Graph-based semantic representations are popular in natural language processing, where it is often convenient to model linguistic concepts as nodes and relations as edges between them. Several attempts have been made to find a generative device that is sufficiently powerful to describe languages of semantic graphs, while at the same allowing efficient parsing. We contribute to this line of work by introducing graph extension grammar, a variant of the contextual hyperedge replacement grammars proposed by Hoffmann et al. Contextual hyperedge replacement can generate graphs with non-structural reentrancies, a type of node-sharing that is very common in formalisms such as abstract meaning representation, but that context-free types of graph grammars cannot model. To provide our formalism with a way to place reentrancies in a linguistically meaningful way, we endow rules with logical formulas in counting monadic second-order logic. We then present a parsing algorithm and show as our main result that this algorithm runs in polynomial time on graph languages generated by a subclass of our grammars, the so-called local graph extension grammars.

We show that a previously proposed algorithm for the N-best trees problem can be made more efficient by changing how it arranges and explores the search space. Given an integer N and a weighted tree automaton (wta) M over the tropical semiring, the algorithm computes N trees of minimal weight with respect to M. Compared with the original algorithm, the modifications increase the laziness of the evaluation strategy, which makes the new algorithm asymptotically more efficient than its predecessor. The algorithm is implemented in the software Betty, and compared to the state-of-the-art algorithm for extracting the N best runs, implemented in the software toolkit Tiburon. The data sets used in the experiments are wtas resulting from real-world natural language processing tasks, as well as artificially created wtas with varying degrees of nondeterminism. We find that Betty outperforms Tiburon on all tested data sets with respect to running time, while Tiburon seems to be the more memory-efficient choice.