# Contractibility for Open Global Constraints

## TL;DR

This paper introduces the concept of contractibility for open global constraints, providing a way to determine when filtering remains sound and how to approximate non-contractible constraints, with applications to hard and soft constraints.

## Contribution

It defines the property of contractibility for open global constraints and offers a method to identify or approximate non-contractible constraints, enhancing constraint programming flexibility.

## Key findings

- Contractibility characterizes when filtering remains sound for open constraints.
- A method to derive contractible approximations for non-contractible constraints.
- Application of the concept to both hard and soft constraints.

## Abstract

Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved, and fit naturally within constraint logic programming. However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open. This paper provides a simple characterization, called contractibility, of the constraints where filtering remains sound when the constraint is open. With this characterization we can easily determine whether a constraint has this property or not. In the latter case, we can use it to derive a contractible approximation to the constraint. We demonstrate this work on both hard and soft constraints. In the process, we formulate two general classes of soft constraints.