2022

Grzegorz Chajewski

# Horizontal flux growth as an efficient preparation method of CeRh2As2 single
crystals

<EMAIL_ADDRESS>Damian Szymański<EMAIL_ADDRESS>Dariusz
Kaczorowski<EMAIL_ADDRESS>Institute of Low Temperature and
Structure Research, Polish Academy of Sciences, Okólna 2, 50-422 Wrocław,
Poland

###### Abstract

We report on the efficient method of obtaining CeRh2As2 single crystals with
the use of a bismuth flux growth method in a horizontal configuration. The
crystals thus obtained exhibit much sharper phase transitions and distinctly
higher characteristic temperatures $T_{\rm c}$ and $T_{0}$, compared to
previous reports. Moreover, in our numerous attempts, we found this technique
to be scalable and repeatable.

###### keywords:

CeRh2As2, superconductivity, single crystal growth

## Introduction

The discovery of multi-phase superconductivity in CeRh2As2 Khim2021  turned
the eyes of many solid-state researchers toward this compound and made it
almost instantly one of the most highlighted materials of recent times. In
consequence, numerous papers, both experimental Khim2021 ; Hafner2022 ;
Landaeta2022 ; Onishi2022 ; Kimura2021 ; Kibune2022 ; Kitagawa2022  and
theoretical Schertenleib2021 ; Mockli2021a ; Mockli2021b ; Ptok2021 ;
Cavanagh2022 ; Nogaki2021 ; Hazra2022 ; Nogaki2022 ; Machida2022 , have been
published in the last few years. The most remarkable features of this material
are related to the phase transition within the superconducting state, which
occurs upon applying a magnetic field along the c-axis and leads to an
extraordinarily distinct violation of the Pauli-Clogston limit. Although
several different phenomena including local inversion symmetry breaking
Khim2021 , quadrupole density wave Khim2021 ; Hafner2022 , and coexisting
antiferromagnetism Machida2022  have been proposed as the origin of this
behavior, there is no clear experimental evidence in favor of either of them
so far. One of the most essential difficulties to resolve the puzzle of the
CeRh2As2 nature is related to the quality of samples available up to date. In
all of them, a significant blurring of the anomalies associated with the phase
transitions at $T_{0}$ and $T_{\rm c}$ is observed. This, in consequence,
makes the analysis of collected experimental data more complicated.

In order to properly understand the physics governing the unusual behavior of
CeRh2As2, addressing two basic issues seems to be crucial: 1) Which physical
mechanism is responsible for the phase transition within the superconducting
state? 2) What is the origin of the $T_{0}$-anomaly observed at temperatures
slightly above the superconducting state? It appears that both these questions
might be closely intertwined, and the first, most important step to get
answers to them is to obtain single-crystalline samples of excellent quality.

Because our attempt to achieve prime-quality single crystals by repeating the
previously reported procedure Khim2021  gave unsatisfactory results (more
details in the text below), we have turned to a slightly different technique,
namely flux growth in a horizontal configuration. Here we present our
adaptation of this relatively rarely utilized but technically simple and in
our opinion extremely powerful method for growing high-quality single crystals
of CeRh2As2.

## Synthesis design

As a starting point of our study and a reference for new crystal growth
attempts, we decided to repeat the synthesis of CeRh2As2 single crystals
following the previously reported routine (see supplementary material to Ref.
Khim2021  for details) and using a total charge of about 10 g. In this manner,
we were able to obtain relatively small platelet-like single crystals with
masses occasionally exceeding 1 mg. Their chemical composition, according to
scanning electron microscopy with energy dispersive x-ray spectroscopy (SEM-
EDS), is in good agreement with nominal stoichiometry 1:2:2, and their powder
x-ray diffraction pattern matches with the tetragonal CeBe2Ge2-type crystal
structure (not shown here). Despite this, the characteristic features of
CeRh2As2 observed in the specific heat data, are distinctly smeared (see the
discussion below), as was also reported before Khim2021 ; Hafner2022 . Most
probably, it is due to the existence of some homogeneity range in the Ce-Rh-As
system around 1:2:2 composition and the relatively large temperature interval
(1150 – 700∘C) used during the synthesis process. In the conventional flux
method, where crystal growth is mainly driven by temperature change and
related shift of phase equilibrium, this may lead to crystallization of
material with slightly divergent chemical composition, and therefore also with
somewhat different properties, during subsequent stages of the growth. As a
final product of such a process one obtains crystals possessing nonuniform
bulk properties. Nevertheless, it should be noticed that also distinct
temperature variations in the concentration of structural defects in the
material grown may strongly influence its physical properties and lead to a
similar inhomogeneity effect.

In order to avoid such problems, we propose a different synthesis strategy
based on the flux growth method in a horizontal configuration. In this
technique, the starting material is sealed in an evacuated quartz tube with
the constituent elements (in this case Ce, Rh, and As) placed at one end of
the ampoule and flux material (here Bi) filling about 70-80 % of the remaining
volume. Such prepared ampoule is placed in a horizontal temperature gradient
with substrate elements situated on the hot side. During the synthesis,
several different zones, where various processes take place, can be
distinguished in the ampule. On the hot side (the dissolution zone) all
constituent elements dissolve in the flux. Because there is a huge excess of
these ingredients in comparison to their solubility limit in the flux metal
and to the flux amount itself, only a small part of them is dissolved at a
given moment. The remaining material serves as a reservoir of ingredients for
further growth. In the intermediate section (the transport zone), dissolved
elements are transported towards the other side of the ampule. The flux metal
plays the role of transporting agent, and the concentration gradient is a
driving force in this process. It is important to note that at the same time
flux moves in the opposite direction allowing further dissolving of substrates
in the hot zone. In the colder end of the ampule, due to lowering the
solubility limit, desired material precipitates out of the flux and the
formation of crystals takes place. Thus, this area can be called the
crystallization (or growth) zone.

The most important feature of the horizontal flux growth method is its
temperature stability. After some initial heating-up period, the temperature
in each ampoule zone is kept stable throughout the whole synthesis process.
Consequently, the crystals form in unaltered thermal conditions, which in turn
leads to the crystallization of material showing highly uniform properties.
Furthermore, because of the presence of opposite concentration gradients of
the flux and the constituent elements, the feed material is continuously
supplied to the crystallization zone. Hence, the growth process can be easily
extended in time, making it possible to obtain crystals of large size.
Theoretically, synthesis can be conducted until all substrates are dissolved
in the flux and their concentration is equalized in the whole ampoule or until
the transport channel becomes clogged by the previously grown crystals.

Figure 1: Synthesis design. a The photograph of a three-part test ampoule
prepared for a horizontal flux growth; b Schematic picture of the ampoule with
a marked arrangement of the elements inside; c temperature profile of the
furnace; Three temperature sub-zones, where the ampoules were located during
the growth, are marked by different colors.

