Toward a complete representation of graphoids in graphs — Abridged Version

Abstract

We consider 3 place relations I(X,Z,Y) where X, Y and Z are sets of propositional variables and I(X,Z,Y) stands for the statement "Knowing Z renders X independent of Y". These relations are called graphoids. The theory of graphoids uncovers the axiomatic basis of informational dependencies and ties it to vertex separation in graphs. In this paper we advance towards a characterization of graphoids by families of graphs. Given two graphs R and B, let M be the set of all independencies which are implied by R and B under closure by the 5 graphoid axioms (defined in the text). We show the following results:

• 1 An algorithm which generates a family of graphs that represent M.

• 2 The number of graphs needed to represent M might be exponential.

• 3 A polynomial time algorithm for the following problem: given an independency t is t ∈ M?

• 4 We define annotated graphs and show an efficient representation of M by this model.

The results shown in this report are part of the MSc Thesis of the first author done under the supervision of the second author — to be presented to Graduate School of the Technion, Israel Institute of Technology. Contribution of the second author supported by the fundation for the promotion of research at the Technion.