
(Received 24 June 1999; accepted 25 August 2002)
Abstract. The reactive planning algorithms proposed by P. Maes in Artiﬁcial Intelligence for real-
time decision making and other derived algorithms are empirically known to converge, but no math-
ematical proofhas been proposed. The evolution equations ofthese incremental algorithms can
be translated into discrete time non-linear dynamical systems. We prove the convergence ofsuch
dynamical systems, provided the necessary assumptions are fulﬁlled, and we give an expression for
Key words: reactive planning, Maes, l
-norm, anharmonic ratio, ﬁxed point
One ofthe main research themes in Artiﬁcial Intelligence is to construct algorithms
for planning. The aim is to build a series of actions for a system so that it reaches a
given conﬁguration in a ﬁnite time. This is a very general problem, whose solution
depends on a good knowledge ofthe consequences ofthe actions, and whether the
system is deterministic or not. Unfortunately, for the great majority of the systems
which are to be controlled, little is known about the consequences ofthe actions.
Quite often, an original plan becomes inapplicable because of an unforeseen event,
or because the knowledge about the system was too incomplete.
Replanning is therefore a central problem, widely discussed in AI literature.
One solution, proposed by P. Maes, is called “reactive planning”. Reactive plan-
ning enables the plan ofactions to be corrected instantly whenever an unexpected
event occurs or one ofthe planned action fails. We propose here a new version
ofthis algorithm, more suitable to deal with autonomous robot decision-making
The mathematical expression ofthis algorithm shows that the choice ofthe
action which is made to control the system depends on the convergence ofa discrete
time dynamical system. The algorithm which was used ﬁrst by P. Maes was not
expressed in mathematical terms. Therefore, the convergence of the underlying
dynamical system had not been investigated. The expression ofthe potential limit
was unknown, even though its expression would have been ofgreat interest in order
to verify the proposed analysis of empirical data.