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Motivated by applications in natural language processing, we study the uniform membership problem for hyperedge-replacement grammars that generate directed acyclic graphs. Our major result is a low-degree polynomial-time algorithm that solves the uniform membership problem for a restricted type of such grammars. We motivate the necessity of the restrictions by two different NP-completeness results.

Order-Preserving DAG Grammars (OPDGs) is a subclass of Hyper-Edge Replacement Grammars that can be parsed in polynomial time. Their associated class of languages is known as Ordered DAG Lan- guages, and the graphs they generate are characterised by being acyclic, rooted, and having a natural order on their nodes. OPDGs are useful in natural-language processing to model abstract meaning representa- tions. We state and prove a Myhill-Nerode theorem for ordered DAG languages, and translate it into a MAT-learning algorithm for the same class. The algorithm infers a minimal OPDG G for the target language in time polynomial in G and the samples provided by the MAT oracle. 

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.

It is well known that hyperedge-replacement grammars can generate NP-complete graph languages even under seemingly harsh restrictions. This means that the parsing problem is difficult even in the non-uniform setting, in which the grammar is considered to be fixed rather than being part of the input. Little is known about restrictions under which truly uniform polynomial parsing is possible. In this paper we propose a low-degree polynomial-time algorithm that solves the uniform parsing problem for a restricted type of hyperedge-replacement grammars which we expect to be of interest for practical applications.

We introduce a weighted extension of the recently proposed notion of order-preserving hyperedge-replacement grammars and prove that the weight of a graph according to such a weighted graph grammar can be computed uniformly in quadratic time (under assumptions made precise in the paper).