Matrix Graph Grammars and Monotone Complex Logics

TL;DR

This paper introduces Monotone Complex Logic to connect Matrix Graph Grammars with complex analysis, enabling geometric and analytic interpretations of graph transformation rules and exploring their relation to fractal structures like the Sierpinski gasket.

Contribution

It generalizes Matrix Graph Grammars by establishing a link with complex analysis through Monotone Complex Logic, facilitating geometric and static analysis of graph transformations.

Findings

MGGs can be interpreted as complex numbers.

The subset of MGG rules forms the Sierpinski gasket.

New geometric and analytic tools for graph transformation analysis.

Abstract

Graph transformation is concerned with the manipulation of graphs by means of rules. Graph grammars have been traditionally studied using techniques from category theory. In previous works, we introduced Matrix Graph Grammars (MGGs) as a purely algebraic approach for the study of graph grammars and graph dynamics, based on the representation of graphs by means of their adjacency matrices. MGGs have been succesfully applied to problems such as applicability of rule sequences, sequentialization and reachability, providing new analysis techniques and generalizing and improving previous results. Our next objective is to generalize MGGs in order to approach computational complexity theory and "static" properties of graphs out of the "dynamics" of certain grammars. In the present work, we start building bridges between MGGs and complexity by introducing what we call "Monotone Complex Logic", which allows establishing a (bijective) link between MGGs and complex analysis. We use this logic to recast the formulation and basic building blocks of MGGs as more proper geometric and analytic concepts (scalar products, norms, distances). MGG rules can also be interpreted - via operators - as complex numbers. Interestingly, the subset they define can be characterized as the Sierpinski gasket.