On the arity gap of finite functions: results and applications

Journal of Multiple-Valued Logic and Soft Computing, 27 (2016), 193–207
http://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-27-number-2-3-2016/mvlsc-27-2-3-p-193-207/

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.