1 Introduction
Over the past few years, Abstract Meaning Representations (AMRs, amBanarescuBCGGHKKPS13) have become a popular target representation for semantic parsing. AMRs are graphs which describe the predicateargument structure of a sentence. Because they are graphs and not trees, they can capture reentrant semantic relations, such as those induced by control verbs and coordination. However, it is technically much more challenging to parse a string into a graph than into a tree. For instance, grammarbased approaches Peng et al. (2015); Artzi et al. (2015) require the induction of a grammar from the training corpus, which is hard because graphs can be decomposed into smaller pieces in far more ways than trees. Neural sequencetosequence models, which do very well on stringtotree parsing Vinyals et al. (2014), can be applied to AMRs but face the challenge that graphs cannot easily be represented as sequences van Noord and Bos (2017a, b).
In this paper, we tackle this challenge by making the compositional structure of the AMR explicit. As in our previous work, graphalgebra17, we view an AMR as consisting of atomic graphs representing the meanings of the individual words, which were combined compositionally using linguistically motivated operations for combining a head with its arguments and modifiers. We represent this structure as terms over the AM algebra as defined in graphalgebra17. This previous work had no parser; here we show that the terms of the AM algebra can be viewed as dependency trees over the string, and we train a dependency parser to map strings into such trees, which we then evaluate into AMRs in a postprocessing step. The dependency parser relies on type information, which encodes the semantic valencies of the atomic graphs, to guide its decisions.
More specifically, we combine a neural supertagger for identifying the elementary graphs for the individual words with a neural dependency model along the lines of kiperwasser16:_simpl_accur_depen_parsin_using for identifying the operations of the algebra. One key challenge is that the resulting term of the AM algebra must be semantically welltyped. This makes the decoding problem NPcomplete. We present two approximation algorithms: one which takes the unlabeled dependency tree as given, and one which assumes that all dependencies are projective. We evaluate on two data sets, achieving stateoftheart results on one and near stateoftheart results on the other (Smatch fscores of 71.0 and 70.2 respectively). Our approach clearly outperforms strong but noncompositional baselines.
2 Related Work
Recently, AMR parsing has generated considerable research activity, due to the availability of largescale annotated data Banarescu et al. (2013) and two successful SemEval Challenges May (2016); May and Priyadarshi (2017).
Methods from dependency parsing have been shown to be very successful for AMR parsing. For instance, the JAMR parser Flanigan et al. (2014, 2016) distinguishes concept identification (assigning graph fragments to words) from relation identification (adding graph edges which connect these fragments), and solves the former with a supertaggingstyle method and the latter with a graphbased dependency parser. foland2017abstract use a variant of this method based on an intricate neural model, yielding stateoftheart results. We go beyond these approaches by explicitly modeling the compositional structure of the AMR, which allows the dependency parser to combine AMRs for the words using a small set of universal operations, guided by the types of these AMRs.
Other recent methods directly implement a dependency parser for AMRs, e.g. the transitionbased model of E171051, or postprocess the output of a dependency parser by adding missing edges Du et al. (2014); Wang et al. (2015). In contrast to these, our model makes no strong assumptions on the dependency parsing algorithm that is used; here we choose that of kiperwasser16:_simpl_accur_depen_parsin_using.
The commitment of our parser to derive AMRs compositionally mirrors that of grammarbased AMR parsers Artzi et al. (2015); Peng et al. (2015). In particular, there are parallels between the types we use in the AM algebra and CCG categories (see Section 3 for details). As a neural system, our parser struggles less with coverage issues than these, and avoids the complex grammar induction process these models require.
More generally, our use of semantic types to restrict our parser is reminiscent of kwiatkowski2010inducing, krishnamurthy2017neural and zhang2017macro, and the idea of deriving semantic representations from dependency trees is also present in reddy2017universal.
3 The AM algebra
A core idea of this paper is to parse a string into a graph by instead parsing a string into a dependencystyle tree representation of the graph’s compositional structure, represented as terms of the ApplyModify (AM) algebra Groschwitz et al. (2017).
The values of the AM algebra are annotated sgraphs, or asgraphs: directed graphs with node and edge labels in which certain nodes have been designated as sources Courcelle and Engelfriet (2012) and annotated with type information. Some examples of asgraphs are shown in Fig. 1. Each asgraph has exactly one root, indicated by the bold outline. The sources are indicated by red labels; for instance, has an ssource and an osource. The annotations, written in square brackets behind the red source names, will be explained below. We use these sources to mark open argument slots; for example, in Fig. 1 represents an intransitive verb, missing its subject, which will be added at the ssource.
