For those not in NLP:
L-BFGS is a convex optimizer designed for large-scale optimization problems
NIPS (Neural Information Processing Systems) is an annual machine learning conference
A logical form is a symbolic representation of a sentence, such as those in Figure 4 of this paper
Beam search is just a modification to breadth-first search or some similar algorithm where you only partially expand each state (due to time / memory constraints). In this context, we are building up parse trees for sentences and we only keep track of the 30 best subtrees for each sub-span of the sentence (as opposed to all of the subtrees, which would be an exponentially large set in the size of the sentence).