Exploration of a solution space structured by finite constraints Laure Brisoux-Devendeville Caroline Essert-Villard Pascal Schreck Laboratoire des Sciences de l’Image, de l’Informatique, et de la Télédétection (LSIIT, URA CNRS 1871) Université Louis Pasteur, Boulevard Sébastien Brant 67400 Illkirch, France Abstract Formal CAD solvers often produce many solutions for a constraint system. It is very time-consuming to examine each of them to determine which one is the closest to the user’s will. In our formal approach a construction plan is produced as a mean to represent all the solutions. In this paper, we show how a construction plan can be seen as a set of constraints of finite type. Then, we use some techniques derived from the SAT problem to efficiently explore the solution space. keywords: Formal geometric constructions; Symbolic constraint solving; SAT problem; Tree pruning; Computer-aided design. 1 Introduction In Computer-Aided Design (CAD), geometric constraints are used to specify rigid bod- ies. In order to actually handle such specified objects, different kinds of solvers have been used in CAD to compute solutions. The complete solvers are able to yield all the solutions according to the user’s constraints. Unfortunately, the number of solutions is often huge, so the complete exploration can be very tedious and time consuming. We propose to use the constraint solving paradigm one more time to efficiently browse the solution space. This paper is focused on this subject: we adopt the point of view of the discrete finite constraints problems to describe and to structure the solution space. Let us recall briefly the issues of geometric construction in CAD. When sketch- ing an object, a draughtsman does not give an exact geometry to his drawing, but he expresses it graphically by the way of some dimensional constraints like in the ele- mentary example of Fig. 1. Given such a dimensioned sketch, a CAD solver produces one, some or all the solutions meeting the requirements. In our example, we have two solutions given Fig. 2. Of course, the problem of solving constrained figures has been studied by many authors [44, 2, 8, 10, 11, 28, 31, 38, 39, 43] following very different approaches which we divide in two main classes: numerical approaches and formal methods. The first ones, mainly used in CAD, consist in solving numerically the equation system related to the dimensions [23, 30, 31, 6, 14]. A formal resolution of the symbolic constraints system allows to efficiently manipulate the defined figure