For our approach to the synthesis of CeRh2As2 single crystals via horizontal
flux growth technique, we prepared three separate quartz tube ampoules of 6 mm
inner diameter. All of them were filled in with total charges of about 8 g
having the same initial composition as in the previous growth protocols,
namely Ce:Rh:As:Bi = 1:2:2:30. All the ingredients were placed in the ampoules
in the manner described above and sealed in a vacuum. Such prepared ampoules
were lined up in a row, enclosed in another, larger tube (see Fig. 1a and Fig.
1b), and located in the two-zone horizontal tube furnace with the temperature
gradient inside set in such a way as to cover almost the entire temperature
range used for the conventional flux growth. Fig. 1c shows the temperature
profile along the furnace. This way, we were able to distinguish three
different temperature subranges: I - high-temperature range, II - medium-
temperature range, and III - low-temperature range (red, yellow and green
areas in Fig. 1c, respectively), and to conduct in one furnace three separate
syntheses at the same time. Such a procedure was aimed at determination of the
most favorable temperature conditions for the formation of the CeRh2As2
crystals. The approximate temperature ranges in zones I, II, and III were
1150–1020 ∘C, 1000–890 ∘C, and 860–770 ∘C, respectively. The tube was removed
from the hot furnace after 15 days of growth.

## Results and Discussion

### Crystal growth

Figure 2: Images of the grown single crystals. Scanning electron microscope
images of the representative single crystals obtained using horizontal flux
growth method: a, b RhAs2 crystals grown in the high-temperature zone; c, d
RhAs2 and e, f CeRh2As2 crystals grown in the medium-temperature zone; g
CeRh2As2 crystal grown in the low-temperature zone; h CeRh2As2 crystal
obtained in the repeated synthesis (see the text for details).

Fig. 2 shows scanning electron microscope images of some representative single
crystals grown using the horizontal flux growth method within particular
temperature ranges. In the ampoule from the high temperature zone we found
crystals exhibiting two different morphologies, namely platelet- (Fig. 2a) and
blade-like (Fig. 2b). According to the SEM-EDS results, both types had a
chemical composition very close to RhAs2. In this batch no crystals of
CeRh2As2 were found. This suggests that above about 1000 ∘C synthesis process
is dominated by the crystallization of Rh-As binary compound and the desired
ternary material forms at lower temperatures.

In the batch from the medium temperature zone, our findings were similar.
Again, most of the collected crystals had blade- (see Fig. 2c) or platelet-
like shapes, but also some prismatic ones (Fig. 2d) could be found. All of
them are chemically composed only of Rh and As in a molar ratio close to 1:2.
However, the crystals grown in the range II were slightly smaller than those
synthesized at higher temperatures. Importantly, apart from the binary
compounds, in this batch, we were also able to find small crystals with masses
of about 0.5 mg and shapes suggesting tetragonal crystal structure (see Fig.
2e and Fig. 2f). Based on the results of SEM-EDS measurements they were
confirmed to be crystals of CeRh2As2. As can be seen in Fig. 2f, some of them
have distinctly rounded edges, which may be the consequence of growth in the
closeness to the melting temperature of the compound.

The best conditions for the growth of CeRh2As2 were found within the low-
temperature range. In the latter batch, we obtained almost solely single
crystals of the desired compound (except for very few small crystals of
RhAs2). They are visibly larger (see Fig. 2g) than those grown at higher
temperatures and the masses of the biggest ones are about 2.5 mg. Moreover,
instead of common for tetragonal crystal structure platelet-like morphology,
many of the crystals have comparable sizes in all three directions, which is
very beneficial in the context of the characterization of their physical
properties taking into account the crystal anisotropy.

In order to verify the repeatability as well as scalability of the proposed
synthesis method, we prepared an additional batch using a three times larger
amount of materials (about 24 g of the total charge and the same composition
as previously). For the growth, we set the same temperature at the hot end of
the ampoule as it was used for the synthesis in the low-temperature range
process (860 ∘C). Due to the use of a longer quartz tube, the temperature at
the cold end of the ampoule was distinctly lower and was approximately 720 ∘C.
Furthermore, we extended the synthesis time up to 25 days. As a result, we
obtained larger CeRh2As2 single crystals (see Fig. 2h) with masses reaching 8
mg. Moreover, because after the synthesis, at the hot end of the ampoule, we
found significant amount of starting materials undissolved in the flux, we
expect that the process could be continued in time in order to achieve even
more massive crystals.

### Physical properties characterization

Figure 3: Specific heat characterization of the crystals. Low-temperature
variations of the specific heat of the CeRh2As2 crystals grown by a standard
flux method and b–d horizontal flux method in the medium-temperature zone,
low-temperature zone, and during repeated growth, respectively. Arrows mark
the phase transitions and vertical dashed and dotted lines are the guides for
the eye. For the sake of clarity, in each panel, phase transitions were marked
only for the curves with the highest transition temperatures.

Due to the fact that in the previous reports on CeRh2As2 the crystal quality
problem most clearly manifested itself through distinctly blurred temperature
characteristics of various physical quantities, we decided to use heat
capacity measurements as a reliable method to evaluate the quality of the
obtained single crystals. The results of our experiments performed on samples
from different batches are gathered in Fig. 3 and displayed as $C_{\rm
p}/T(T)$ curves.

As can be easily inferred from Fig. 3a, the standard flux growth method leads
to the formation of crystals, which exhibit smeared specific heat anomalies
related to the superconducting and ’the other’ phase transitions at $T_{\rm
c}$ and $T_{0}$, respectively. In addition, for some of the crystals (e.g.
sample A2) the anomaly at $T_{0}$, rather than having a peak-like shape,
appears to be only a change in the slope of the $C_{\rm p}/T(T)$ curve.
Furthermore, in so-obtained crystals, the critical temperature $T_{\rm c}$
seems to be a sample-dependent property and varies significantly for different
specimens. The highest values of the transition temperatures for this group of
samples are $T_{\rm c}$ = 0.22 K and $T_{0}$ = 0.32 K.

In contrast, all the CeRh2As2 crystals grown by the horizontal method exhibit
a very sharp transition to the superconducting state at $T_{\rm c}$ and well
defined specific heat anomaly at $T_{0}$ (see panels b–d in Fig. 3).
Furthermore, the magnitudes of both peaks in $C_{\rm p}$ are much larger and
temperatures $T_{\rm c}$ and $T_{0}$ are distinctly higher, compared to those
for samples grown by the standard flux technique. Interestingly, among
numerous crystals tested, we were able to distinguish three groups of samples
characterized by slightly different critical temperatures, namely $T_{\rm
c}~{}=~{}$0.305(5), 0.318(6), and 0.330(8) K (green, red, and blue points in
Fig. 3b–d, respectively). In order to maintain the clarity of the image, only
one curve for each critical temperature observed in the batch is shown. One
can also note, that for all of those crystals, samples with higher $T_{\rm c}$
have also higher $T_{0}$. This suggests that the two phase transitions in
CeRh2As2 are linked to each other. Therefore, the related phenomena presumably
interfere, and the unusual properties of the superconducting state in CeRh2As2
might result from the closeness of the phase transition at $T_{0}$.

