Possibilistic Constraint Satisfaction Problems or "How to handle soft constraints?" (1303.5427v1)

Abstract: Many AI synthesis problems such as planning or scheduling may be modelized as constraint satisfaction problems (CSP). A CSP is typically defined as the problem of finding any consistent labeling for a fixed set of variables satisfying all given constraints between these variables. However, for many real tasks such as job-shop scheduling, time-table scheduling, design?, all these constraints have not the same significance and have not to be necessarily satisfied. A first distinction can be made between hard constraints, which every solution should satisfy and soft constraints, whose satisfaction has not to be certain. In this paper, we formalize the notion of possibilistic constraint satisfaction problems that allows the modeling of uncertainly satisfied constraints. We use a possibility distribution over labelings to represent respective possibilities of each labeling. Necessity-valued constraints allow a simple expression of the respective certainty degrees of each constraint. The main advantage of our approach is its integration in the CSP technical framework. Most classical techniques, such as Backtracking (BT), arcconsistency enforcing (AC) or Forward Checking have been extended to handle possibilistics CSP and are effectively implemented. The utility of our approach is demonstrated on a simple design problem.

• The paper formalizes soft constraints by integrating possibility distributions and necessity measures into traditional CSP frameworks.
• It adapts standard CSP techniques like Backtracking, Arc-Consistency, and Forward Checking to efficiently handle uncertainty in constraint satisfaction.
• Empirical evaluations, including a menu planning example, demonstrate the method's potential for balancing expert input with flexible, real-world decision-making.

Overview of "Possibilistic Constraint Satisfaction Problems or 'How to handle soft constraints?'"

The paper by Thomas Schiex presents a detailed exploration of Possibilistic Constraint Satisfaction Problems (PCSPs), an extension of traditional Constraint Satisfaction Problems (CSPs) tailored to account for soft constraints whose satisfaction is not absolute. Unlike classical CSPs that involve hard constraints requiring solutions to satisfy strict conditions, PCSPs introduce the flexibility of dealing with uncertainty in constraint satisfaction via possibility theory.

The essence of PCSPs is the incorporation of soft constraints through possibility distributions over labelings, combined with necessity measures that provide a framework to express the degree of certainty attributed to each constraint. This method allows for a nuanced representation of problems in domains like job-shop scheduling, where all constraints do not hold equal importance and mandatory satisfaction is not feasible.

Key Contributions

• Formalization of Soft Constraints: The paper refrains from generic theoretical formulations, favoring a specific and operational meaning for soft constraints when defining PCSPs. This specificity leads to efficient solving techniques embedded within CSP frameworks, enhancing their practical applicability.
• Possibility and Necessity Measures: Two central concepts within PCSPs are the possibility and necessity measures, which are formulated to evaluate the extent of constraint satisfaction. These measures originate from a possibility distribution that provides flexibility in handling partial consistency within constraint networks.
• Algorithmic Extensions: Schiex extends traditional CSP solving techniques like Backtracking, Arc-Consistency, and Forward Checking to PCSPs, thereby demonstrating the adaptability of these classical methods to handle the uncertainty inherent in soft constraints. The paper outlines ways to compute the best labelings that meet the defined necessity-bound constraints efficiently.

Numerical Results and Example Application

The utility of PCSPs is demonstrated through a simple design problem involving menu planning in a high-end restaurant. By incorporating hard and soft constraints reflecting both expert knowledge and customer preferences, the resulting CSP is effectively solved using modified search techniques, arriving at optimal solutions that balance constraint satisfaction with preference adherence.

Implications and Future Prospects

This work has significant theoretical and practical implications in artificial intelligence and automated decision-making:

• Theoretical Insights: The elucidation of PCSPs paves the way for more generalized frameworks where constraints might not be entirely rigid, allowing artificial systems to operate in more natural and uncertain environments. This paper thus contributes to a deeper understanding of constraint reasoning under uncertainty.
• Practical Applications: In real-world applications, especially those involving scheduling and planning, PCSPs enable more flexible handling of constraints, making them invaluable in situations where traditional CSPs fall short due to rigidity in constraint satisfaction.
• Future Developments: The prospects for further research include extending various classical CSP techniques (such as backjumping and cycle cutset) to PCSPs and exploring integration with fuzzy constraints for enhanced expressive power. Understanding how PCSPs interact with other models, like probabilistic constraints, remains a fertile avenue for exploration.

This paper's formulation and subsequent empirical evaluation of PCSPs offer a robust foundation for future explorations into soft constraint satisfaction, reaffirming the importance of flexibility and adaptability in constraint-based AI systems.