Projection and unification for conceptual graphs

Abstract

In this paper we will investigate subsumption and unification for structured descripitons by considering conceptual graphs with their projection and (maximal) join. The importance of projection for conceptual graphs is well-known as it essentially defines a partial order (subsumption hierarchy) on the graphs, that allows one to speed up search considerably. We investigate the complexity of projection by introducing a weaker notion of structural similarity, polyprojection. We prove that a polyprojection implies a projection for so-called non-repeating conceptual graphs. Furthermore, we show that a polyprojection can be determined by a polynomial algorithm. Indeed, the algorithm presented generalizes well-known algorithms for subtree isomorphism, and subsumption between feature term graphs.

A maximal join is defined as the join on a maximally extended compatible projection. The operation is closely related to unification in feature logics and logic programming, but it is allows more flexibility of representation. In essence, a maximal join corresponds to the greatest lower bound of two conceptual graphs when the partial order due to projection is a lattice. Finally, unification of structured descriptions as maximal join is shown to be polynomially related to projection.