Journal of the ACM (JACM)
WMP '00 Proceedings of the Workshop on Multiset Processing: Multiset Processing, Mathematical, Computer Science, and Molecular Computing Points of View
A canonical form for piecewise defined functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Reasoning with Generic Cases in the Arithmetic of Abstract Matrices
Calculemus '09/MKM '09 Proceedings of the 16th Symposium, 8th International Conference. Held as Part of CICM '09 on Intelligent Computer Mathematics
Computing with abstract matrix structures
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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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.