Constraint theory: an overview

The purpose of this paper is to develop an analytic foundation—called Constraint Theory—for the systematic determination of mathematical model consistency and computational allowability. Constraint Theory's primary application is the more efficient construction and use of heterogeneous, multidimensional mathematical models, especially when interdisciplinary technical teams, system analysts, and managers are involved.

A rigorous basis for the formerly ill-defined notion of ‘ constraint ’ is established and four interrelated viewpoints of a mathematical model are defined : (a) the set-theoretic relation space, (b) the submodel family, (c) the bipartite graph, and (d) the constraint matrix. The first two viewpoints represent complete models ; the second two represent metamodels.

Many correspondences are proved between the topological properties of a model's graph and its constraint properties, Variables located in different connected components of a graph are always mutually consistent, but computations performed on them are never allowable. If a model graph of universal relations has a tree structure, all its variables are mutually consistent. If a model graph of regular relations has a tree structure, all its variables are consistent and, moreover, none can be point constrained. The circuit-cluster portions of the model graph are the only possible locations for point constraint and overconstraint to occur in a set of regular relations. In these cases, point constraint can always be identified by a ‘ basic nodal square ’. Topological concepts such as circuit rank, circuit index, and constraint potential are applied to extract the basic nodel squares from a large, complex circuit-cluster portion of a model graph.

Several examples applying the principles of constraint theory are provided.     

Tractable Decision for a Constraint Language Implies Tractable Search

A constraint satisfaction problem (CSP) instance has a set of variables, each of which can take values in some domain. It also has a set of constraints, each of which restricts the variables in its scope to take values limited by its constraint relation.The language of a constraint satisfaction problem instance is the set of different constraint relations used in its specification. For a given set of relations over some domain we define the problem CSP () to the set of CSP instances whose language is contained in .The decision problem for a set of CSP instances is, given an instance in the class, to decide whether a solution exists. The search problem is to find such a solution. Here we address the connection between the tractability of the decision and search problems. We prove that given a constraint language over a finite domain for which the decision problem for CSP () is tractable, the search problem is always tractable.We define a surjective language over a finite domain in a standard way. We also show that we can determine in polynomial time whether an instance over a surjective language with a tractable decision problem has fewer than k solutions, and that we can generate all of its solutions with polynomial delay.     

Computer implementation of the constraint method

A computer program based on the constraint method approach to finite elements is described. The program permits the user to select arbitrarily polynomial orders for the various fields to be approximated. A set of nodal variables enforcing C¹ continuity for arbitrary polynomial order is presented, and those aspects of the program algorithm which differ from conventional finite element programs are described, including the numerical determination of a set of independent variables. An example is presented and the results obtained by the constraint method program compared with results from a conventional program. The constraint method approach is seen to be competitive with the conventional approach.     

A new tractable class of constraint satisfaction problems

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras. AMS subject classification 08A70     

Fast algebraic methods for interval constraint problems

We describe an effective generic method for solving constraint problems, based on Tarski’s relation algebra, using path-consistency as a pruning technique. We investigate the performance of this method on interval constraint problems. Time performance is affected strongly by the path-consistency calculations, which involve the calculation of compositions of relations. We investigate various methods of tuning composition calculations, and also path-consistency computations. Space performance is affected by the branching factor during search. Reducing this branching factor depends on the existence of ‘nice’ subclasses of the constraint domain. Finally, we survey the statistics of consistency properties of interval constraint problems. Problems of up to 500 variables may be solved in expected cubic time. Evidence is presented that the ‘phase transition’ occurs in the range 6 ≤ n.c ≤15, where n is the number of variables, and c is the ratio of non-trivial constraints to possible constraints. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cardinal: A Finite Sets Constraint Solver