Figure 4: Comparison of specific heat and electrical resistivity. Low-
temperature specific heat and electrical resistivity (current flowing within
the tetragonal plane) of CeRh2As2 crystal grown in the low-temperature zone.

Fig. 4 shows the comparison of the specific heat and electrical resistivity
data collected on the same crystal grown in the low-temperature range process.
As can be seen, in both measured quantities the superconducting phase
transition occurs at almost ideally the same temperature. This finding differs
from the previous reports Khim2021 ; Hafner2022 , where the drops in the
resistivity were observed at notably higher temperatures, compared to the
anomalies in the specific heat and magnetic susceptibility, probably because
of inhomogeneity in the chemical composition of the specimens. Thus, the
result shown in Fig.4 can be considered as another proof of highly uniform
properties of the crystals grown by the horizontal flux growth method.

### Crystal structure analysis

Figure 5: Crystal structure analysis. a Powder XRD pattern (circles) obtained
for several pulverized single crystals of CeRh2As2 grown by the horizontal
flux growth method in the low-temperature zone, together with the refined
theoretical profile (red solid line), the difference between them (blue solid
curve), and the refined positions of the Bragg reflections for the main phase
and Bi flux residues (vertical ticks); b and c Laue x-ray backscattering
patterns obtained for a single crystal of CeRh2As2 from the same batch for
[100] and [001] directions, respectively; d and e the same images with the
overlays of theoretically generated patterns (blue circles).

The powder x-ray diffraction (XRD) pattern collected on several ground single
crystals from the low-temperature range batch is displayed in Fig. 5a. As
illustrated, all of the Bragg peaks can be ascribed to either CeRh2As2,
crystallizing in the tetragonal CaBe2Ge2-type structure (space group P4/nmm),
or a small amount of residual Bi flux. The Rietveld analysis of this XRD data
yielded the lattice parameters $a$ =4.2833(5) Å and $c$ = 9.8518(4) Å, in good
agreement with the previously reported ones Khim2021 .

The high quality of crystals grown by the horizontal flux method was also
confirmed by the Laue x-ray backscattering technique. The patterns collected
in both [100] and [001] directions (figs. 5b and 5c, respectively) show very
sharp spots and no signatures of crystal twinning or other crystal defects.
Moreover, they perfectly match with the theoretical patterns (figs. 5d and
5e), generated based on the crystal structure parameters determined via the
Rietveld refinement.

## Conclusion

We presented the Bi flux growth technique in a horizontal configuration as an
efficient method of obtaining large and high-quality single crystals of the
unconventional superconductor CeRh2As2. The specific heat and electrical
resistivity measurements, performed on numerous samples, proved that the
synthesized single crystals exhibit highly uniform physical properties.
Moreover, they show much sharper phase transitions, and located at higher
temperatures, compared to the previously published data. The excellent quality
of the grown crystals was also corroborated by the powder x-ray diffraction
and Laue x-ray backscattering, and their chemical composition was confirmed to
be very close to the nominal one by the SEM-EDS analysis. Importantly, this
growth technique was found to be easily scalable and highly repeatable.

## Methods

### Single crystal growth

Single crystals of CeRh2As2 were grown using two different approaches to the
flux growth method: i) the conventional one - according to the procedure
reported previously by Khim et al. Khim2021  and ii) the horizontal flux
growth method Yan2017  described in more detail in the main text. In the
latter method, a horizontal two-zone tube furnace was used, however only in
one of the zones (the hot zone) the heating was turned on during the whole
synthesis process. In each of the batches the initial stoichiometry
Ce:Rh:As:Bi = 1:2:2:30 was used. In order to extract single crystals and
remove flux, the material after synthesis was transferred to alumina crucibles
and closed together with alumina strainers and additional reverted crucibles
in evacuated quartz tubes. Such prepared ampules were reheated up to 450∘C and
centrifuged. The residual flux remaining on the surfaces of crystals was
removed by etching with the use of 2:1 mixture of glacial acetic acid and 30%
hydrogen peroxide.

### X-ray diffraction and elemental analysis

The quality of the samples was checked using a PROTO COS Laue diffraction
camera. The x-ray diffraction measurements were performed on several ground
small single crystals on an X’pert Pro PANalytical diffractometer with Cu Kα
radiation. The obtained powder x-ray diffraction data were analyzed with the
use of the Rietveld refinement method implemented in FullProf software
Carvajal1993

Scanning electron microscope (SEM) images were taken using an FE-SEM FEI
NovaNano SEM 230 equipped with an EDAX Genesis XM4 spectrometer. The same
instrument was used to check the chemical composition of the samples via
electron-microprobe analysis with energy-dispersive x-ray spectroscopy (EDS).
SEM images and EDS measurements presented in the paper were acquired using the
acceleration voltage of 30 kV.

### Specific heat and resistivity measurements

Specific heat measurements were performed in temperatures down to 80 mK using
the standard time-relaxation method Hwang1998  implemented in a commercial
Quantum Design PPMS (Physical Property Measurement System) platform equipped
with a 3He-4He dilution refrigerator. Electrical resistivity was measured
employing the ac four-probe technique on bar-shaped samples with the use of
the same experimental platform.

Author Contributions G.C. synthesized the samples and performed their physical
and crystallographic characterization. D.S. took SEM images of crytals and
performed SEM-EDS based analysis. G.C. and D.K. discussed the results and
prepared the manuscript. G.C., D.S. and D.K. revised and corrected the
manuscript.

Conflict of Interest The authors declare no conflict of interest.

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# Normalizing Compositional Structures Across Graphbanks

Lucia Donatelli, Jonas Groschwitz, Alexander Koller,Matthias Lindemann, Pia
Weißenhorn
Department of Language Science and Technology
Saarland University, Germany
<EMAIL_ADDRESS>
###### Abstract

The emergence of a variety of graph-based meaning representations (MRs) has
sparked an important conversation about how to adequately represent semantic
structure. These MRs exhibit structural differences that reflect different
theoretical and design considerations, presenting challenges to uniform
linguistic analysis and cross-framework semantic parsing. Here, we ask the
question of which design differences between MRs are meaningful and
semantically-rooted, and which are superficial. We present a methodology for
normalizing discrepancies between MRs at the compositional level [Lindemann et
al., 2019], finding that we can normalize the majority of divergent phenomena
using linguistically-grounded rules. Our work significantly increases the
match in compositional structure between MRs and improves multi-task learning
(MTL) in a low-resource setting, demonstrating the usefulness of careful MR
design analysis and comparison.

