Computing the Transitive Closure of a Union of Affine Integer Tuple Relations

  • Anna Beletska
  • Denis Barthou
  • Wlodzimierz Bielecki
  • Albert Cohen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5573)


This paper proposes a method to compute the transitive closure of a union of affine relations on integer tuples. Within Presburger arithmetics, complete algorithms to compute the transitive closure exist for convex polyhedra only. In presence of non-convex relations, there exist little but special cases and incomplete heuristics. We introduce a novel sufficient and necessary condition defining a class of relations for which an exact computation is possible. Our method is immediately applicable to a wide area of symbolic computation problems. It is illustrated on representative examples and compared with state-of-the-art approaches.


transitive closure affine tuple relations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alias, C., Barthou, D.: On domain specific languages re-engineering. In: Glück, R., Lowry, M. (eds.) GPCE 2005. LNCS, vol. 3676, pp. 63–77. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Barthou, D., Feautrier, P., Redon, X.: On the equivalence of two systems of affine recurrence equations. In: Monien, B., Feldmann, R.L. (eds.) Euro-Par 2002. LNCS, vol. 2400, pp. 309–313. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Beletska, A., Bielecki, W., San Pietro, P.: Extracting coarse-grained parallelism in program loops with the slicing framework. In: ISPDC 2007: Proceedings of the Sixth International Symposium on Parallel and Distributed Computing, p. 29. IEEE Computer Society Press, Washington (2007)Google Scholar
  4. 4.
    Bielecki, W., Klimek, T., Trifunovic, K.: Calculating exact transitive closure for a normalized affine integer tuple relation. Journal of Electronic Notes in Discrete Mathematics 33, 7–14 (2009)Google Scholar
  5. 5.
    Boigelot, B.: Symbolic Methods for Exploring Infinite State Spaces. PhD thesis, Université de Liège (1998)Google Scholar
  6. 6.
    Boigelot, B., Wolper, P.: Symbolic verification with periodic sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  7. 7.
    Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and presburger arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    Darte, A., Robert, Y., Vivien, F.: Scheduling and Automatic Parallelization. Birkhaüser (2000)Google Scholar
  9. 9.
    Kelly, W., Pugh, W., Rosser, E., Shpeisman, T.: Transitive closure of infinite graphs and its applications. Int. J. Parallel Programming 24(6), 579–598 (1996)CrossRefGoogle Scholar
  10. 10.
    Shashidhar, K.C., Bruynooghe, M., Catthoor, F., Janssens, G.: An automatic verification technique for loop and data reuse transformations based on geometric modeling of programs. Journal of Universal Computer Science 9(3), 248–269 (2003)Google Scholar
  11. 11.

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Anna Beletska
    • 1
  • Denis Barthou
    • 2
  • Wlodzimierz Bielecki
    • 3
  • Albert Cohen
    • 1
  1. 1.INRIA SaclayFrance
  2. 2.University of Versailles St. QuentinFrance
  3. 3.Technical University of SzczecinPoland

Personalised recommendations