The AM algebra can combine asgraphs with each other using two linguistically motivated operations: apply and modify. Apply (App) adds an argument to a predicate. For example, we can add a subject – the graph in Fig. 1 – to the graph in Fig. 2d using App, yielding the complete AMR in Fig. 2b. Linguistically, this is like filling the subject (s) slot of the predicate wants to sleep soundly with the argument the writer. In general, for a source , , combines the asgraph representing a predicate, or head, with the asgraph , which represents an argument. It does this by plugging the root node of into the source of – that is, the node of marked with source . The root of the resulting asgraph is the root of , and we remove the marking on , since that slot is now filled.
The modify operation (Mod) adds a modifier to a graph. For example, we can combine two elementary graphs from Fig. 1 with Mod (, ), yielding the graph in Fig. 2c. The msource of the modifier attaches to the root of . The root of the result is the same as the root of in the same sense that a verb phrase with an adverb modifier is still a verb phrase. In general, , combines a head with a modifier . It plugs the root of into the source of . Although this may add incoming edges to the root of , that node is still the root of the resulting graph . We remove the marking from .
In both App and Mod, if there is any other source which is present in both graphs, the nodes marked with are unified with each other. For example, when is oapplied to in Fig. 2d, the ssources of the graphs for “want” and “sleep soundly” are unified into a single node, creating a reentrancy. This falls out of the definition of merge for sgraphs which formally underlies both operations (see Courcelle and Engelfriet (2012)).
Finally, the AM algebra uses types to restrict its operations. Here we define the type of an asgraph as the set of its sources with their annotations^{1}^{1}1See Groschwitz et al. (2017) for a more formally complete definition.; thus for example, in Fig. 1, the graph for “writer” has the empty type , has type , and has type . Each source in an asgraph specifies with its annotation the type of the asgraph which is plugged into it via App. In other words, for a source , we may only apply with if the annotation of the source in matches the type of . For example, the osource of (Fig. 1) requires that we plug in an asgraph of type ; observe that this means that the reentrancy in Fig. 2b is lexically specified by the control verb “want”. All other source nodes in Fig. 1 have no annotation, indicating a type requirement of .
Linguistically, modification is optional; we therefore want the modified graph to be derivationally just like the unmodified graph, in that exactly the same operations can apply to it. In a typed algebra, this means Mod should not change the type of the head. Mod therefore requires that the modifier have no sources not already present in the head , except , which will be deleted anyway.
As in any algebra, we can build terms from constants (denoting elementary asgraphs) by recursively combining them with the operations of the AM algebra. By evaluating the operations bottomup, we obtain an asgraph as the value of such a term; see Fig. 2 for an example. However, as discussed above, an operation in the term may be undefined due to a type mismatch. We call an AMterm welltyped if all its operations are defined. Every welltyped AMterm evaluates to an asgraph. Since the applicability of an AM operation depends only on the types, we also write if asgraphs of type and can be combined with the operation f and the result has type .
Relationship to CCG.
There is close relationship between the types of the AM algebra and the categories of CCG. A type specifies that the asgraph needs to be applied to two arguments to be semantically complete, similar a CCG category such as , where a string needs to be applied to two NP arguments to be syntactically complete. However, AM types govern the combination of graphs, while CCG categories control the combination of strings. This relieves AM types of the need to talk about word order; there are no “forward” or “backward” slashes in AM types, and a smaller set of operations. Also, the AM algebra spells out raising and control phenomena more explicitly in the types.
4 Indexed AM terms
In this paper, we connect AM terms to the input string for which we want to produce a graph. We do this in an indexed AM term, exemplified in Fig. 3a. We assume that every elementary asgraph at a leaf represents the meaning of an individual word token in , and write to annotate the leaf with the index of this token. This induces a connection between the nodes of the AMR and the tokens of the string, in that the label of each node was contributed by the elementary asgraph of exactly one token.
We define the head index of a subtree to be the index of the token which contributed the root of the asgraph to which evaluates. For a leaf with annotation , the head index is ; for an App or Mod node, the head index is the head index of the left child, i.e. of the head argument. We annotate each App and Mod operation with the head index of the left and right subtree.
