A geometric constraint over k-dimensional objects and shapes subject to business rules

This report presents a global constraint that enforces rules written in a language based on arithmetic and first-order logic to hold among a set of objects. In a first step, the rules are rewritten to Quantifier-Free Presburger Arithmetic (QFPA) formulas. Secondly, such formulas are compiled to generators of k-dimensional forbidden sets. Such generators are a generalization of the indexicals of cc(FD). Finally, the forbidden sets generated by such indexicals are aggregated by a sweep-based algorithm and used for filtering. The business rules allow to express a great variety of packing and placement constraints, while admitting efficient and effective filtering of the domain variables of the k-dimensional object, without the need to use spatial data structures. The constraint was used to directly encode the packing knowledge of a major car manufacturer and tested on a set of real packing problems under these rules, as well as on a packing-unpacking problem

Generalized Support and Formal Development of Constraint Propagators

Abstract The concept of support is pervasive in constraint programming. Traditionally, when a domain value ceases to have support, it may be removed because it takes part in no solutions. Arc-consistency algorithms such as AC2001 [8] make use of support in the form of a single domain value. GAC algorithms such as GAC-Schema We design a methodology for developing correct propagators using generalized support. A constraint is expressed as a family of support properties, which may be proven correct against the formal semantics of the constraint. Using CurryHoward isomorphism to interpret constructive proofs as programs, we show how to derive correct propagators from the constructive proofs of the support properties. The framework is carefully designed to allow efficient algorithms to be produced. Derived algorithms may make use of dynamic literal triggers or watched literal

Generating propagators for finite set constraints

Abstract. Ideally, programming propagators as implementations of constraints should be an entirely declarative specification process for a large class of constraints: a high-level declarative specification is automatically translated into an efficient propagator. This paper introduces the use of existential monadic secondorder logic as declarative specification language for finite set propagators. The approach taken in the paper is to automatically derive projection propagators (involving a single variable only) implementing constraints described by formulas. By this, the paper transfers the ideas of indexicals to finite set constraints while considerably increasing the level of abstraction available with indexicals. The paper proves soundness and completeness of the derived propagators and presents a run-time analysis, including techniques for efficiently executing projectors for n-ary constraints.