Paper published in a book (Scientific congresses, symposiums and conference proceedings)
On the arity gap of finite functions: results and applications
COUCEIRO, Miguel; LEHTONEN, Erkko
2008 • In Boudabbous, Youssef; Zaguia, Nejib (Eds.) Proceedings of the First International Conference on Relations, Orders and Graphs: Interaction with Computer Science (ROGICS '08)
finite function; Boolean function; variable substitution; essential variable; arity gap
Abstract :
[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.
Disciplines :
Mathematics
Identifiers :
UNILU:UL-CONFERENCE-2010-167
Author, co-author :
COUCEIRO, Miguel ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit