Strategic directions in constraint programming

Abstract

A constraint can be thought of intuitively as a restriction on a space of possibilities. Mathematical constraints are precisely specifiable relations among several unknowns (or variables), each taking a value in a given domain. Constraints restrict the possible values that variables can take, representing some (partial) information about the variables of interest. For instance, “The second side of a sheet of a paper must be imaged 9000 milliseconds after the time at which the first side is imaged,” relates two variables without precisely specifying the values they must take. One can think of such a constraint as standing for (a possibly infinite) set of values, in this case the set {^0, 9000&, ^1500, 10500&, . . .}. Constraints arise naturally in most areas of human endeavor. They are the natural medium of expression for formalizing regularities that underlie the computational and (natural or designed) physical worlds and their mathematical abstractions, with a rich tradition going back to the days of Euclidean geometry, if not earlier. For instance, the three angles of a triangle sum to 180 degrees; the four bases that make up DNA strands can only combine in particular orders; the sum of the currents flowing into a node must equal zero; the trusses supporting a bridge can only carry a certain static and dynamic load; the pressure, volume, and temperature of an enclosed gas must obey the “gas law”; Mary, John, and Susan must have different offices; the relative position of the scroller in the window scroll-bar must reflect the position of the current text in the underlying document; the derivative of a function is positive at zero; the function is monotone in its first argument, and so on. Indeed, whole subfields of mathematics (e.g., theory of Diophantine equations, group theory) and many celebrated conjectures of mathematics (e.g., Fermat’s Last Theorem) deal with whether certain constraints are satisfiable. Constraints naturally enjoy several interesting properties. First, as previously remarked, constraints may specify partial information—a constraint need not uniquely specify the value of its variables. Second, they are additive: given a constraint c1, say, X 1 Y $ Z, another constraint c2 can be added, say, X 1 Y # Z. The order of imposition of constraints does not matter; all that matters at the end is that the conjunction of constraints is in effect. Third,