Frédéric Benhamou

Applying interval arithmetic to real, integer, and boolean constraints

Abstract We present in this paper a unified processing for real, integer, and Boolean constraints based on a general narrowing algorithm which applies to any n-ary relation on R. The basic idea is to define, for every such relation ρ, a narrowing function ρ based on the approximation of ρ by a Cartesian product of intervals whose bounds are floating-point numbers. We then focus on nonconvex relations and establish several properties. The more important of these properties is applied to justify the computation of usual relations defined in terms of intersections of simpler relations. We extend the scope of the narrowing algorithm used in the language BNR-Prolog to integer and disequality constraints, to Boolean constraints, and to relations mixing numerical and Boolean values. As a result, we propose a new Constraint Logic Programming language called CLP(BNR), where BNR stands for Booleans, Naturals, and Reals. In this language, constraints are expressed in a unique structure, allowing the mixing of real numbers, integers, and Booleans. We end with the presentation of several examples showing the advantages of such an approach from the point of view of the expressiveness, and give some preliminary computational results from a prototype.

Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques

RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreover, they may have different natures, being square or nonsquare, sparse or dense, linear, polynomial, or involving transcendental functions.The modeling language permits stating constraint models and tuning parameters of solving algorithms which efficiently combine interval methods and constraint satisfaction techniques. Several consistency techniques (box, hull, and 3B) are implemented. The distribution includes C sources, executables for different machine architectures, documentation, and benchmarks. The portability is ensured by the GNU C compiler.

Principles and Practice of Constraint Programming - CP 2006

Invited Papers.- Global Optimization of Probabilistically Constrained Linear Programs.- Algorithms and Constraint Programming.- Interval Analysis and Robotics.- Constraint Based Resilience Analysis.- Regular Papers.- Infinite Qualitative Simulations by Means of Constraint Programming.- Algorithms for Stochastic CSPs.- Graph Properties Based Filtering.- The ROOTS Constraint.- CoJava: Optimization Modeling by Nondeterministic Simulation.- An Algebraic Characterisation of Complexity for Valued Constraint.- Typed Guarded Decompositions for Constraint Satisfaction.- Propagation in CSP and SAT.- The Minimum Spanning Tree Constraint.- Impact of Censored Sampling on the Performance of Restart Strategies.- Watched Literals for Constraint Propagation in Minion.- Inner and Outer Approximations of Existentially Quantified Equality Constraints.- Performance Prediction and Automated Tuning of Randomized and Parametric Algorithms.- Adaptive Clause Weight Redistribution.- Localization of an Underwater Robot Using Interval Constraint Propagation.- Approximability of Integer Programming with Generalised Constraints.- When Constraint Programming and Local Search Solve the Scheduling Problem of Electricite de France Nuclear Power Plant Outages.- Generalized Arc Consistency for Positive Table Constraints.- Stochastic Allocation and Scheduling for Conditional Task Graphs in MPSoCs.- Boosting Open CSPs.- Compiling Constraint Networks into AND/OR Multi-valued Decision Diagrams (AOMDDs).- Distributed Constraint-Based Local Search.- High-Level Nondeterministic Abstractions in C++.- A Structural Characterization of Temporal Dynamic Controllability.- When Interval Analysis Helps Inter-block Backtracking.- Randomization in Constraint Programming for Airline Planning.- Towards an Efficient SAT Encoding for Temporal Reasoning.- Decomposition of Multi-operator Queries on Semiring-Based Graphical Models.- Dynamic Lex Constraints.- Generalizing AllDifferent: The SomeDifferent Constraint.- Mini-bucket Elimination with Bucket Propagation.- Constraint Satisfaction with Bounded Treewidth Revisited.- Preprocessing QBF.- The Theory of Grammar Constraints.- Constraint Programming Models for Graceful Graphs.- A Simple Distribution-Free Approach to the Max k-Armed Bandit Problem.- Generating Propagators for Finite Set Constraints.- Compiling Finite Linear CSP into SAT.- Differentiable Invariants.- Revisiting the Sequence Constraint.- BlockSolve: A Bottom-Up Approach for Solving Quantified CSPs.- General Symmetry Breaking Constraints.- Poster Papers.- Inferring Variable Conflicts for Local Search.- Reasoning by Dominance in Not-Equals Binary Constraint Networks.- Distributed Stable Matching Problems with Ties and Incomplete Lists.- Soft Arc Consistency Applied to Optimal Planning.- A Note on Low Autocorrelation Binary Sequences.- Relaxations and Explanations for Quantified Constraint Satisfaction Problems.- Static and Dynamic Structural Symmetry Breaking.- The Modelling Language Zinc.- A Filter for the Circuit Constraint.- A New Algorithm for Sampling CSP Solutions Uniformly at Random.- Sports League Scheduling: Enumerative Search for Prob026 from CSPLib.- Dynamic Symmetry Breaking Restarted.- The Effect of Constraint Representation on Structural Tractability.- Failure Analysis in Backtrack Search for Constraint Satisfaction.- Heavy-Tailed Runtime Distributions: Heuristics, Models and Optimal Refutations.- An Extension of Complexity Bounds and Dynamic Heuristics for Tree-Decompositions of CSP.- Clique Inference Process for Solving Max-CSP.- Global Grammar Constraints.- Constraint Propagation for Domain Bounding in Distributed Task Scheduling.- Interactive Distributed Configuration.- Retroactive Ordering for Dynamic Backtracking.