##  1 Introduction

Graph-based representations of sentence meaning offer an expressive and
flexible means of modeling natural language semantics. In recent years, a
number of different _graphbanks_ have annotated large corpora with graph-based
semantic representations of various types, for example Elementary Dependency
Structures (EDS) [Oepen and Lønning, 2006]; DELPH-IN MRS Bi-Lexical
Dependencies (DM) [Ivanova et al., 2012]; Abstract Meaning Representation
(AMR) [Banarescu et al., 2013]; and Universal Conceptual Cognitive Annotation
(UCCA) [Abend and Rappoport, 2013]. Because of differences in the design
principles underlying the different graphbanks, these graphs differ greatly
and often in fundamental strategies such as how lexical items are anchored in
graph nodes [Oepen et al., 2019]. As an example, the graphs shown in Fig. 1
correspond to the three graphbanks of the SemEval 2015 Shared Task on Semantic
Dependency Parsing, which we take as our focus in this paper. The graphs
visibly differ with respect to edge labels, edge directions, the treatment of
a periphrastic verb construction, and coordination.

Understanding the nature of these differences is important both from a
linguistic and from a parsing perspective. With respect to linguistics, one
can ask which design differences between graphbanks are meaningful (e.g.
understanding coordinate structures as headed by a conjunct or the
conjunction); which ones are artifacts that can be equalized by simple
transformations (e.g. flipping the direction of edges); and which ones are
worth spending time on when discussing the design of future corpora. With
respect to parsing, the differences between graphbanks make it hard to develop
parsers that are accurate across many graphbanks [Oepen et al., 2019]. A more
unified approach to certain structures across graphbanks has the potential to
facilitate multi-task learning (MTL) to train parsers on smaller graphbanks by
exploiting the training set of a larger graphbank, in addition to being
linguistically parsimonious.

In this paper, we show how annotations from different graphbanks can be
normalized at the level of their compositional structures. We build upon the
work of ?), who used _AM dependency trees_ to represent the compositional
structure of graph-based meaning representations (MRs); they developed
graphbank-specific heuristics for automatically constructing AM dependency
trees for the graphs in a number of different graphbanks built upon the AM
algebra [Groschwitz et al., 2018] (Section 2). We develop a new methodology
for identifying and quantifying mismatches between the compositional
structures assigned to the same sentence by different graphbanks; we then
present our extended AM+ algebra, which is able to systematically reshape
these compositional structures in order to make them more uniform across
graphbanks (Section 3). Finally, we walk through how our methodology can be
applied to normalize specific linguistic phenomena that differ in
representation across graphbanks (Section 4). Our work is focused on the three
graphbanks of the SemEval 2015 Shared Task on Semantic Dependency Parsing
[Oepen et al., 2015] – DM, PAS, and PSD – as these are annotated over the same
base text corpus. Our methods can in principle be applied to more complex
graphbanks, as well.

Using our methods, we increase the match between the AM dependency trees
(compositional structure) of the different graphbanks to 76.3 (DM-PAS), 78.8
(DM-PSD), and 82.0 (PAS-PSD) directed unlabeled F-score, compared to 63.5,
55.7, and 57.0 for Lindemann et al.’s AM dependency trees (Section 5). This is
a drastic improvement over the unlabeled F-scores of 64.2, 26.1, and 29.6 for
the original graphs [Oepen et al., 2015]. We additionally demonstrate that
when training a graph parser on a tiny graphbank combined with larger
graphbanks using multi-task learning (MTL), our methods improve parsing
accuracy by up to 2.6 points F-score over MTL without normalized AM dependency
trees. Finally, we assess the hierarchy of design differences with respect to
their theoretical and quantitative impacts and discuss phenomena which cannot
be made uniform using the techniques developed here (Section 6).

##  2 Background

We first describe the graphbanks we use for our experiment in this paper
(Section 2.1). We then motivate our use of the compositional AM algebra and
parser that we use for our implementation (Section 2.2).

###  2.1 The Graphbanks

(a) DM

(b) PAS

(c) PSD

Figure 1: Examples for DM, PAS and PSD graphs. The PSD graph includes
Lindemann et al.’s (2019) reversible preprocessing of coordination.

We focus our efforts on the three graphbanks of the SemEval 2015 Shared Task
on Semantic Dependency Parsing (SDP): (i) DELPH-IN MRS-Derived Semantic
Dependencies (DM), (ii) Enju Predicate–Argument Structures (PAS), and (iii)
Prague Semantic Dependencies (PSD) [Oepen et al., 2015]. All graphbanks have
parallel semantic annotations over the same common English texts: the Wall
Street Journal (WSJ) and Brown segments of the Penn Treebank (PTB; ?)). These
graphbanks are all bilexical, node-ordered, labeled, directed graphs and
represent core semantic information such as predicate-argument relations.
Nothing in our paper hinges on the choice of these particular graphbanks; the
methods presented here can be applied to other graphbanks, though we assume
that the graphs are over the same base texts.

The SDP graphbanks make different choices about which linguistic information
to represent and how to represent it [Oepen et al., 2016] (Table 1). Graphs in
the style of the three graphbanks for the sentence “Cats and cute dogs are
playing in the house” are shown in Fig. 1. While in all three graphs each
token of the sentence corresponds to at most one node, there may also be
tokens which are not part of the graph. These ‘ignored’ tokens differ across
graphbanks; for instance, in DM coordinators (“and”) and temporal auxiliaries
(“are”) are not nodes in the graph, while PSD ignores temporal auxiliaries,
prepositions, and determiners. The edges are also different; for instance,
edges that point from adjectives to nouns in DM and PAS go the other way in
PSD.

| Directed F | Undirected F
---|---|---
| PAS | PSD | PAS | PSD
DM | 64.2 | 26.1 | 67.2 | 56.8
PAS |  | 29.6 |  | 54.9
Table 1: Unlabeled F-scores across formalisms including punctuation according
to ?).

The fundamental question we discuss in this paper is the extent to which these
variations are superficial or, alternatively, represent irreconcilable design
differences. To this end, we make the standard assumption from theoretical
semantics that MRs are derived _compositionally_ from the sentence, i.e. by
systematically computing MRs for each part of the sentence from its immediate
subparts. We say that a difference between two graphs is ‘superficial’ if it
can be derived using the same compositional structure. To give an example,
Fig. 4(4(a), 4(b)) show the compositional structure which ?) assign to the DM
and PSD graphs in Fig. 1. We explain the technical details in Section 2.2; for
now, observe that certain differences between DM and PSD have been normalized
(e.g. both dependency trees analyze the noun as the head and the adjective as
the modifier), while others are still present (e.g. ignored tokens). In this
paper, we present a method for deriving compositional structures for SDP
sentences that are more consistent across graphbanks and still linguistically
solid. This allows us to distinguish superficial design differences more
cleanly from deeper ones, and is useful for parsing.

