Representing CSPs with Set-Labeled Diagrams: A Compilation Map

  • Alexandre Niveau
  • Hélène Fargier
  • Cédric Pralet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7205)


Constraint Satisfaction Problems (CSPs) offer a powerful framework for representing a great variety of problems. Unfortunately, most of the operations associated with CSPs are NP-hard. As some of these operations must be addressed online, compilation structures for CSPs have been proposed, e.g. finite-state automata and Multivalued Decision Diagrams (MDDs).

The aim of this paper is to draw a compilation map of these structures. We cast all of them as fragments of a more general framework that we call Set-labeled Diagrams (SDs), as they are rooted, directed acyclic graphs with variable-labeled nodes and set-labeled edges; contrary to MDDs and Binary Decision Diagrams, SDs are not required to be deterministic (the sets labeling the edges going out of a node are not necessarily disjoint), ordered nor even read-once.

We study the relative succinctness of different subclasses of SDs, as well as the complexity of classically considered queries and transformations. We show that a particular subset of SDs, satisfying a focusing property, has theoretical capabilities very close to those of Decomposable Negation Normal Forms (DNNFs), although they do not satisfy the decomposability property stricto sensu.


Model Check Variable Order Constraint Satisfaction Problem Outgoing Edge Label Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AFM02]
    Amilhastre, J., Fargier, H., Marquis, P.: Consistency restoration and explanations in dynamic CSPs — application to configuration. Artificial Intelligence 135(1-2), 199–234 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [AHHT07]
    Andersen, H.R., Hadzic, T., Hooker, J.N., Tiedemann, P.: A Constraint Store Based on Multivalued Decision Diagrams. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 118–132. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. [APV99]
    Amilhastre, J., Janssen, P., Vilarem, M.-C.: FA Minimisation Heuristics for a Class of Finite Languages. In: Boldt, O., Jürgensen, H. (eds.) WIA 1999. LNCS, vol. 2214, pp. 1–12. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. [Bry86]
    Bryant, R.E.: Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers 35(8), 677–691 (1986)zbMATHCrossRefGoogle Scholar
  5. [Dar01]
    Darwiche, A.: Decomposable Negation Normal Form. Journal of the ACM 48(4), 608–647 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [DM02]
    Darwiche, A., Marquis, P.: A Knowledge Compilation Map. JAIR 17, 229–264 (2002)MathSciNetzbMATHGoogle Scholar
  7. [FV04]
    Fargier, H., Vilarem, M.-C.: Compiling CSPs into tree-driven automata for interactive solving. Constraints 9, 263–287 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [KS92]
    Kautz, H.A., Selman, B.: Forming concepts for fast inference. In: Proc. of AAAI 1992, San Jose, CA, pp. 786–793 (1992)Google Scholar
  9. [KVBSV97]
    Kam, T., Villa, T., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: Synthesis of Finite State Machines: Functional Optimization. Kluwer Academic Publishers, Norwell (1997)zbMATHGoogle Scholar
  10. [KVBSV98]
    Kam, T., Villa, T., Brayton, R.K., Sangiovanni-Vincentelli, A.: Multi-valued Decision Diagrams: Theory and Applications. Multiple-Valued Logic 4(1-2), 9–62 (1998)MathSciNetzbMATHGoogle Scholar
  11. [MD06]
    Mateescu, R., Dechter, R.: Compiling Constraint Networks into AND/OR Multi-valued Decision Diagrams (AOMDDs). In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 329–343. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. [MT98]
    Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design: OBDD — Foundations and Applications. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. [NFPV10]
    Niveau, A., Fargier, H., Pralet, C., Verfaillie, G.: Knowledge compilation using interval automata and applications to planning. In: ECAI, pp. 459–464 (2010)Google Scholar
  14. [PD08]
    Pipatsrisawat, K., Darwiche, A.: New compilation languages based on structured decomposability. In: AAAI 2008, pp. 517–522 (2008)Google Scholar
  15. [RBW06]
    Rossi, F., van Beek, P., Walsh, T.: Handbook of Constraint Programming (Foundations of Artificial Intelligence). Elsevier Science Inc., New York (2006)Google Scholar
  16. [SKMB90]
    Srinivasan, A., Kam, T., Malik, S., Brayton, R.K.: Algorithms for discrete function manipulation. In: ICCAD 1990, pp. 92–95 (November 1990)Google Scholar
  17. [Vem92]
    Vempaty, N.R.: Solving Constraint Satisfaction Problems Using Finite State Automata. In: AAAI, pp. 453–458 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexandre Niveau
    • 1
  • Hélène Fargier
    • 2
  • Cédric Pralet
    • 3
  1. 1.CRIL/Université d’ArtoisLensFrance
  2. 2.IRIT/CNRSToulouseFrance
  3. 3.Onera/DCSDToulouseFrance

Personalised recommendations