This paper deals with the problem of safety verification of non-linear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and improve it by developing an additional refinement step that employs interval constraint propagation to add information to the abstraction without introducing new grid elements. Moreover, the resulting method allows switching conditions, initial states and unsafe states to be described by complex constraints instead of sets that correspond to grid elements. Nevertheless, the method can be easily implemented since it is based on a well-defined set of constraints, on which one can run any constraint propagation based solver. Tests of such an implementation are promising.

In this paper, a new search technique over numeric csps is presented: dynamic domain splitting. The usual search technique over numeric csps is a dichotomic search interleaved with a consistency filtering, which is called domain splitting. This paper proposes to replace chronological backtracking at the core of domain splitting by a non destructive backtracking technique.

Abstract. This paper presents a model and implementation techniques for speeding up constraint propagation. Two fundamental approaches to improving constraint propagation are explored: keeping track of which propagators are at fixpoint, and choosing which propagator to apply next. We show how idempotence reasoning and events help track fixpoints more accurately. We improve these methods by using them dynamically (taking into account current domains to improve accuracy). We define priority-based approaches to choosing a next propagator and show that dynamic priorities can improve propagation. We illustrate that the use of multiple propagators for the same constraint can be advantageous with priorities, and introduce staged propagators which combine the effects of multiple propagators with priorities for greater efficiency. 1

This paper presents a model and implementation techniques for speeding up constraint propagation. Three fundamental approaches to improving constraint propagation based on propagators as implementations of constraints are explored: keeping track of which propagators are at fixpoint, choosing which propagator to apply next, and how to combine several propagators for the same constraint. We show how idempotence reasoning and events help track fixpoints more accurately. We improve these methods by using them dynamically (taking into account current variable domains to improve accuracy). We define priority-based approaches to choosing a next propagator and show that dynamic priorities can improve propagation. We illustrate that the use of multiple propagators for the same constraint can be advantageous with priorities, and introduce staged propagators that combine the effects of multiple propagators with priorities for greater efficiency.

CLIP is an implementation of CLP(Intervals) which has been designed to be verifiably correct in the sense that the answers it returns are mathematically correct solutions to the underlying arithmetic constraints. This fundamental design criteria affects many aspects of the implementation from the input and output of decimal constants to the design of the interval arithmetic libraries and the constraint solving algorithms. In particular, to enhance verifiability, CLIP employs the simplest model of constraint solving in which constraints are decomposed into sets of primitive constraints which are then solved using a library of primitive constraint contractors. This approach results in a simple constraint solver whose correctness is relatively straightforward to verify, but the solver is only able to solve relatively simple constraints. In this paper, we present the syntax, semantics, and implementation of CLIP, and we show how to use metalevel techniques to enhance the power of the CLIP constraint solver while preserving the simple structure of the system. In particular, we demonstrate that several of the box-narrowing algorithms from the Newton and Numerica systems can be easily implemented in CLIP. The principal advantages of this approach are (1) the resulting solvers are relatively easy to prove correct, (2) new solvers can be rapidly prototyped since the code is more concise and declarative than for imperative languages, and (3) contractors can be implemented directly from mathematical formulae without having to first prove results about interval arithmetic operators. Finally, the source code for the system is publicly available, which is a clear prerequisite for public, independent verifiability.

Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis. 2002 Elsevier Science B.V. All rights reserved.

Interval constraint solvers use local consistencies— among which one worth mentioning is box consistency— for computing verified solutions of real constraint systems. Though among the most efficient ones, the algorithm for enforcing box consistency suffers from the use of timeconsuming operators. This paper first introduces box consistency, a weakening of box consistency; this new notion then allows us to devise an adaptive algorithm that computes box consistency by enforcing box consistency, decreasing the parameter as variables ’ domains get tightened, then achieving eventually box consistency, which is equivalent to box consistency. A new propagation algorithm is also given, that intensifies the use of the most contracting pruning functions based on box consistency. The resulting algorithm is finally shown to outperform the original scheme for enforcing box consistency on a set of standard benchmarks. 1.

Abstract. Constraints generated for Program Verification tasks very often involve integer variables ranging on all the machine-representable integer values. Thus, if the propagation takes a time that is linear in the size of the domains, it will not reach a fix point in practical time. Indeed, the propagation time needed to reduce the interval domains for as simple equations as x = 2y + 1 and x = 2z is proportional to the size of the initial domains of the variables. To avoid this slow convergence phenomenon, we propose to enrich a Constraint Programming Solver (CP Solver) with congruence domains. This idea has been introduced by [1] in the abstract interpretation community and we show how a CP Solver can benefit from it, for example in discovering immediately that 12x + |y | = 3 and 4z + 7y = 0 have no integer solution. 1

An e#cient parallel interval narrowing algorithm for solving numerical problems is designed, implemented and tested. Di#erences with the corresponding sequential algorithm are clearly stated. The algorithm's performance is analyzed in the Bulk-Synchronous Parallel (BSP) cost model which suggests speed-ups on highbandwidth architectures. Experimental results on a massively parallel machine Cray T3E-1200 validate the model and show the parallel algorithm's e#ciency as well as its limitations. Key words: parallel algorithms, numerical algorithms, interval narrowing, constraint propagation, BSP model. Appears in: Information Processing Letters, 74(3-4):141--146, 2000. Elsevier Science. 1 Introduction Parallel processing of numerical problems via interval constraints has been proposed as a general framework for high-performance numerical computation in [5]. Its two potential advantages over classical methods are 1. the guarantee of numerically correct answers through interval arithme...

This paper proposes a generic CP algorithm that is more general than the classical one but closer to practical requirements (re-use, prototyping, . . . ). A CP strategy is given by a set of four algorithms that are seen as black-boxes specified by some minimal properties to ensure termination and confluence. This framework is well-suited for the design of a CP library in which the components can be plugged in any new algorithm. Let us remark that in the framework of constraint satisfaction, the SCP3 algorithm can implement the AC1 and AC3 algorithms for arc consistency, and the BC3 algorithm for box consistency; nevertheless, the ACn (with n > 3) algorithms use particular data structures and then cannot be realized (except in the meaningless case where CP implements ACn).