Characterization of Threshold Functions : State of the Art, Some New Contributions and Open Problems

The study of the characterization of threshold functions within the class of switching functions is an important problem that goes back at least to the mid-20th century. Due to different motivations switching and threshold functions have been investigated in a variety of different mathematical contexts: Boolean or switching functions, neural networks, hypergraphs, coherent structures, Sperner families, clutters, secret sharing and simple games or binary voting systems. The paper revises the state of the art about this significant problem and proposes some new contributions concerning asummability and invariant asummability, a refinement of asummability. It also includes several questions and conjectures for future research whose solution would mean a new breakthrough