## Abstract

Let A be a finite set and B an arbitrary set with at least two elements. The arity gap of a function f : A^{n} → 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.

Original language | British English |
---|---|

Pages (from-to) | 193-207 |

Number of pages | 15 |

Journal | Journal of Multiple-Valued Logic and Soft Computing |

Volume | 27 |

Issue number | 2-3 |

State | Published - 2016 |