###  2.2 Compositional Semantic Parsing with the AM Algebra

To normalize discrepancies between the three formalisms, we use the Apply-
Modify (AM) Algebra initially presented in ?) for compositional AMR parsing.
Utilizing the AM algebra allows us to modify graph structure at the
compositional level, represented as AM dependency trees (explained below). ?)
(hereafter referred to as L19) implemented this methodology and extended the
AM algebra to four additional graph-based MRs (DM, PAS, PSD, and EDS); ?)
extended this work to UCCA. L19 achieved state of the art results for all
formalisms in question with pretrained BERT embeddings and manual heuristics
tailored to each graphbank. We adapt these heuristics to find uniform
structures that generalize across graphbanks.

The AM algebra builds sentence-level graphs from graph fragments called
annotated s-graphs, or as-graphs. Fig. 2(a) shows as-graphs from which the PAS
graph in Fig. 2(d) for the sentence “the cat is not lazy” can be built. Note
that Fig. 2(d) is the same as the graph in Fig. 3(b) minus the ordering among
the nodes. As-graphs possess two kinds of markers: (i) an obligatory _root
source_ on the _root node_ (indicated with bold outline); and (ii) optionally
further sources (marked in red, e.g. s and o) on other nodes, which allow us
to combine as-graphs. Every node can bear at most one source, and each source
can occur at most once in an as-graph.

The AM algebra defines two operations for combining as-graphs. (i) Apply for a
source X (App${}_{\text{{X}}}$) combines head and argument as-graphs by
filling the head’s X-source with the root of the argument. For example, the
result of
$\text{{App}}_{\text{{o}}}\left(G_{\text{be}},G_{\text{lazy}}\right)$ is shown
in Fig. 2(b).111Note: “lazy” is represented in Fig. 2(b) with an s-source for
ease of understanding here; L19 analyze it as a mod-source. We use s-sources
for adjectives in PAS in the presentation of all examples. Nodes in both
graphs with the same source (here s) are merged. The type annotation
$[\text{{s}}]$ at the o-source of $G_{\text{be}}$ requests the argument to
have an s-source (which $G_{\text{lazy}}$ has). This way, type annotations
lead to reentrancies in the graph (like the triangle structure here). (ii)
Modify allows an as-graph to modify a head. For example,
$\text{{mod}}_{\text{{det}}}\left(G_{\text{cat}},G_{\text{the}}\right)$ and
its result is shown in Fig. 2(c). Here, $G_{\text{cat}}$ is the head and
$G_{\text{the}}$ the modifier; $G_{\text{the}}$ attaches with its det-source
at $G_{\text{cat}}$ and loses its own root source. We combine the two partial
results we have built so far using App${}_{\text{{s}}}$. The final graph is
the result of the mod${}_{\text{{mod}}}$ operation at the top of the term in
Fig. 2(e).

(a) Constants $G_{\text{be}}$, $G_{\text{lazy}}$, $G_{\text{cat}}$,
$G_{\text{the}}$, and $G_{\text{not}}$

(b) $\text{{App}}_{\text{{O}}}\left(G_{\text{be}},G_{\text{lazy}}\right)$

(c) $\text{{mod}}_{\text{{det}}}\left(G_{\text{cat}},G_{\text{the}}\right)$

(d) PAS graph (cf. Fig. 3(b))

for tree=align=center, l=l sep=5pt [mod${}_{\text{{mod}}}$, for
tree=fill=green!30 [App${}_{\text{{s}}}$, for tree=fill=green!30
[App${}_{\text{\text{{o}}}}$ [$G_{\text{be}}$,tier=word] [$G_{\text{lazy}}$,
for tree=fill=red!30,tier=word] ] [mod${}_{\text{{det}}}$, for
tree=fill=brown!10!yellow!60 [$G_{\text{cat}}$,tier=word] [$G_{\text{the}}$,
for tree=fill=blue!30,tier=word] ] ] [$G_{\text{not}}$, for
tree=fill=orange!30,tier=word] ]

(e) AM term

[amdep, label style=font=] [column sep=.3cm, row sep=0.2ex]The & cat & is &
not &[-2mm] lazy
$G_{\text{the}}$ & $G_{\text{cat}}$ & $G_{\text{be}}$ & $G_{\text{not}}$ &
$G_{\text{lazy}}$
& & & &
[edge unit distance=2ex]3root [edge unit distance=2ex, label
style=fill=brown!10!yellow!60]21mod${}_{\text{{det}}}$ [edge unit
distance=2ex, label style=fill=green!30]32App${}_{\text{{s}}}$ [edge unit
distance=2.5ex, label style=fill=green!30]35App${}_{\text{{o}}}$ [edge unit
distance=2ex, label style=fill=green!30]34mod${}_{\text{{mod}}}$

(f) AM dependency tree

Figure 2: PAS for “the cat is not lazy” with its AM analysis according to L19.

By tracking the ‘semantic heads’ of each subtree of an AM term as color coded
in Fig. 2(e), L19 encode AM terms as AM dependency trees (Fig. 2(f)): whenever
the AM term combines two graphs with some operation, a dependency edge is
added from one semantic head to the other. The graph constants of the AM term
become the _lexical as-graphs_ for the tokens in the sentence (if a token has
no graph constant associated with it, we write ‘$\bot$’). Such an AM
dependency tree evaluates deterministically to a graph.

A central challenge with AM dependency parsing is that the AM dependency trees
in the training corpus are latent. L19 formulated graphbank-specific
decomposition heuristics to address this. In the context of normalizing the
graphbanks, this challenge turns into an opportunity: we can normalize the
graphbanks at their compositional level without having to directly change the
graphs. In fact, even the AM dependency trees of L19 naturally normalize some
differences of the MRs. Consider the example “is not” in the PAS and PSD
graphs in Fig. 3(3(b), 3(c)) and their L19 AM trees in Fig. 3(3(e), 3(f)). In
the graphs, the edges between “is” and “not” have a different label and
different direction, whereas both AM trees have a mod${}_{\text{{mod}}}$ edge
with the same direction. Furthermore, in the same graphs, PAS has a triangle
“is”-“lazy”-“cat”, while PSD does not. In the L19 AM trees (Fig. 3(3(e),
3(f))), this difference is reduced to the lexical as-graphs.