In this paper we present Cardinal, a general finite sets constraint solver just made publicly available in ECLiPSe Constraint System, suitable for combinatorial problem solving by exploiting inferences over sets cardinality. In fact, to deal with set variables and set constraints, existing set constraint solvers are not adequate to handle a number of problems, as they do not actively use important information about the cardinality of the sets, a key feature in such problems. Cardinal is formally presented as a set of rewriting rules on a constraint store and we illustrate its efficiency with experimental results. We show the importance of propagating constraints on sets cardinality, by comparing Cardinal with other solvers. Another contribution of this paper is on modelling: we focus essentially on digital circuits problems, for which we present new modelling approaches and prove that sets alone can be used to model these problems in a clean manner and solve them efficiently using Cardinal. Results on a set of diagnostic problems show that Cardinal obtains a speed up of about two orders of magnitude over Conjunto, a previous available set constraint solver, which uses a more limited amount of constraint propagation on cardinalities. Additionally, to further extend modelling capabilities and efficiency, we generalized Cardinal to actively consider constraints over set functions other than cardinality. The Cardinal version just released allows declaring union, minimum and maximum functions on set variables, and easily constraining those functions, letting Cardinal especial inferences efficiently take care of different problems. We describe such extensions and discuss its potentialities, which promise interesting research directions.     

A new approach to partial constraint satisfaction problems

A finite binary Constraint Satisfaction Problem (CSPs) is defined as consisting of a set of n problem variables, a domain of d potential values for each variable and a set of m binary constraints involving only two variables at a time. A solution to such a CSP is specified by assignment of a value to each variable that does not violate any of the constraints. The CSPs belong to the class of NP-Complete Problems. Backtracking and its variants have been generally used for solving CSPs. The class of Partial Constraint Satisfaction Problems (PCSPs) is a subclass of CSPs that are either too difficult to solve or are unsolvable. Near optimal solutions are always desired to these problems.

In this article, we have considered only finite binary CSPs or PCSPs and developed a method of time complexity O(n ² d ²) to obtain a near optimal solution for them. The performance of the method in terms of the average number of consistency checks and the average number of constraint violations is measured on various randomly generated binary CSPs and compared with the Branch and Bound (BB) method used to obtain the same solution. The BB method is a widely used optimization technique that may be viewed as a variation of backtracking. Thus, it was a natural choice in seeking an analog of backtracking to find optimal partial solutions for PCSPs. The proposed method moves much faster to the solution. The performance results indicate that in terms of the number of consistency checks, the proposed method has much less consistency checks than BB whereas in terms of average number of constraint violations both methods are same. An upper bound on the distance of the solution from the optimal solution is obtained analytically as ?n(n???2)(d???2)/(d???1)?.     

Optimal Filters with Multiresolution Apertures

This paper presents an extension to the recently introduced class of nonlinear filters known as Aperture Filters. By taking a multiresolution approach, it can be shown that more accurate filtering results (in terms of mean absolute error) may be achieved compared to the standard aperture filter given the same size of training set. Most optimisation techniques for nonlinear filters require a knowledge of the conditional probabilities of the output. These probabilities are estimated from observations of a representative training set. As the size of the training set is related to the number of input combinations of the filter, it increases very rapidly as the number of input variables increases. It can be impossibly large for all but the simplest binary filters. In order to design nonlinear filters of practical use, it is necessary to limit the size of the search space i.e. the number of possible filters (and hence the training set size) by the application of filter constraints. Filter constraints take several different forms, the most general of which is the window constraint where the output filter value is estimated from only a limited range of input variables.Aperture filters comprise a special case of nonlinear filters in which the input window is limited not only in its domain (or duration) but also in its amplitude. The reduced range of input signal leads directly to a reduction in the size of training set required to produce accurate output estimates. However in order to solve complex filtering problems, it is necessary for the aperture to be sufficiently large so as to observe enough of the signal to estimate its output accurately.In this paper it is shown how the input range of the aperture may be expanded without increasing the size of the search space by adopting a multiresolution approach. The constraint applied in this case is the resolution constraint. This paper presents both theoretical and practical results to demonstrate and quantify the improvement.     

