  This paper introduces a framework for formally establishing a connection
between a portion of an algebraic language and a Graph Neural Network (GNN).
The framework leverages Context-Free Grammars (CFG) to organize algebraic
operations into generative rules that can be translated into a GNN layer model.
As CFGs derived directly from a language tend to contain redundancies in their
rules and variables, we present a grammar reduction scheme. By applying this
strategy, we define a CFG that conforms to the third-order Weisfeiler-Lehman
(3-WL) test using MATLANG. From this 3-WL CFG, we derive a GNN model, named
G$^2$N$^2$, which is provably 3-WL compliant. Through various experiments, we
demonstrate the superior efficiency of G$^2$N$^2$ compared to other 3-WL GNNs
across numerous downstream tasks. Specifically, one experiment highlights the
benefits of grammar reduction within our framework.
