Thick Separators

  • Luc JaulinEmail author
  • Benoit Desrochers
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 276)


If an interval of \(\mathbb {R}\) is an uncertain real number, a thick set is an uncertain subset of \(\mathbb {R^{\text {n}}}\). More precisely, a thick set is an interval of the powerset of \(\mathbb {R}^{n}\) equipped with the inclusion \(\subset \) as an order relation. It can generally be defined by parameters or functions which are not known exactly, but are known to belong to some intervals. In this paper, we show how to use constraint propagation methods in order to compute efficiently an inner and an outer approximations of a thick set. The resulting inner/outer contraction are made using an operator which is called a thick separator. Then, we show how thick separators can be combined in order to compute with thick sets.


  1. 1.
    Brefort, Q., Jaulin, L., Ceberio, M., Kreinovich, V.: If we take into account that constraints are soft, then processing constraints becomes algorithmically solvable. In: Proceedings of the IEEE Series of Symposia on Computational Intelligence SSCI’2014. Orlando, Florida, 9–12 Dec 2014Google Scholar
  2. 2.
    Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Desrochers, B., Jaulin, L.: Computing a guaranteed approximation the zone explored by a robot. IEEE Trans. Autom. Control (2016)Google Scholar
  4. 4.
    Desrochers, B., Lacroix, S., Jaulin, L.: Set-membership approach to the kidnapped robot problem. In: IROS 2015 (2015)Google Scholar
  5. 5.
    Goldsztejn, A., Chabert, G.: On the approximation of linear ae-solution sets. In: 12th International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Duisburg, Germany, (SCAN 2006) (2006)Google Scholar
  6. 6.
    Jaulin, L., Desrochers, B.: Robust localisation using separators. In: COPROD 2014 (2014)Google Scholar
  7. 7.
    Jaulin, L., Desrochers, B.: Introduction to the algebra of separators with application to path planning. Eng. Appl. Artif. Intell. 33, 141–147 (2014)CrossRefGoogle Scholar
  8. 8.
    Kreinovich, V., Shary, S.: Interval methods for data fitting under uncertainty: a probabilistic treatment. Reliab. Comput. (2016)Google Scholar
  9. 9.
    Schvarcz Franco, G., Jaulin, L.: How to avoid fake boundaries in contractor programming. In: SWIM’16 (2016)Google Scholar
  10. 10.
    Shary, S.: On optimal solution of interval linear equations. SIAM J. Numer. Anal. 32(2), 610–630 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shary, S.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput. 8, 321–418 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Lab-STICCENSTA BretagneBrestFrance

Personalised recommendations