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On the arity gap of finite functions : results and applications.

Miguel Couceiro 1 Erkko Lehtonen 2
1 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
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.
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Submitted on : Saturday, July 11, 2015 - 5:49:04 PM
Last modification on : Tuesday, October 13, 2020 - 11:14:02 AM


  • HAL Id : hal-01175695, version 1



Miguel Couceiro, Erkko Lehtonen. On the arity gap of finite functions : results and applications.. Journal of Multiple-Valued Logic and Soft Computing, Old City Publishing, 2016, 27 (2-3), pp.15. ⟨hal-01175695⟩



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