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.