Symbolic Domain Decomposition

  • Jacques Carette
  • Alan P. Sexton
  • Volker Sorge
  • Stephen M. Watt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)


Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity.


Domain Decomposition Hybrid Function Integer Matrix Piecewise Function Generalise Partition 
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. 1.
    Blizard, W.D.: Multiset theory. Notre Dame Journal of Formal Logic 30(1), 36–66 (Winter 1989)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blizard, W.D.: Negative membership. Notre Dame Journal of Formal Logic 31(3), 346–368 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boole, G.: An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. Walton and Maberly, London (1854), Google Scholar
  4. 4.
    Burgin, M.S.: Concept of multisets in cybernetics. Cybernetics and Systems Analysis 28(3), 371–469 (1992)CrossRefGoogle Scholar
  5. 5.
    Carette, J.: A canonical form for piecewise defined functions. In: Proc. of ISSAC 2007, pp. 77–84. ACM Press, New York (2007)CrossRefGoogle Scholar
  6. 6.
    Hailperin, T.: Boole’s Logic and Probability, 1st/2nd edn. North-Holland Publishing Company, Amsterdam (1976)zbMATHGoogle Scholar
  7. 7.
    Kahl, W.: Compositional syntax and semantics of tables. SQRL Report 15, Department of Computing and Software, McMaster University (2003),
  8. 8.
    Karr, M.: Summation in finite terms. J. ACM 28(2), 305–350 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Loeb, D.: Sets with a negative number of elements. Advances in Mathematics 91(1), 64–74 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sexton, A.P., Sorge, V., Watt, S.M.: Computing with abstract matrix structures. In: Proc. of ISSAC 2009, pp. 325–332. ACM Press, New York (2009)CrossRefGoogle Scholar
  11. 11.
    Sexton, A.P., Sorge, V., Watt, S.M.: Reasoning with generic cases in the arithmetic of abstract matrices. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) Calculemus 2009. LNCS (LNAI), vol. 5625, pp. 138–153. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Syropoulos, A.: Mathematics of multisets. In: Calude, C.S., Pun, G., Rozenberg, G., Salomaa, A. (eds.) Multiset Processing. LNCS, vol. 2235, pp. 347–358. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Whitney, H.: A logical expansion in mathematics. Bulletin of the American Mathematical Society 34(8), 572–579 (1932)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Whitney, H.: Characteristic functions and the algebra of logic. Annals of Mathematics 34(3), 405–414 (1933), CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jacques Carette
    • 1
  • Alan P. Sexton
    • 2
  • Volker Sorge
    • 2
  • Stephen M. Watt
    • 3
  1. 1.Department of Computing and SoftwareMcMaster University 
  2. 2.School of Computer ScienceUniversity of Birmingham 
  3. 3.Department of Computer ScienceUniversity of Western Ontario 

Personalised recommendations