##  3 Quantifying Compositional Mismatches

The AM dependency trees of L19 (henceforth, L19 AM trees) constitute a first
step towards normalizing MRs across graphbanks by making compositional
structures explicit. This has a positive result: the AM dependency trees in
Fig. 4(4(a)-4(c)) are more consistent with each other than the original graphs
in Fig. 3. Yet, there remain differences between the L19 AM trees. The
question we ask here is, can we find a single compositional structure which
will correspond to all three MR graphs by changing only the lexical as-graphs
for the individual tokens and not the L19 AM trees themselves? This would
suggest that the designs and annotators of the three graphbanks tacitly agree
on compositional structure; surface differences between the graphs can then be
explained by low-level representations of individual words.

In this section, we present our methodology to apply transformations to the
L19 AM trees that do not change the graphs they evaluate to. We show how to
detect and quantify mismatches between the L19 AM trees for the same sentence
in terms of _pattern signatures_ (Section 3.1). As we will then see, the AM
algebra has limitations which prevent us from fully normalizing the
compositional structures. We will therefore slightly extend it to the _AM+
algebra_ (Section 3.2), which we will use in Section 4 to automatically
normalize AM dependency trees, making them more consistent with each other.

(a) DM graph

(b) PAS graph

(c) PSD graph

(d) L19 AM tree for DM

(e) L19 AM tree for PAS

(f) L19 AM tree for PSD

(g) normalized tree for DM

(h) normalized tree for PAS

(i) normalized tree for PSD

Figure 3: SDP graphs, L19 AM trees and normalized trees for the sentence “The
cat is not lazy”

(a) L19 AM tree for DM

(b) L19 AM tree for PSD

(c) L19 AM tree is equal to normalized tree for PAS

(d) normalized tree for DM

(e) normalized tree for PSD

Figure 4: L19 AM trees and normalized trees for SDP graphs of “Cats and cute
dogs are playing in the house” presented in Fig. 1.

###  3.1 Detecting Local Patterns

[amdep] $w$
$\bot$

(a) Pattern empty

[amdep] $w$
[edge unit distance=1ex]1modx

(b) Pattern mod/cmod

[amdep] [column sep=2em] $w$ & $v$
[edge unit distance=2ex]1Appx 12Appy

(c) Pattern app1

[amdep] [column sep=2em] $u$ & $w$ & $v$
[edge unit distance=2ex]2Appx 21Appy 23Appz

(d) Pattern app2/capp2

[amdep] $w$
[edge unit distance=1ex]1Appx

(e) Pattern basic

[amdep] [column sep=2em] $w$ & $v$
[edge unit distance=2ex]1modx 12Appy

(f) Pattern conn

Figure 5: The set of local patterns we distinguish based on their structures
in L19 AM trees.

In order to normalize the compositional structures of the graphbanks
automatically, we must be able to detect and quantify discrepancies between
their L19 AM trees automatically. We do this by identifying ‘local’ patterns
in the L19 AM trees trees that correspond to individual tokens (Fig. 5). We
then compare these local patterns across the L19 AM trees for the three
graphbanks to uncover a pattern signature, with which we can identify and
normalize differences across the graphbanks.

To illustrate how this works, take the case of the determiner “the” in Fig. 3.
In the graphs themselves (3(a)–3(c)), DM (3(a)) and PAS (3(b)) mark the
determiner with different relations while PSD (3(c)) ignores it. This is
reflected in L19 AM trees (3(d)–3(f)): the DM (3(d)) and PAS (3(e)) trees
combine the determiner with a mod relation, while the PSD tree (3(f)) leaves
it absent ($\bot$). We name these patterns mod for DM and PAS, and empty for
PSD, and create a pattern signature with the triple [DM / PAS / PSD]: [mod /
mod / empty]. We use this order for all pattern signatures. Table 2 shows that
the determiner pattern signature is the most common across the MRs, alongside
the other most frequent pattern signatures we find and fix.

The set of patterns shown in Fig. 5 are the basis for the pattern signatures
we utilize to capture a range of linguistic phenomena.222 The patterns do not
take source names or outgoing mod edges into account; for example, “cat” in
Fig. 3 would still exhibit the basic pattern if it had an incoming
App${}_{\text{{o}}}$ edge instead of App${}_{\text{{s}}}$, or if it were
modified. The empty pattern corresponds to elements not present in the trees
(Fig. 5(a)). If the word has an incoming mod edge and no outgoing App edges in
its L19 AM tree, like determiners in DM and PAS (Fig. 4(b), “the”), we assign
it the mod pattern, shown in Fig. 5(b). We have several patterns for words
that are heads in the AM dependency tree, words that are either the root of
the AM dependency tree or have an incoming App edge (Fig. 5(5(c)-5(e))). If
such a head has no arguments of its own, i.e. no outgoing App edges, we use
the basic pattern in Fig. 5(e). This pattern applies most commonly to nouns
(e.g. “cat” as $G_{\text{cat}}$ in Fig. 2(a) or the nouns in Figs. 3 and 4).
If the head has one or two arguments (= outgoing App edges), as seen in Fig.
5(5(c), 5(d)), we assign it the app1 or app2 pattern respectively.
Intransitive and transitive verbs typically have these respective patterns; we
find that the coordination structures of PAS and PSD in Fig. 4 show the app2
pattern as well.

If a modifier (here defined as a word with an incoming mod edge in the L19 AM
trees) itself has an argument (an outgoing App edge), we assign it the conn
pattern (Fig. 5(f)). This pattern is typical for prepositions in DM and PAS.
We also find the following distinctions useful: sometimes a modifier, such as
the temporal auxiliary “are” in the PAS analysis in Fig. 4(c), carries
additional sources with it that create reentrancies at the time of
modification. We use the cmod pattern (Fig. 5(b)) instead of mod for such
complex modifications. When a head with two children has an annotation that
will lead to a reentrancy between the children, such as the copula “is” with
the constant $G_{\text{be}}$ in Fig. 2(a), we use the capp2 pattern rather
than app2. Besides copula in PSD, all MRs have this pattern for control verbs.
We summarize all other patterns as other.

Examining our pattern signatures, we find that about half of the nodes have
the same pattern in all three graphbanks: for example, adjectives often
display the signature [mod / mod / mod] and nouns [basic / basic / basic].
These aligned pattern signatures represent a priori shared compositional
structure across graphbanks, typically in the form of open-class words and
predicate-argument structure. In contrast, Table 2 shows the most common
pattern signatures with differences between the MRs. These pattern signatures
map quite precisely to linguistic phenomena, specifically those involving
closed-class functional words.

