Order-preserving DAG grammars (OPDGs) is a formalism for processing semantic infor- mation in natural languages [5, 4]. OPDGs are sufficiently expressive to model abstract meaning representations, a graph-based form of semantic representation in which nodes en- code objects and edges relations. At the same time, they allow for efficient parsing in the uniform setting, where both the grammar and subject graph are taken as part of the input.

In this article, we introduce an initial algebra semantic for OPDGs, which allows us to view them as regular tree grammars. This makes it possible to transfer a number of results from that domain to OPDGs, both in the unweighted and the weighted case. In particular, we show that deterministic OPDGs can be minimised efficiently, and that they are learnable in the so-called MAT setting. To conclude, we show that the languages generated by OPDGs are MSO-definable.