4.1 AM dependency trees
We can represent indexed AM terms more compactly as AM dependency trees, as shown in Fig. 3b. The nodes of such a dependency tree are the tokens of . We draw an edge with label f from to if there is a node with label in the indexed AM term. For example, the tree in 3b has an edge labeled Mod from 5 () to 6 () because there is a node in the term in 3a labeled . The same AM dependency tree may represent multiple indexed AM terms, because the order of apply and modify operations is not specified in the dependency tree. However, it can be shown that all welltyped AM terms that map to the same AM dependency tree evaluate to the same asgraph. We define a welltyped AM dependency tree as one that represents a welltyped AM term.
Because not all words in the sentence contribute to the AMR, we include a mechanism for ignoring words in the input. As a special case, we allow the constant , which represents a dummy asgraph (of type ) which we use as the semantic value of words without a semantic value in the AMR. We furthermore allow the edge label ignore in an AM dependency tree, where if and is undefined otherwise; in particular, an AM dependency tree with ignore edges is only welltyped if all ignore edges point into nodes. We keep all other operations as is, i.e. they are undefined if either or is , and never yield as a result. When reconstructing an AM term from the AM dependency tree, we skip ignore edges, such that the subtree below them will not contribute to the overall AMR.
4.2 Converting AMRs to AM terms
In order to train a model that parses sentences into AM dependency trees, we need to convert an AMR corpus – in which sentences are annotated with AMRs – into a treebank of AM dependency trees. We do this in three steps: first, we break each AMR up into elementary graphs and identify their roots; second, we assign sources and annotations to make elementary asgraphs out of them; and third, combine them into indexed AM terms.
For the first step, an aligner uses handwritten heuristics to identify the string token to which each node in the AMR corresponds (see Section C in the Supplementary Materials for details). We proceed in a similar fashion as the JAMR aligner
Flanigan et al. (2014), i.e. by starting from highconfidence tokennode pairs and then extending them until the whole AMR is covered. Unlike the JAMR aligner, our heuristics ensure that exactly one node in each elementary graph is marked as the root, i.e. as the node where other graphs can attach their edges through App and Mod. When an edge connects nodes of two different elementary graphs, we use the “blob decomposition” algorithm of graphalgebra17 to decide to which elementary graph it belongs. For the example AMR in Fig. 2b, we would obtain the graphs in Fig. 1 (without source annotations). Note that ARG edges belong with the nodes at which they start, whereas the “manner” edge in goes with its target.In the second step we assign source names and annotations to the unlabeled nodes of each elementary graph. Note that the annotations are crucial to our system’s ability to generate graphs with reentrancies. We mostly follow the algorithm of graphalgebra17, which determines necessary annotations based on the structure of the given graph. The algorithm chooses each source name depending on the incoming edge label. For instance, the two leaves of can have the source labels s and o because they have incoming edges labeled ARG0 and ARG1. However, the Groschwitz algorithm is not deterministic: It allows object promotion (the sources for an ARG3 edge may be o3, o2, or o), unaccusative subjects (promoting the minimal object to s if the elementary graph contains an ARGiedge () but no ARG0edge Perlmutter (1978)), and passive alternation (swapping o and s). To make our asgraphs more consistent, we prefer constants that promote objects as far as possible, use unaccusative subjects, and no passive alternation, but still allow constants that do not satisfy these conditions if necessary. This increased our Smatch score significantly.
Finally, we choose an arbitrary AM dependency tree that combines the chosen elementary asgraphs into the annotated AMR; in practice, the differences between the trees seem to be negligible.^{2}^{2}2Indeed, we conjecture that for a fixed set of constants and a fixed AMR, there is only one dependency tree.
5 Training
We can now model the AMR parsing task as the problem of computing the best welltyped AM dependency tree for a given sentence . Because is welltyped, it can be decoded into an (indexed) AM term and thence evaluated to an asgraph.
We describe in terms of the elementary asgraphs it uses for each token and of its edges . We assume a nodefactored, edgefactored model for the score of :
(1) 
where the edge weight further decomposes into the sum of a score for the presence of an edge from to and a score for this edge having label f. Our aim is to compute the welltyped with the highest score.
