A Unified Framework for Interval Constraints and Interval Arithmetic

We are concerned with interval constraints: solving constraints among real unknowns in such a way that soundness is not aaected by rounding errors. The contraction operator for the constraint x + y = z can simply be expressed in terms of interval arithmetic. An attempt to use the analogous deenition for x y = z fails if the usual deenitions of interval arithmetic are used. We propose an alternative to the interval arithmetic deenition of interval division so that the two constraints can be handled in an analogous way. This leads to a uniied treatment of both interval constraints and interval arithmetic that makes it easy to derive formulas for other constraint contraction operators. We present a theorem that justiies simulating interval arithmetic evaluation of complex expressions by means of constraint propagation. A naive implementation of this simulation is ineecient. We present a theorem that justiies what we call the totality optimization. It makes simulation of expression evaluation by means of constraint propagation as eecient as in interval arithmetic. It also speeds up the contraction operators for primitive constraints.