Continuous and Interval Constraints

Publisher Summary This chapter reviews that continuous constraint solving has been widely studied in several fields of applied mathematics and computer science. In computer algebra, continuous constraints are viewed as formulas from first-order logic interpreted over the real numbers. The symbolic algorithms transform the constraint systems within the same equivalence class in the interpretation domain according to some simplification ordering. The chapter also discusses the interval analysis, which is a set extension of numerical analysis such that the floating-point numbers are replaced with the intervals. The interval approximations are defined so as to enclose the computed real quantities and the algorithms are said to be complete. In constraint programming, continuous constraints are viewed as relations. The complete solving of nonlinear systems is implemented by exhaustive search techniques that compute solution space coverings by means of multi-dimensional boxes. The search is commonly accelerated through propagation-based algorithms. It reviews that continuous and interval constraints are generally contrasted with non negative integer or more generally discrete constraints. These last constraints, sometimes also called finite domain constraints, are studied in the constraint satisfaction problems (CSP) framework and are basic components of most current constraint-based languages.

Heterogeneous Constraint Solving

Most CLP languages designed in the past few years feature at least some combination of constraint solving capabilities. These combinations can take multiple forms since they achieve either the mixing of different domains or the use of different algorithms over the same domain. These solvers are also very different in nature. Some of them perform complete constraint solving while others are based on propagation methods. This paper is an attempt to design a unified framework describing the cooperation of constraint solving methods. Most techniques used in constraint-based systems are shown to be implementations of operators called constraint narrowing operators. A generalized notion of arc-consistency, called weak arc-consistency is proposed and is used to model heterogeneous constraint solving. We provide conditions on the constraint solving algorithms which guarantee termination, correctness and confluence of the resulting combined solver. This framework is shown to be general enough to describe the operational semantics of the basic constraint solving mechanisms in a number of current CLP systems.

Interval Constraint Logic Programming

In this paper, we present an overview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate n-ary relations over IR with Cartesian products of intervals whose bounds are taken in a finite subset of IR. Variables represent real values whose domains are intervals defined in the same manner. Narrowing operators are defined from approximations. These operators compute, from an interval and a relation, a set included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each step values from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators. We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong ling between approximations and local consistency notions and show that arc-consistency is an instance of the approximation framework. We finally describe recent work on different variants of the initial algorithm proposed by John Cleary and developped by W. Older and A. Vellino which have been proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, efficiency of the computation and finally, cooperation with other solvers. Some open questions are also identified.

