Declarative programming languages base on the idea that programs should be as close as possible to the problem specification and domain. Programs of these languages usually consist of directly formulated mathematical objects, i.e. relations and functions. Accordingly, declarative languages are distinguished into logic, functional, functional-logic, and constraint programming languages.

We propose a general scheme for the cooperation of di#erent constraint solvers. A uniform interface for constraint solvers allows to formally specify information exchange between them and it enables the development of an open and very flexible combination mechanism. This mechanism allows the definition of a wide range of di#erent cooperation strategies according to the current requirements such that our overall system forms a general framework for cooperating constraint solvers.

In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the meta-language needed to manipulate the constraints. This has the advantage to provide descriptions that are very close to the proof theoretical setting used now to describe constraint manipulations like unification or numerical constraint solving. We examplify the approach by presenting examples of constraint solvers descriptions and combinations written in the ELAN language. 1

Constraints are a powerful general paradigm for representing knowledge in intelligent systems. The standard constraint satisfaction paradigm involves variables over a discrete value domain and constraints which restrict the solutions to allowed value combinations. This standard paradigm is inapplicable to problems which are either: (a) mixed, involving both numeric and discrete variables, or (b) conditional? containing variables whose existence depends on the values chosen for other variables, or (c) both, conditional and mixed.

We propose a general scheme for the cooperation of different constraint solvers. On top of a uniform interface for constraint we stepwise develop reduction systems which describe the behaviour of an overall combined system. The modularity of our definitions of reduction relations at different levels allows the definition of cooperation strategies for the solvers according to the current requirements such that our overall system forms a general framework for cooperating solvers.

. Constraint solving has been successfully employed in diverse areas such as Operation Research, Planning, Logic Programming, and Automated Deduction. This has led to the development of a number of specialised approaches as well as to the adoption of different integration schemes and methodologies. In this paper we introduce an approach to incorporate constraint solving in term rewriting and we compare it with the Constraint Logic Programming scheme. We compare the two approaches both at the theoretical and at the implementational level and discuss potentials for cross-fertilisation. 1 Introduction Constraint solving aims at the development of efficient procedures tailored to restricted domains as well as to the integration of such procedures into general purpose reasoners. Constraint solving has been successfully employed in diverse areas such as Operation Research, Planning, Logic Programming, and Automated Deduction. This has led to the development of a number of specialised approa...

Constraints provide declarative descriptions of important requirements related to engineering projects. Most existing algorithms for constraint satisfaction require input consisting of binary constraints on variables that have discrete values. Such restrictions limit their use in engineering since complex constraints, involving several variables that have discrete and numeric values, are common. This paper provides an approach for decision support through approximating solution spaces that are defined by constraints. Our algorithm is not limited to a specific type of constraint but handles numeric and discrete variables in the same framework. Since a new type of local consistency narrows down the search space effectively, full-scale engineering tasks, such as designs involving hundreds of variables, are accommodated without excessive computational complexity. The approach is demonstrated for selection of appropriate wind bracing for single story steel-framed buildings. Results may be used for input into other tools containing algorithms such as those o ering i) higher levels of consistency, ii) optimally directed point-solution search and iii) simulation behavior. Finally, extension to dynamic constraint satisfaction using different combinations of activation conditions is straightforward. It is expected that this approach will improve the performance of many existing and future computer-aided engineering tools.

. We propose a general way of combining background reasoners in theory reasoning. Using a restricted version of the Craig Interpolation Lemma, we show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures. In this case, constraint information is propagated across reasoners by exchanging residues over a common signature. As an application of our approach, we describe a multi-theory version of the semantic tableau calculus. 1 Introduction Theory reasoning is a powerful deduction paradigm in which a general-purpose reasoner is complemented by a specialized procedure, the background reasoner, which (semi-)decides formula satisfiability with respect to a certain background theory. After the pioneering work of Stickel who devised a theory version of resolution [8], nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see [2] for a surve...

Design tasks in structural engineering have always involved the use of constraints to formulate design requirements. Most existing algorithms for constraint satisfaction require input consisting of binary constraints on variables that have discrete values. Such restrictions limit their use in structural engineering since typical structural design tasks involve discrete and numerical variables. This paper provides an approach for decision support through approximating solution spaces of such constraint problems by local consistency. The approach is demonstrated for the selection of appropriate wind bracing for single story steel-framed buildings involving more than hundred variables.