We present three models for : one for the graph scores and two for the edge scores. All of these are based on a twolayer bidirectional LSTM, which reads inputs
token by token, concatenating the hidden states of the forward and the backward LSTMs in each layer. On the second layer, we thus obtain vector representations
for the individual input tokens (see Fig. 4). Our models differ in the inputs and the way they predict scores from the .5.1 Supertagging for elementary asgraphs
We construe the prediction of the asgraphs for each input position as a supertagging task Lewis et al. (2016). The supertagger reads inputs , where is the word token, its POS tag, and is a characterbased LSTM encoding of . We use pretrained GloVe embeddings Pennington et al. (2014) concatenated with learned embeddings for , and learned embeddings for .
To predict the score for each elementary asgraph out of a set of options, we add a dimensional output layer as follows:
and train the neural network using a crossentropy loss function. This maximizes the likelihood of the elementary asgraphs in the training data.
5.2 Kiperwasser & Goldberg edge model
Predicting the edge scores amounts to a dependency parsing problem. We chose the dependency parser of kiperwasser16:_simpl_accur_depen_parsin_using, henceforth K&G, to learn them, because of its accuracy and its fit with our overall architecture. The K&G parser scores the potential edge from to and its label from the concatenations of and :
We use inputs including the type of the supertag at position , using trained embeddings for all three. At evaluation time, we use the best scoring supertag according to the model of Section 5.1. At training time, we sample from , where , for any and
is a hyperparameter controlling the bias towards the aligned supertag. We train the model using K&G’s original DyNet implementation. Their algorithm uses a hinge loss function, which maximizes the score difference between the gold dependency tree and the best predicted dependency tree, and therefore requires parsing each training instance in each iteration. Because the AM dependency trees are highly nonprojective, we replaced the projective parser used in the offtheshelf implementation by the ChuLiuEdmonds algorithm implemented in the TurboParser
Martins et al. (2010), improving the LAS on the development set by 30 points.5.3 Local edge model
We also trained a local edge score model, which uses a crossentropy rather than a hinge loss and therefore avoids the repeated parsing at training time. Instead, we follow the intuition that every node in a dependency tree has at most one incoming edge, and train the model to score the correct incoming edge as high as possible. This model takes inputs .
We define the edge and edge label scores as in Section 5.2
, with tanh replaced by ReLU. We further add a learned parameter
for the “LSTM embedding” of a nonexistent node, obtaining scores for having no incoming edge.To train , we collect all scores for edges ending at the same node into a vector . We then minimize the crossentropy loss for the gold edge into under , maximizing the likelihood of the gold edges. To train the labels , we simply minimize the crossentropy loss of the actual edge labels f of the edges which are present in the gold AM dependency trees.
The PyTorch code for this and the supertagger are available at
bitbucket.org/tclup/amrdependency.6 Decoding
Given learned estimates for the graph and edge scores, we now tackle the challenge of computing the best welltyped dependency tree
for the input string , under the score model (equation (1)). The requirement that must be welltyped is crucial to ensure that it can be evaluated to an AMR graph, but as we show in the Supplementary Materials (Section A), makes the decoding problem NPcomplete. Thus, an exact algorithm is not practical. In this section, we develop two different approximation algorithms for AM dependency parsing: one which assumes the (unlabeled) dependency tree structure as known, and one which assumes that the AM dependency tree is projective.6.1 Projective decoder
The projective decoder assumes that the AM dependency tree is projective, i.e. has no crossing dependency edges. Because of this assumption, it can recursively combine adjacent substrings using dynamic programming. The algorithm is shown in Fig. 5 as a parsing schema Shieber et al. (1995), which derives items of the form with scores . An item represents a welltyped derivation of the substring from to with head index , and which evaluates to an asgraph of type .
The parsing schema consists of three types of rules. First, the Init rule generates an item for each graph fragment that the supertagger predicted for the token , along with the score and type of that graph fragment. Second, given items for adjacent substrings and , the Arc rules apply an operation f to combine the indexed AM terms for the two substrings, with ArcR making the lefthand substring the head and the righthand substring the argument or modifier, and ArcL the other way around. We ensure that the result is welltyped by requiring that the types can be combined with f. Finally, the Skip rules allow us to extend a substring such that it covers tokens which do not correspond to a graph fragment (i.e., their AM term is ), introducing ignore edges. After all possible items have been derived, we extract the best welltyped tree from the item of the form with the highest score, where .
Because we keep track of the head indices, the projective decoder is a bilexical parsing algorithm, and shares a parsing complexity of with other bilexical algorithms such as the Collins parser. It could be improved to a complexity of using the algorithm of eisner99:_effic.
