Abstract
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.
Highlights
The study of switching functions goes back at least to Dedekind’s 1897 work [9], in which he determined the exact number of simple games with four or fewer players
This paper looks at the characterization of threshold functions within the class of switching functions
The new results presented in this paper have been exposed in the simple game terminology since some significant advances have been held in this area in the last two decades
Summary
The study of switching functions goes back at least to Dedekind’s 1897 work [9], in which he determined the exact number of simple games with four or fewer players. One of the most fundamental questions in all of the above mentioned areas is to characterize which monotonic switching functions (simple games) are weighted threshold functions (weighted games) In threshold logic this is known as the linear separability problem. In their book [63] they adapted, for simple games, the most important results of threshold logic in relation to the linear separability problem. Their property of trade-robustness is equivalent to the property of asummability. We treat the extreme case of dimension 1, i.e., we consider the relavant issue whether a given simple game (switching function) is weighted and propose some new characterizations.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
