Link inheritance in abstract clause graphs

Abstract

Clause graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form: the nodes are labelled with literals and the edges (links) connect complementary unifiable literals, i.e. they provide an explicit representation of the resolution possibilities. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are ‘unifiable’ relative to a given relation between literals. This relation is not explicitely defined, only certain abstract properties are required. For instance the existence of a special purpose unification algorithm is assumed, which computes substitutions, the application of which makes the relation hold for two literals. When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. The definition of abstract clause graphs and the theory about link inheritance is general enough to provide a framework so that as new ideas are proposed for graph based theorem provers, the methodology for both implementing links and proving their properties will be readily available.