6.2 Fixedtree decoder
The fixedtree decoder computes the best unlabeled dependency tree for , using the edge scores , and then computes the best AM dependency tree for whose unlabeled version is . The ChuLiuEdmonds algorithm produces a forest of dependency trees, which we want to combine into . We choose the tree whose root has the highest score for being the root of the AM dependency tree and make the roots of all others children of .
At this point, the shape of is fixed. We choose supertags for the nodes and edge labels for the edges by traversing bottomup, computing types for the subtrees as we go along. Formally, we apply the parsing schema in Fig. 6. It uses items of the form , where is a node of , is the set of children of for which we have already chosen edge labels, and is a type. We write for the set of children of in .
The Init rule generates an item for each graph that the supertagger can assign to each token in , ensuring that every token is also assigned as a possible supertag. The Edge rule labels an edge from a parent node in to one of its children , whose children already have edge labels. As above, this rule ensures that a welltyped AM dependency tree is generated by locally checking the types. In particular, if all types that can be derived for are incompatible with , we fall back to an item for with (which always exists), along with an ignore edge from to .
The complexity of this algorithm is , where is the maximal arity of the nodes in .
7 Evaluation
We evaluate our models on the LDC2015E86 and LDC2017T10^{3}^{3}3https://catalog.ldc.upenn.edu/LDC2017T10, identical to LDC2016E25. datasets (henceforth “2015” and “2017”). Technical details and hyperparameters of our implementation can be found in Sections B to D of the Supplementary Materials.
7.1 Training data
The original LDC datasets pair strings with AMRs. We convert each AMR in the training and development set into an AM dependency tree, using the procedure of Section 4.2. About 10% of the training instances cannot be split into elementary asgraphs by our aligner; we removed these from the training data. Of the remaining AM dependency trees, 37% are nonprojective.
Furthermore, the AM algebra is designed to handle shortrange reentrancies, modeling grammatical phenomena such as control and coordination, as in the derivation in Fig. 2. It cannot easily handle the longrange reentrancies in AMRs which are caused by coreference, a noncompositional phenomenon.^{4}^{4}4As E171051 comment: “A valid criticism of AMR is that these two reentrancies are of a completely different type, and should not be collapsed together.” We remove such reentrancies from our training data (about 60% of the roughly 20,000 reentrant edges). Despite this, our model performs well on reentrant edges (see Table 2).
7.2 Pre and postprocessing
We use simple pre and postprocessing steps to handle rare words and some AMRspecific patterns. In AMRs, named entities follow a pattern shown in Fig. 7. Here the named entity is of type “person”, has a name edge to a “name” node whose children spell out the tokens of “Agatha Christie”, and a link to a wiki entry. Before training, we replace each “name” node, its children, and the corresponding span in the sentence with a special NAME token, and we completely remove wiki edges. In this example, this leaves us with only a “person” and a NAME node. Further, we replace numbers and some date patterns with NUMBER and DATE tokens. On the training data this is straightforward, since names and dates are explicitly annotated in the AMR. At evaluation time, we detect dates and numbers with regular expressions, and names with Stanford CoreNLP Manning et al. (2014). We also use Stanford CoreNLP for our POS tags.
Each elementary asgraph generated by the procedure of Section 4.2 has a unique node whose label corresponds most closely to the aligned word (e.g. the “want” node in and the “write” node in ). We replace these node labels with LEX in preprocessing, reducing the number of different elementary asgraphs from 28730 to 2370. We factor the supertagger model of Section 5.1 such that the unlexicalized version of and the label for LEX are predicted separately.
At evaluation, we relexicalize all LEX nodes in the predicted AMR. For words that were frequent in the training data (at least 10 times), we take the supertagger’s prediction for the label. For rarer words, we use simple heuristics, explained in the Supplementary Materials (Section D). For names, we just look up name nodes with their children and wiki entries observed for the name string in the training data, and for unseen names use the literal tokens as the name, and no wiki entry. Similarly, we collect the type for each encountered name (e.g. “person” for “Agatha Christie”), and correct it in the output if the tagger made a different prediction. We recover dates and numbers straightforwardly.
7.3 Supertagger accuracy
All of our models rely on the supertagger to predict elementary asgraphs; they differ only in the edge scores. We evaluated the accuracy of the supertagger on the converted development set (in which each token has a supertag) of the 2015 data set, and achieved an accuracy of 73%. The correct supertag is within the supertagger’s 4 best predictions for 90% of the tokens, and within the 10 best for 95%.
