A complete classification of equational classes of threshold functions included in clones

The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investig...

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Bibliographic Details
Published inarXiv.org
Main Authors Couceiro, Miguel, Lehtonen, Erkko, Schölzel, Karsten
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.10.2013
Subjects
Boolean algebra
Boolean functions
Classification
Functional equations
Intersections
Mathematics - Combinatorics
Mathematics - Rings and Algebras
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Summary:The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as intersections of the class of all threshold functions with clones of Boolean functions, and provide a complete classification of such intersections in respect to whether they have finite characterizations. Moreover, we provide a characterizing set of relational constraints for each class of threshold functions arising in this way.
ISSN:2331-8422
DOI:10.48550/arxiv.1310.7041