Rank |  Pattern signature ([DM / PAS / PSD]) |  Most commonly associated phenomena | Count |  Percentage of total
---|---|---|---|---
1 | [mod / mod / empty] | Determiners | $49\,965$ |  $13.38$%
2 | [empty / mod / empty] | Punctuation* | $45\,544$ |  $12.20$%
3 | [conn / conn / empty] | Prepositions | $42\,395$ |  $11.35$%
4 | [basic / basic / mod] | NPs with prep.* | $30\,202$ |  $8.09$%
5 | [empty / conn / empty] | Prepositions | $22\,324$ |  $5.98$%
8 | [empty / cmod / empty] | Temporal auxiliaries | $14\,074$ |  $3.77$%
10 | [app1 / mod / mod] | Verbal Negation* | $7215$ |  $1.93$%
14 | [empty / app2 / app2] | Coordination | $5709$ |  $1.53$%
20 | [empty / capp2 / app2] | Copula (adjectival) | $2915$ |  $0.78$%
21 | [empty / other / other] | Coordination | $2828$ |  $0.76$%
Table 2: Most frequent pattern signatures we fix. Asterisks indicate that
additional phenomena comprise a significant portion of the pattern. See
Appendix A for complete list of pattern signatures we detect.

###  3.2 The AM+ Algebra

Now that we can detect mismatches of the compositional structures across
graphbanks, we would like to modify these compositional structures such that
the AM dependency trees for all three graphs contain the same edges. This is
illustrated in Fig. 3(3(g)-3(i)): All three trees have the same edges and
ignore the same tokens (namely, none); all three analyses agree on the
compositional structure, and all graphbank-specific differences have been
pushed into the graphs for the individual tokens.

However, there is one technical complication: These AM dependency trees are
invalid because some graphs have a root node which is also marked with some
other source (e.g. the one-node as-graph for “is” in Fig. 3(g)). This is
necessary to make the copula semantically vacuous – we have a node for it in
the AM dependency tree, but there is no node in the DM as-graph to which the
tree evaluates – but the AM algebra does not allow root nodes to also have
another source besides the root source.

We therefore extend the AM algebra to the _AM+ algebra_ , which is exactly
like the AM algebra but allows root nodes to contain additional sources. In
all other ways, the AM+ algebra is defined exactly like the AM algebra, and
the concept of an AM dependency tree generalizes naturally to the AM+ algebra.
Thus Fig. 3(3(g)-3(i)) are valid AM dependency trees of the AM+ algebra.

The seemingly innocent step from the AM algebra to the AM+ algebra is not
without risks. On a technical level, the AM algebra is grounded in the HR
graph algebra [Courcelle and Engelfriet, 2012]. In this translation, root
nodes in the AM algebra are mapped to sources with a special name (“root”) in
the HR algebra, and a node in a graph of the HR algebra may not be marked with
more than one source at once. This is to prevent situations where e.g. an
“apply” operation could merge together two nodes of the same graph. Thus we
sacrifice the grounding in the HR algebra and with it, certain technical
guarantees. From a modeling perspective, the extra flexibility of the AM+
algebra means that the same graph can be described with more terms of the AM+
algebra than with terms of the AM algebra. This can be useful – in fact, we
rely on this flexibility here – but it also means that the choice of
compositional structure must be made with care and, ideally, in a
linguistically informed manner. Making such a choice is the main theme of the
next section.

##  4 Normalizing Compositional Structures Across Graphbanks

We normalize the AM dependency trees using the AM+ algebra by applying
transformations directly to the AM dependency trees of L19. The
transformations described below are carefully designed not to change the graph
the AM dependency trees evaluate to. We do this for all AM dependency trees by
iterating over the sentences in the graphbanks and applying a transformation
whenever its corresponding pattern signature matches. To make sure that we
only apply a transformation when it matches the linguistic phenomenon we
intend to address, we impose further restrictions based on POS tags and
lemmas. The transformations are always applied in the same order (see Table 3
in Section 5 below). In the remainder of this section, we describe in detail
how the individual transformations work and how they relate to the pattern
signatures and linguistic phenomena.

###  4.1 Category 1: Ignored Elements

A common difference among the SDP graphs is that PSD does not represent the
meaning of some tokens – the tokens are ‘ignored’ in the sense that they do
not have incident edges – whereas PAS includes nearly all tokens in the graph;
DM falls somewhere in the middle. Section 3.1 discussed one such example: in
Fig. 1 the determiner “the” is part of the PAS and DM graphs, but not PSD. In
the corresponding L19 AM trees, ignored tokens are not part of the dependency
tree: they have no incoming edge (“the” in Fig. 4). Our fixes for these
elements thus relies on creating vacuous lexical as-graphs that do not change
the graph when combined. We address the following cases, which account for
roughly 30% of the divergent phenomena detected.

#### Determiners: [mod / mod / empty].

As seen in Section 3.1, determiners show up as modifiers in the AM dependency
trees for DM and PAS, but not in PSD, yielding the [mod / mod / empty]
signature.
Our fix: We can modify the AM dependency tree for PSD to match the edges of
the other AM trees (Fig. 3(i)): we use a semantically vacuous constant for the
determiner “the” that consists of only an unlabeled node, marked with a det-
source and as the root at the same time. Performing the mod${}_{\text{{det}}}$
operation with this constant does not change the graph of the noun (here
“cat”), and thus the AM dependency tree in Fig. 3(i) still evaluates to the
graph in Fig. 3(c). This construction makes sense: the mod${}_{\text{{det}}}$
operation reflects a relation on the sentence surface, and the empty
determiner constant for PSD reflects the fact that the information carried by
the determiner is ignored in the deeper PSD graph structure.

Note that since this empty lexical as-graph does not change the graph, we
could attach it anywhere and still obtain the same graph as evaluation result.
In other words, just looking at the PSD graph is not enough to tell us where
the determiner should attach. We attach the new vacuous PSD determiners to the
token where the determiner attaches in DM.
Further restrictions: To make sure we exactly fix determiners this way, the
token belonging to the local pattern must have the POS tag DT. This covers 95%
of the pattern.

#### Punctuation, Particles & Co.: [empty / mod / empty].

PAS is the only SDP MR to broadly annotate tokens that are mostly syntactic
markers. This pattern includes punctuation, infinitival “to”, passive “by” and
particles of particle verbs.
Our fix: As for determiners, we add empty modifiers, here to DM and PSD.
Further restrictions: POS tag indicates punctuation or is IN, TO or RP (covers
97% of pattern).

#### Temporal and Aspectual Auxiliary Verbs: [empty / cmod / empty].

An example of this is “are” in Fig. 4. These auxiliaries are annotated only in
PAS, where the L19 AM trees (Fig. 4(c)) use complex modification to create the
reentrancy observed in Fig. 1(b).
Our fix: While the cmod pattern is different from the mod pattern as far as
the resulting graph is concerned, both patterns simply have an incoming mod
edge when only looking at the AM tree edges. Thus, we can apply the same fix
here as for the punctuation case.
Further restrictions: POS tag starts with V (covers 82% of pattern).

