Constraint Problems: Computability Is Equivalent to Continuity

In many practical situations, we would like to compute the set of all possible values that satisfy given constraints. It is known that even for computable (constructive) constraints, computing such set is not always algorithmically possible. One reason for this algorithmic impossibility is that sometimes, the dependence of the desired set on the parameters of the problem is not continuous, while all computable functions of real variables are continuous. In this paper, we show that this discontinuity is the only case when the desired set cannot be computed. Specifically, we provide an algorithm that computes such a set for all the cases when the dependence is continuous. 1 Constraint Satisfaction and Constraint Optimization: From a Practical Problems to Constructive Mathematics Constraints are ubiquitous. To describe a state of a physical system, we measure the values of the physical quantities characterizing this system – its coordinates, its velocity, its temperature, etc. The state can be then characterized by the tuple x = (x1, . . . , xn) consisting of the results x1, . . . , xn of these measurements. Not every tuple of n real numbers can represent a state, there are restrictions (constraints) on possible combinations x = (x1, . . . , xn). Some of these constraints are inequalities, i.e., have the type f(x) ≥ c or f(x) ≤ c: e.g., the velocity of a system cannot exceed the speed of light; the entropy of the closed system cannot be smaller than the initial value of its entropy, etc. Other constraints are equalities, i.e., have the type g(x) = v: e.g., the energy of a closed system must be always equal to the initial value of this energy.