Matrix Graph Grammars with Application Conditions

TL;DR

This paper enhances the matrix graph grammar framework by explicitly representing negative application conditions with a nihilation matrix and integrating complex conditions using monadic second order logic, enabling more flexible and concise graph transformation analysis.

Contribution

It introduces a nihilation matrix for negative conditions and a new application condition formalism combining graph diagrams with monadic second order logic, embedding these into rules.

Findings

Nihilation matrix explicitly encodes forbidden elements.

Application conditions can be embedded into rules.

Existing sequence analysis techniques apply to new conditions.

Abstract

In the Matrix approach to graph transformation we represent simple digraphs and rules with Boolean matrices and vectors, and the rewriting is expressed using Boolean operators only. In previous works, we developed analysis techniques enabling the study of the applicability of rule sequences, their independence, state reachability and the minimal graph able to fire a sequence. In the present paper we improve our framework in two ways. First, we make explicit (in the form of a Boolean matrix) some negative implicit information in rules. This matrix (called nihilation matrix) contains the elements that, if present, forbid the application of the rule (i.e. potential dangling edges, or newly added edges, which cannot be already present in the simple digraph). Second, we introduce a novel notion of application condition, which combines graph diagrams together with monadic second order logic. This allows for more flexibility and expressivity than previous approaches, as well as more concise conditions in certain cases. We demonstrate that these application conditions can be embedded into rules (i.e. in the left hand side and the nihilation matrix), and show that the applicability of a rule with arbitrary application conditions is equivalent to the applicability of a sequence of plain rules without application conditions. Therefore, the analysis of the former is equivalent to the analysis of the latter, showing that in our framework no additional results are needed for the study of application conditions. Moreover, all analysis techniques of [21, 22] for the study of sequences can be applied to application conditions.