Interestingly, supertags that introduce grammatical reentrancies are predicted quite reliably, although they are relatively rare in the training data. The elementary asgraph for subject control verbs (see in Fig. 1) accounts for only 0.8% of supertags in the training data, yet 58% of its occurrences in the development data are predicted correctly (84% in 4best). The supertag for VP coordination (with type ) makes up for 0.4% of the training data, but 74% of its occurrences are recognized correctly (92% in 4best). Thus the prediction of informative types for individual words is feasible.
7.4 Comparison to Baselines
Typeunaware fixedtree baseline. The fixedtree decoder is built to ensure welltypedness of the predicted AM dependency trees. To investigate to what extent this is required, we consider a baseline which just adds the individually highestscoring supertags and edge labels to the unlabeled dependency tree , ignoring types. This leads to AM dependency trees which are not welltyped for 75% of the sentences (we fall back to the largest welltyped subtree in these cases). Thus, an offtheshelf dependency parser can reliably predict the tree structure of the AM dependency tree, but correct supertag and edge label assignment requires a decoder which takes the types into account.
JAMRstyle baseline. Our elementary asgraphs differ from the elementary graphs used in JAMRstyle algorithms in that they contain explicit source nodes, which restrict the way in which they can be combined with other asgraphs. We investigate the impact of this choice by implementing a strong JAMRstyle baseline. We adapt the AMRtodependency conversion of Section 4.2 by removing all unlabeled nodes with source names from the elementary graphs. For instance, the graph in Fig. 1 now only consists of a single “want” node. We then aim to directly predict AMR edges between these graphs, using a variant of the local edge scoring model of Section 5.3 which learns scores for each edge in isolation. (The assumption for the original local model, that each node has only one incoming edge, does not apply here.)
When parsing a string, we choose the highestscoring supertag for each word; there are only 628 different supertags in this setting, and 1best supertagging accuracy is high at 88%. We then follow the JAMR parsing algorithm by predicting all edges whose score is over a threshold (we found 0.02 to be optimal) and then adding edges until the graph is connected. Because we do not predict which node is the root of the AMR, we evaluated this model as if it always predicted the root correctly, overestimating its score slightly.
Model  2015  2017 
Ours  
local edge + projective decoder 
71.0


local edge + fixedtree decoder  
K&G edge + projective decoder  
K&G edge + fixedtree decoder  
Baselines  
fixedtree (typeunaware)  
JAMRstyle  66.2  
Previous work  
CAMR Wang et al. (2015)  66.5   
JAMR Flanigan et al. (2016)  67   
E171051  64   
van2017neural  68.5  71.0 
foland2017abstract  70.7   
buys2017oxford    61.9 
7.5 Results
Table 1 shows the Smatch scores (Cai and Knight, 2013) of our models, compared to a selection of previously published results. Our results are averages over 4 runs with confidence intervals (JAMRstyle baselines are single runs). On the 2015 dataset, our best models (local + projective, K&G + fixedtree) outperform all previous work, with the exception of the Foland and Martin (2017) model; on the 2017 set we match state of the art results (though note that van2017neural use 100k additional sentences of silver data). The fixedtree decoder seems to work well with either edge model, but performance of the projective decoder drops with the K&G edge scores. It may be that, while the hinge loss used in the K&G edge scoring model is useful to finding the correct unlabeled dependency tree in the fixedtree decoder, scores for bad edges – which are never used when computing the hinge loss – are not trained accurately. Thus such edges may be erroneously used by the projective decoder.
As expected, the typeunaware baseline has low recall, due to its inability to produce welltyped trees. The fact that our models outperform the JAMRstyle baseline so clearly is an indication that they indeed gain some of their accuracy from the type information in the elementary asgraphs, confirming our hypothesis that an explicit model of the compositional structure of the AMR can help the parser learn an accurate model.
Table 2 analyzes the performance of our two best systems (PD = projective, FTD = fixedtree) in more detail, using the categories of E171051, and compares them to Wang’s, Flanigan’s, and Damonte’s AMR parsers on the 2015 set and , and van2017neural for the 2017 dataset. (foland2017abstract did not publish such results.) The good scores we achieve on reentrancy identification, despite removing a large amount of reentrant edges from the training data, indicates that our elementary asgraphs successfully encode phenomena such as control and coordination.
