Dualisation, decision lists and identification of monotone discrete functions

Many data-analysis algorithms in machine learning, datamining and a variety of other disciplines essentially operate on discrete multi-attribute data sets. By means of discretisation or binarisation also numerical data sets can be successfully analysed. Therefore, in this paper we view/introduce the theory of (partially defined) discrete functions as an important theoretical tool for the analysis of multi-attribute data sets. In particular we study monotone (partially defined) discrete functions. Compared with the theory of Boolean functions relatively little is known about (partially defined) monotone discrete functions. It appears that decision lists are useful for the representation of monotone discrete functions. Since dualisation is an important tool in the theory of (monotone) Boolean functions, we study the interpretation and properties of the dual of a (monotone) binary or discrete function. We also introduce the dual of a pseudo-Boolean function. The results are used to investigate extensions of partially defined monotone discrete functions and the identification of monotone discrete functions. In particular we present a polynomial time algorithm for the identification of so-called stable discrete functions.