Robust H∞ dynamic output feedback control for spacecraft rendezvous with poles and input constraint

This paper is concerned with the output feedback control problem for spacecraft rendezvous subject to target angular velocity uncertainty and controller uncertainty, external disturbance and input constraint. A general full-order dynamic output feedback (DOF) controller is proposed. As a stepping-stone, the H_∞ performance requirement, poles and input constraint are analysed separately via linear matrix inequalities (LMIs). Then, with the obtained results, the controller design problem is cast into a convex problem subject to a set of LMI constraints through a critical change of controller variables. Furthermore, when the system states are all available, a reduced sufficient condition of the non-fragile state feedback controller is given. Compared with existing results, the designed controller has overcome the disadvantage of strictly proper DOF controller, where the initial value of the control input is zero. Besides, the constraint on poles placement is relaxed. A numerical simulation is performed to verify the effectiveness of the proposed method.     

Managing dynamic CSPs with preferences

We present a new framework, managing Constraint Satisfaction Problems (CSPs) with preferences in a dynamic environment. Unlike the existing CSP models managing one form of preferences, ours supports four types, namely: unary and binary constraint preferences, composite preferences and conditional preferences. This offers more expressive power in representing a wide variety of dynamic constraint applications under preferences and where the possible changes are known and available a priori. Conditional preferences allow some preference functions to be added dynamically to the problem, during the resolution process, if a given condition on some variables is true. A composite preference is a higher level of preference among the choices of a composite variable. Composite variables are variables whose possible values are CSP variables. In other words, this allows us to represent disjunctive CSP variables. The preferences are viewed as a set of soft constraints using the fuzzy CSP framework. Solving constraint problems with preferences consists in finding a solution satisfying all the constraints while optimizing the global preference value. This is handled by four variants of the branch and bound algorithm, we propose in this paper, and where constraint propagation is used to improve the time efficiency in practice. In order to evaluate and compare the performance of these four strategies, we conducted an experimental study on randomly generated dynamic CSPs with quantitative preferences. The results are reported and discussed in the paper.     

Least Squares Sub-pixel Registration Refinement Using Area Sampler Model

Super-resolution applications require sub-pixel registrations of low resolution images to be almost exact due to the deterioration caused by inaccurate image registration. A linear-least-squares technique is proposed to refine sub-pixel translation parameters, which can be employed when the images are registered but just where there is not enough sub-pixel accuracy. In the technique, it is assumed that low resolution pixels are obtained by area sampling high resolution pixel field which have twice the density of their low resolution correspondents. Using this downsampling schema, a set of equations is formed. Assumed geometry and layout provide a constraint set to be used with the equation set. The sub-pixel translations are then found using least-squares-solution-with-equality-constraints. The method is shown to improve the registration accuracy.     

Filtering Algorithms for the NValue Constraint

The NValue constraint counts the number of different values assigned to a vector of variables. Propagating generalized arc consistency on this constraint is NP-hard. We show that computing even the lower bound on the number of values is NP-hard. We therefore study different approximation heuristics for this problem. We introduce three new methods for computing a lower bound on the number of values. The first two are based on the maximum independent set problem and are incomparable to a previous approach based on intervals. The last method is a linear relaxation of the problem. This gives a tighter lower bound than all other methods, but at a greater asymptotic cost.     

Don’t care in SMT: building flexible yet efficient abstraction/refinement solvers

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.     

Consistency techniques for continuous constraints

We consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed.Our approach is based on approximating regions of feasible solutions by 2 ^k -trees, a representation commonly used in computer vision and image processing. We give simple and stable algorithms for computing labelings with arbitrary degrees of consistency. The algorithms can process constraints and solution spaces of arbitrary complexity, but with a fixed maximal resolution.Previous work has shown that when constraints are convex and binary, path-consistency is sufficient to ensure global consistency. We show that for continuous domains, this result can be generalized to ternary and in fact arbitrary n-ary constraints using the concept of (3,2)-relational consistency. This leads to polynomial-time algorithms for computing globally consistent labelings for a large class of constraint satisfaction problems with continuous variables.     

Fixed-order H∞ compensator design

A method is presented for the construction of fixed-order compensators to provide H_∞ norm constraint for linear control systems with exogenous disturbances. The method is based on the celebrated bounded-real lemma that predicates the H_∞ norm constraint via a Riccati inequality. The synthesis of fixed-order controllers whose dimensions are less than the order of a given plant, is demonstrated by a set of sufficient conditions along with a numerical algorithm.     

