On outward and inward productions in the categorical graph-grammar approach and Δ-grammars

Abstract

We consider the relationship between three ways of defining graph derivability. That the traditional double-pushout approach and Banach's inward version are equivalent in the case of injective left-hand sides is proved in a purely categorical setting. In the case of noninjective left-hand sides, equivalence can be shown in special categories if the right-hand side is injective. Both approaches have the same generative power in the category of graphs if the pushout connecting the outward production with the inward one is a pullback as well. Finally, it is shown that Banach's point of view establishes a close relationship between the categorical approach and Kaplan's Δ-grammars, allowing a slight generalization of Δ-grammars and making them an operational description of the categorical approach.