On Minimum Elementary-Triplet Bases for Independence Relations

Abstract

A semi-graphoid independence relation is a set of independence statements, called triplets, and is typically exponentially large in the number of variables involved. For concise representation of such a relation, a subset of its triplets is listed in a so-called basis; its other triplets are defined implicitly through a set of axioms. An elementary-triplet basis for this purpose consists of all elementary triplets of a relation. Such a basis however, may include redundant information. In this paper we provide two lower bounds on the size of an elementary-triplet basis in general and an upper bound on the size of a minimum elementary-triplet basis. We further specify the construction of an elementary-triplet basis of minimum size for restricted relations.