Propagation algorithms for lexicographic ordering constraints

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finite-domain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq.     

Design by example for SQL table definitions with functional dependencies

A database is C-Armstrong for a given set of constraints in a class C if it satisfies every constraint of the set and violates every constraint in C not implied by the set. Therefore, Armstrong databases are test data that perfectly illustrate the current perceptions about the semantics of a schema. We extend the existing theory of Armstrong relations to a toolbox of Armstrong tables. That is, we investigate structural and computational properties of Armstrong tables for the class of functional dependencies (FDs) over SQL tables. Relations are special instances of SQL tables with no duplicate rows and no null value occurrences. While FDs do not enjoy Armstrong tables, the combined class of standard FDs and NOT NULL constraints does enjoy Armstrong tables. The problem of finding an Armstrong table is shown to be precisely exponential for this combined class. However, we establish an algorithm that computes Armstrong tables with a size at most quadratic in that of a minimum-sized Armstrong table. Our resulting toolbox of Armstrong tables can be applied by data engineers to concisely visualize constraints on SQL data. Such support can lead to designs that guarantee efficient data management in practice.     

HairControl: A Tracking Solution for Directable Hair Simulation

We present a method for adding artistic control to physics‐based hair simulation. Taking as input an animation of a coarse set of guide hairs, we constrain a subsequent higher‐resolution simulation of detail hairs to follow the input motion in a spatially‐averaged sense. The resulting high‐resolution motion adheres to the artistic intent, but is enhanced with detailed deformations and dynamics generated by physics‐based simulation. The technical core of our approach is formed by a set of tracking constraints, requiring the center of mass of a given subset of detail hair to maintain its position relative to a reference point on the corresponding guide hair. As a crucial element of our formulation, we introduce the concept of dynamically‐changing constraint targets that allow reference points to slide along the guide hairs to provide sufficient flexibility for natural deformations. We furthermore propose to regularize the null space of the tracking constraints based on variance minimization, effectively controlling the amount of spread in the hair. We demonstrate the ability of our tracking solver to generate directable yet natural hair motion on a set of targeted experiments and show its application to production‐level animations.     

Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instantiations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instantiations are provided owing to the relaxation of some constraints. Fuzzy constraints are thus flexible. CSP notions of consistency and k-consistency can be extended to this framework and the classical algorithms used in CSP resolution (e.g., tree search and filtering) can be adapted without losing much of their efficiency. Most classical theoretical results remain applicable to FCSPs. In the paper, various types of constraints are modelled in the same framework. The handling of uncertain parameters is carried out in the same setting because possibility theory can account for both preference and uncertainty. The presence of uncertain parameters leads to ill-defined CSPs, where the set of constraints which defines the problem is not precisely known.     

Concurrent behavior: A construct to specify the external behavior of objects in object databases

We present a linguistic construct to define concurrency control for the objects of an object database. This construct, calledconcurrent behavior, allows to define a concurrency control specification for each object type in the database; in a sense, it can be seen as a type extension. The concurrent behavior is composed by two parts: the first one, calledcommutativity specification, is a set of conditional rules, by which the programmer specifies when two methods do not conflict each other. The second part, the constraint specification, is a set of guarded regular expressions, calledconstraints, by which the programmer defines the allowed sequences of method calls. At each time during an actual execution, a subset of constraints may be active so limiting the external behavior of the object. A constraint becomesactive when its guard is verified, where a guard is composed of the occurrence of some method callm along with the verification of a boolean expression on the object state and the actual parameters ofm. A constraint dies when a string of the language corresponding to the regular expression has been recognized. While the commutativity specification is devoted to specify the way in which the external behavior of an object is influenced by the existence of concurrent transactions in the system, the constraint specification defines the behavior of the object, independently from the transactions. Since the two parts of the concurrent behavior are syntactically distinct and, moreover, each of them consists of a set of independent rules, modularity in specifying the objects is enhanced, with respect to a unique specification. We outline an implementation of the construct, which is based on a look-ahead policy: at each method execution, we foresee the admissible successive behaviors of the object, instead of checking the admission of each request at the time it is actually made.