The projective decoder is given 4, and the fixedtree decoder 6, supertags for each token. We trained the supertagging and edge scoring models of Section 5 separately; joint training did not help. Not sampling the supertag types during training of the K&G model, removing them from the input, and removing the characterbased LSTM encodings from the input of the supertagger, all reduced our models’ accuracy.
7.6 Differences between the parsers
Although the Smatch scores for our two best models are close, they sometimes struggle with different sentences. The fixedtree parser is at the mercy of the fixed tree; the projective parser cannot produce nonprojective AM dependency trees. It is remarkable that the projective parser does so well, given the prevalence of nonprojective trees in the training data. Looking at its analyses, we find that it frequently manages to find a projective tree which yields an (almost) correct AMR, by choosing supertags with unusual types, and by using modify rather than apply (or vice versa).
8 Conclusion
We presented an AMR parser which applies methods from supertagging and dependency parsing to map a string into a welltyped AM term, which it then evaluates into an AMR. The AM term represents the compositional semantic structure of the AMR explicitly, allowing us to use standard treebased parsing techniques.
The projective parser currently computes the complete parse chart. In future work, we will speed it up through the use of pruning techniques. We will also look into more principled methods for splitting the AMRs into elementary asgraphs to replace our handcrafted heuristics. In particular, advanced methods for alignments, as in lyu2018amr, seem promising. Overcoming the need for heuristics also seems to be a crucial ingredient for applying our method to other semantic representations.
Acknowledgements
We would like to thank the anonymous reviewers for their comments. We thank Stefan Grünewald for his contribution to our PyTorch implementation, and want to acknowledge the inspiration obtained from nguyen2017novel. We also extend our thanks to the organizers and participants of the Oslo CAS Meaning Construction workshop on Universal Dependencies. This work was supported by the DFG grant KO 2916/21 and a Macquarie University Research Excellence Scholarship for Jonas Groschwitz.
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Appendix A NPcompleteness of the decoding problem
We prove NPcompleteness for the welltyped decoding problem by reduction from HamiltonianPath.
Let be a directed graph with nodes and edges . A Hamiltonian path in is a sequence that contains each node of exactly once, such that for all . We assume w.l.o.g. that . Deciding whether has a Hamiltonian path is NPcomplete.
Given , we construct an instance of the decoding problem for the sentence as follows. We assume that the first graph fragments shown in Fig. 8a (with node label “i”) is the only graph fragment the supertagger allows for 1, …, , and the second one (with node label “f”) is the only graph fragment allowed for . We let if , and zero otherwise.
Under this construction, every welltyped AM dependency tree for corresponds to a linear sequence of nodes connected by edges with label (see Fig. 8c for an example) More specifically, is a leaf, and every node except for has precisely one outgoing edge; this is enforced by the welltypedness. Because of the edge scores, the score of such a dependency tree is iff it only uses edges that also exist in ; otherwise the score is less than . Therefore, we can decide whether has a Hamiltonian path by running the decoder, i.e. computing the highestscoring welltyped AM dependency tree for , and checking whether the score of is .
Appendix B Neural Network Details
We implemented the supertagger (Section 5.1) and the local dependency model (Section 5.3) in PyTorch, and used the original DyNet implementation of Kiperwasser and Goldberg (2016) (short K&G) for the K&G model. Further details are:

As pretrained embeddings, we use GloVE Pennington et al. (2014). The vectors we use have 200 dimensions and are trained on Wikipedia and Gigaword. We add randomly initialized vectors for the name, date and number tokens and for the unknown word token (if no GloVE vector exists). We keep these embeddings fixed and do not train them.

For the learned word embeddings, we follow K&G in all our models in using a word dropout of . That is, during training, for a word that occurs
times in the training data, with probability
we instead use the word embedding for the unknown word token instead of . 
The characterbased encodings for the supertagger are generated by a single layer LSTM with 100 hidden dimensions, reading the word left to right. If a word (or sequence of words) is replaced by e.g. a name
token during preprocessing, the characterbased encoding reads the original string instead (this helps to classify names correctly as country, person etc.).

To prevent overfitting, we add dropout of 0.5 in the LSTM layers of all the models except for the K&G model which we keep as implemented by the authors. We also add 0.5 dropout to the MLPs in the supertagger and local dependency model.

