Reasoning on the properties of numerical constraints Lucas Bordeaux, Eric Monfroy, and Fr´ ed´ eric Benhamou {bordeaux,monfroy,benhamou@irin.univ-nantes.fr} Institut de Recherche en Informatique de Nantes (IRIN), France Abstract. Numerical Constraint-Satisfaction Problems (NCSPs) are stated as (in)-equalities between numerical functions. The properties of these functions are a most useful information, which determines the use of spe- cialized algorithms. We propose a framework to reason on a set of func- tion properties which integrates monotonicity and convexity, and aims at being general and extensible. Properties are seen as abstractions of the function curves, and we propose deduction rules to reason on these abstractions. We give guidelines on how this tool can cooperate with existing or customized constraint-solvers. 1 Introduction Among the most fundamental computational problems involving numerical func- tions are optimization, i.e., the task of finding an optimal value w.r.t. a numerical criterion, and constraint solving, i.e., the task of finding values that satisfy a set of equalities or inequalities. Arising from the field of combinatorial search in ar- tificial intelligence, constraint-propagation methods [13] recently turned-out to be a general and competitive approach for solving these continuous problems. The transition between discrete and continuous domains was made possible us- ing intervals to approximate the ranges of the values permitted for the variables of the problem [5, 11, 1]. It is well-known that the tractability of constraint-satisfaction strongly de- pends on the properties satisfied by the functions involved. For instance, lin- ear functions (i.e., of the form ∑ i a i x i ) can be handled efficiently using linear programming techniques. More general though still not too difficult classes of functions are the monotonic and convex ones, for which there exists specialized algorithms in the non-linear programming [10, 16], and constraint programming literature [17]. However, and despite the constant interest in solving particular classes of functions, the issue of devising logical tools to represent properties and perform reasoning on these properties has, as far as we know, not been considered in the literature. We define a framework where properties are seen as constraint abstractions. The Abstract Interpretation framework [3] helps defining a sound representation