Newton—constraint programming over nonlinear constraints

This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an effort to reconcile the declarative nature of constraint logic programming languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of box-consistency, which approximates arc-consistency, a notion well-known in artificial intelligence. Box-consistency achieves an effective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equation-solving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equation-solving benchmarks and outperforms the interval-based methods we are aware of on optimization problems.

Symbolic-interval cooperation in constraint programming

This paper surveys the field of cooperative constraint solving for a constraint programming perspective with an emphasis on combinations of symbolic and interval methods. On the one hand, symbolic methods provide adapted representations of the constraint expressions. On the other hand, interval methods compute verified enclosures of solution sets. Using cooperation of solvers, one can take advantage of both techniques in a unified framework: symbolic algorithms generally need to be combined with root extraction methods, and the efficiency of interval algorithms strongly depends on constraint expressions.

Trends in constraint programming

Introduction. Part I. The Past, Present and Future of Constraint Programming. Chapter 1. Constraint Programming as Declarative Algorithmics. Chapter 2. Constraint Programming Tools. Chapter 3. The Next 10 Years of Constraint Programming. Chapter 4. Constraint Propagation and Implementation. Chapter 5. On the First SAT/CP Integration Workshop. Chapter 6. Constraint-based Methods for Bioinformatics. Part II. Constraint Modeling and Reformulation. Chapter 7. Improved Models and Reformulation. Chapter 8. The Automatic Generation of Redundant Representations and Channeling Constraints. Part III. Symmetry in Constraint Satisfaction Problems. Chapter 9. GAPLex: Generalized Static Symmetry Breaking. Chapter 10. Symmetry Breaking in Subgraph Pattern Matching. Part IV. Interval Analysis, Constraint Propagation and Applications. Chapter 11. Modeling and Solving of a Radio Antenna Deployment Support Application. Chapter 12. Guaranteed Numerical Injectivity Test via Interval Analysis. Chapter 13. An Interval-based Approximation Method for Discrete Changes in Hybrid cc. Part V. Local Search Techniques in Constraint Satisfaction. Chapter 14. Combining Adaptive Noise and Look-Ahead in Local Search for SAT. Chapter 15. Finding Large Cliques using SAT Local Search. Chapter 16. Multi-Point Constructive Search for Constraint Satisfaction: An Overview. Chapter 17. Boosting SLS Using Resolution. Chapter 18. Growing COMET. Part VI. Preferences and Soft Constraints. Chapter 19. The Logic Behind Weighted CSP. Chapter 20. Dynamic Heuristics for Branch and Bound on Tree-Decomposition of Weighted CSPs. Part VII. Constraints in Software Testing, Verification and Analysis. Chapter 21. Extending a CP Solver with Congruences as Domains for Program Verification. Chapter 22. Generating Random Values Using Binary Decision Diagrams and Convex Polyhedra. Chapter 23. A Symbolic Model for Hash-Collision Attacks. Chapter 24. Strategy for Flaw Detection Based on a Service-driven Model for Group Protocols. Part VIII. Constraint Programming for Graphical Applications. Chapter 25. Trends and Issues in using Constraint Programming for Graphical Applications. Chapter 26. A Constraint Satisfaction Framework for Visual Problem Solving. Chapter 27. Computer Graphics and Constraint Solving: An Application to Virtual Camera Control. Index.

Progress in the Solving of a Circuit Design Problem

This paper describes a new global branch-and-prune algorithm dedicated to the solving of nonlinear systems. The pruning technique combines a multidimensional interval Newton method with HC4, a state of the art constraint satisfaction algorithm recently proposed by the authors. From an algorithmic point of view, the main contributions of this paper are the design of a fine-grained interaction between both algorithms which avoids some unnecessary computation and the description of HC4 in terms of a chain rule for constraint projections. Our algorithm is experimentally compared, on a particular circuit design problem proposed by Ebers and Moll in 1954, with two global methods proposed in the last ten years by Ratschek and Rokne and by Puget and Van Hentenryck. This comparison shows an improvement factor of five with respect to the fastest of these previous implementations on the same machine.