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

Miguel Couceiro (1, 2) , Erkko Lehtonen (3) , Karsten Schölzel (4)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
2 ORPAILLEUR - Knowledge representation, reasonning
3 CSC - Computer Science and Communications Research Unit [Luxembourg]
4 Mathematics Research Unit

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.

Domaines: Mathématiques [math]; Informatique [cs]

Miguel Couceiro, Erkko Lehtonen, Karsten Schölzel. A complete classification of equational classes of threshold functions included in clones. RAIRO - Operations Research, 2015, 49 (1), pp.39-66. ⟨10.1051/ro/2014034⟩. ⟨hal-01090621⟩