Decompositions of set functions

Pivotal decompositions of functions

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary section of a function only depends upon its values at two given elements. Pivotal decompositions appear to hold for various function classes, such as the class of lattice polynomial functions or the...

On decompositions of multivariate functions

We present formulas that allow us to decompose a function f of d variables into a sum of 2d terms fu indexed by subsets u of {1, . . . , d}, where each term fu depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {Pj}j=1, where Pj(f) does not depend on the variable xj . We present an explicit formula for fu, which is new even for th...

Decompositions of Substochastic Transition Functions

Three decompositions of a substochastic transition function are shown to yield substochastic parts. These are the Lebesgue decomposition with respect to a finite measure, the decomposition into completely atomic and continuous parts, and on Rn, a decomposition giving a part with continuous distribution function and a part with discontinuous distribution function. Introduction. The present paper...

Width Functions for Hypertree Decompositions

World-Set Decompositions: Expressiveness and Efficient Algorithms

Uncertain information is commonplace in real-world data management scenarios. The ability to represent large sets of possible instances (worlds) while supporting efficient storage and processing is an important challenge in this context. The recent formalism of worldset decompositions (WSDs) provides a space-efficient representation for uncertain data that also supports scalable processing. WSD...