This paper shows how to use constraint programming techniques for solving first-order constraints over the reals (i.e., formulas in the first-order predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., box-consistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of first-order consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1

Quantified constraints (i.e., first-order formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the behavior of quantified constraints under perturbation by showing that one can formulate the problem of solving quantified constraints as a nested parametric optimization problem followed by one sign computation. Using the fact that minima and maxima are stable under perturbation, but the sign of a real number is stable only for non-zero inputs, we derive practically useful conditions for the stability of quantified constraints under perturbation.

In this paper we study search heuristics for box decomposition methods that solve problems such as global optimization, minimax optimization, or quantified constraint solving. For this we unify these methods as nested branch-and-bound algorithms, and develop a model that is more convenient for studying heuristics for these algorithms than the traditional models from Artificial Intelligence. We use the result to prove various theorems about heuristics and apply the outcome to the box decomposition methods under consideration. We support the findings with timings for the method of quantified constraint solving developed by the author.

do not contain solutions. When we cannot easily prune more elements, we do branching by splitting a bound into pieces (for quantified variables this means replacing sub-constraints of the form ∀x ∈ I φ by ∀x ∈ I1 φ ∧ ∀x ∈ I2 φ where I = I1 ∪ I2, or the corresponding existential case). This gives us new possibilities for pruning. We repeat the two steps until we have pruned all elements (or disproved the constraint). For computing elements of the bounds that do contain solutions we take the negation of the input constraint and again apply the above branch-and-prune approach. In the paper we formalize this approach, study its properties in detail, improve it for an implementation, and do timings that show its efficiency. As a side-effect, this paper even improves the current methods for numerical constraint satisfaction problems in the case where the solution set does not consist of finitely many, isolated solutions, which—up to now—was essential for their efficiency. For example, the book describing the system Numerica [56] explicitly states that for inputs not fulfilling that property the method creates

Let a real first-order inequality constraint be a formula in the first-order predicate language over the reals, with the predicate symbols sin, . . . .

In this paper a new methodology to solve the pole clustering problem for parametric uncertain systems is introduced: The problem of clustering the closed loop poles into prescribed D-regions in the complex plane is stated as a quantified constraint problem that represents bounded uncertain parameters by intervals; and an engineering-oriented approach based on interval methods is developed to solve this quantified constraint problem. The result is a new, robust, reliable and design oriented method to deal with parametric uncertain systems. The

This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to tackle these continuous CSPs with parametric constraints, where the pruning and the solution identification processes are dedicated to universally quantified inequalities. Special rules are proposed to handle the parameter domains of the constraints. The originality of our framework lies in the fact that it solves the QCSP as a non-quantified CSP where the quantifiers are handled locally, at the level of each constraint. Experiments show that our algorithm outperforms the state of the art methods based on constraint techniques.