For the K&G model with the fixedtree decoder, we perform early stopping computing the Smatch score on the development set with 2 best supertags after each epoch.
Optimizer  Adam 
Learning Rate  0.004 
Epochs  37 
Pretrained word embeddings  glove.6B 
Pretrained word emb. dimension  200 
Learned word emb. dimension  100 
POS embedding dimension  32 
Character encoding dimension  100 
(word dropout)  0.25 
BiLSTM layers  2 (stacked) 
Hidden dimensions in each LSTM  256 
Hidden units in MLPs  256 
Internal dropout of LSTMs, MLPs  0.5 
Input vector dropout  0.8 
Optimizer  Adam 
Learning rate  default 
Epochs  16 
Word embedding dimension  100 
POS embedding dimension  20 
Type embedding dimension  32 
(word dropout)  0.25 
BiLSTM layers  2 (stacked) 
Hidden dimensions in each LSTM  128 
0.2  
Hidden units in MLPs  100 
Optimizer  Adam 
Learning Rate  0.004 
Epochs  35 
Pretrained word embeddings  glove.6B 
Pretrained word emb. dimension  200 
POS embedding dimension  25 
BiLSTM layers  2 (stacked) 
Hidden dimensions in each LSTM  256 
Hidden units in MLPs  256 
Internal dropout of LSTMs, MLPs  0.5 
Input vector dropout  0.8 
Appendix C Decoding Details
The goal item of the decoders is one with empty type that covers the complete sentence. In practice, the projective decoder always found such a derivation. However, in in a few cases, this cannot be achieved by the fixedtree decoder with the given supertags. Thus, we take instead the item which minimizes the number of open sources in the resulting graph.
When the fixedtree decoder takes longer than 20 minutes using best supertags, it is rerun with best supertags. If , a dummy graph is used instead. Typically, the limit of 20 minutes is exceeded one or more times by the same sentence of the test set.
With the projective decoder, in most runs, 1 or 2 sentences took too long to parse and we used a dummy graph instead.
We trained the supertagger and all models 4 times with different initializations. For evaluation, we paired each edge model with a supertag model such that every run used a different edge model and different supertags. The reported confidence intervals are 95% confidence intervals according to the tdistribution.
Appendix D Pre and postprocessing Details
d.1 Aligner
We use a heuristic process to generate alignments satisfying the conditions above. Its core principles are similar to the JAMR aligner of Flanigan et al. (2014). There are two types of actions:
Action 1: Align a word to a node (based on the word and the node label, using lexical similarity, handwritten rules^{5}^{5}5E.g. the node label “havecondition91” can be aligned to “if” and “otherwise”. and WordNet neighbourhood; we align some name and date patterns directly). That node becomes the lexical node of the alignment.
Action 2: Extend an existing alignment to an adjacent node, such as from “write” to “person” in the example graph in the main paper. Such an extension is chosen on a heuristic based on

the direction and label of the edge along which the alignment is split,

the labels of both the node we spread from, and the node we spread to, and

the word of the alignment.
We disallow this action if the resulting alignment would violate the singleroot constraint of Section 4.2 in the main paper.
Each action has a basic heuristic score, which we increase if a nearby node is already aligned to a nearby word, and decrease if other potential operations conflict with this one. We remove We iteratively execute the highest scoring action until all heuristic options are exhausted or all nodes aligned. We then align remaining unaligned nodes to words near adjacent alignments.
d.2 Postprocessing
Having obtained an AM dependency tree, we can recover an AM term and evaluate it. During postprocessing we have to relexicalize the resulting graph according to the input string. For relatively frequent words in the training data (occurring at least 10 times), we take the supertagger’s prediction for the label. For rarer words, the neural label prediction accuracy drops, and we simply take the node label observed most often with the word in the training data. For unseen words, if the lexicalized node has outgoing ARGx edges, we first try to find a verb lemma for the word in WordNet Miller (1995) (we use version 3.0). If that fails, we try, again in WordNet, to find the closest verb derivationally related to any lemma of the word. If that also fails, we take the word literally. In any case, we add “01” to the label. If the lexicalized node does not have an outgoing ARGx edge, we try fo find a noun lemma for the word in Wordnet, and otherwise take the word literally.
For names, we again simply look up name nodes and wiki entries observed for the word in the training data, and for unseen names use the literal tokens as the name and no wiki entry. We recover dates and numbers straightforwardly.
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