Proceedings of the First International Conference on Relations, Orders and Graphs: Interaction with Computer Science (ROGICS '08)
Boudabbous, Youssef
Zaguia, Nejib
Nouha Editions
65-72
Yes
No
International
978-0-9809498-0-3
Sfax
Tunisia
First International Conference on Relations, Orders and Graphs: Interaction with Computer Science (ROGICS '08)
12-17 May 2008
Mahdia
Tunisia
[en] finite function ; Boolean function ; variable substitution ; essential variable ; arity gap
[en] Let A be a finite set and B an arbitrary set with at least two elements. The arity gap of a function f:Aⁿ→B is the minimum decrease in the number of essential variables when essential variables of f are identified. A non-trivial fact is that the arity gap of such B-valued functions on A is at most |A|. Even less trivial to verify is the fact that the arity gap of B-valued functions on A with more than |A| essential variables is at most 2. These facts ask for a classification of B-valued functions on A in terms of their arity gap. In this paper, we survey what is known about this problem. We present a general characterization of the arity gap of B-valued functions on A and provide explicit classifications of the arity gap of Boolean and pseudo-Boolean functions. Moreover, we reveal unsettled questions related to this topic, and discuss links and possible applications of some results to other subjects of research.
[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