#### Copula: [empty / capp2 / app2].

While all three SDP MRs treat the copula verb as a transitive verb in case of
a nominal predicate (e.g. “Cats are mammals”), the treatment of copula with an
adjectival predicate (e.g. “The cat is not lazy”, see Fig. 3(3(a)-3(c)))
differs: DM treats this construction equivalent to adnominal adjectives (“cute
dogs” in Fig. 1(a)), therefore the copula does not contribute directly to the
graph, corresponding to the empty pattern. PSD and PAS, however, still encode
the copula verb as a transitive verb thus differing from the structure of
adnominal adjectives. PAS includes an edge from the adjective to the subject,
resulting in a reentrancy in the graph and the capp2 pattern in the AM tree.
PSD does not include such an edge and expresses the relation between the
adjective and the subject only via the copula. This results in the app2
pattern in the AM tree.

While the app2 pattern used in PSD differs from the capp2 pattern employed by
PAS, both MRs use the same AM dependency tree: the verb is the root and has
outgoing App${}_{\text{{s}}}$ and App${}_{\text{{o}}}$ edges to the subject
and the predicate respectively (Fig. 3(3(e), 3(f))). Thus, only DM exhibits a
different compositional structure.
Our fix: We normalize the AM dependency trees by adding a vacuous lexical as-
graph for the copula verb in DM. This lexical as-graph consists of one root
node with o-source, requiring an s-source in its argument (see “is” in Fig.
3(g)). We remove the App${}_{\text{{s}}}$ edge between the adjective (“lazy”)
and the subject (“cat”), and instead make the subject and adjective
App${}_{\text{{s}}}$ and App${}_{\text{{o}}}$ children of the copula
respectively. Further, we move the incoming dependency edge of the adjective
to the copula, making it the new head of the phrase. When evaluating this new
AM tree, first the adjective (“lazy”) is plugged into the o-source of the
copula (“is”), which results in an as-graph identical to the lexical as-graph
of the adjective. Then the s-source is filled, plugging the subject into the
correct slot of the adjective. Note that this solution allows us to use the
same DM adjective graph fragment for adnominal and predicative uses.

In practice, the key to automating this transformation is to find the
App${}_{\text{{s}}}$ edge between the adjective and the subject in DM, since
this is the edge we need to replace with the above construction. To find it,
we look at the App${}_{\text{{s}}}$ and App${}_{\text{{o}}}$ children of the
PSD L19 AM tree to identify adjective and subject, and use the
App${}_{\text{{s}}}$ edge between them in the DM L19 AM tree. However, such an
edge does not always exist, for example in cases where subject or adjective
are complex phrases rather than single tokens, and PSD and DM disagree on the
heads of these phrases. In these cases we cannot apply the fix automatically;
the fix works in 86% of the cases.
Further restrictions: “be” lemma in PSD; the two outgoing App edges in PSD AM
dependency tree must be App${}_{\text{{s}}}$ and App${}_{\text{{o}}}$; and the
App${}_{\text{{o}}}$ child of PSD must have a POS tag starting with JJ
(adjective). This covers 64% of the pattern.

###  4.2 Category 2: Headedness

For some constructions, there is disagreement across MRs about which token is
the head of the construction. One such construction is verbal negation
(roughly 2% of divergent phenomena detected), for which we present a pattern
and transformation here:

#### Verbal Negation: [app1 / mod / mod].

For negation of verbs, we find that PSD and PAS treat the negation as
modifier, while DM usually considers the negation to be the head (see Fig.
3(3(a)-3(f))).
Our fix: We make the negation the head in the AM dependency trees for PAS and
PSD graphs. Consider the example PSD dependency tree in Fig. 3(f) and its
transformation in Fig. 3(i) (the process is the same for PAS). In the
transformation of this example, we first reverse the dependency edge between
the negation “not” and negated word “is” and use App instead of mod. We then
shift the incoming dependency edge (here the ‘root’ marker) of “is” to “not”,
making “not” the head of the phrase. To make this work, we also need to change
the graph fragment for “not”. This is because in the actual PSD graph (Fig.
3(c)), “is” is still the root. We can make “is” the root in the resulting
graph, even though “not” is the root of the AM dependency tree, by shifting
the root in the graph fragment for “not” from the lexical label to the neg-
source. This ensures that after the App${}_{\text{{neg}}}$ operation fills
“is” into the neg-source, the root of the resulting graph is actually at “is”.
This transformation resembles type raising in lambda calculus where functor
and argument are exchanged.
Further restrictions: Lemma is “#Neg” or “never” in PSD. This covers 35% of
the pattern; other phenomena adhering to this pattern include some discourse
connectors and adverbs.

Besides conceptual differences in headedness like negation, some phrases
simply do not have an obvious head, e.g. a date like “Nov. 29”. In such cases,
the MRs also often disagree on whether “Nov.” or “29” is the head. We leave
the automatic detection and transformation of such cases to future work.

###  4.3 Category 3: Relational Elements

As a third category, the SDP graphbanks differ in how they mark certain
linguistic relations such as coordination or preposition. We observe two
strategies: (i) the relation is marked using only an edge between the two more
‘contentful’ elements, ignoring the functional word; or (ii) the relation is
marked using a node for the functional word with outgoing edges to the more
contentful elements. These decisions permeate through to the AM dependency
trees. We address the two most common phenomena of this category here:
prepositions and coordination, accounting for roughly 30% of the divergent
phenomena we detect. In both cases, we normalize the AM dependency trees
towards strategy (ii).

#### Prepositions and Their Effects: [conn / conn / empty], [empty / conn /
empty], [basic / basic / mod].

DM and PAS use strategy (ii) for most prepositions and encode them as nodes,
whereas PSD uses strategy (i) and encodes them as edges. See the preposition
“in” in Fig. 1, to which the PSD ‘LOC’ edge corresponds. In the AM trees,
these prepositions have the [conn / conn / empty] signature. Note that in the
PSD graph in this example, the ‘LOC’ edge corresponds to this preposition
instead, which we find to be a general rule: if a preposition is not a node in
the graph, the relation it expresses is instead encoded as an edge. In the
original decomposition of L19, the ‘LOC’ edge here is attached to the lexical
as-graph of “house” yielding the mod pattern, a common cause for the [basic /
basic / mod] signature. Thus, normalizing the preposition case addresses two
of the most common pattern differences at once. Some prepositions such as
possessives (“of”) are encoded as edges in DM, too, yielding the [empty / conn
/ empty] signature.
Our fix: In the graphs that encode the preposition as an edge, we use that
edge as the lexical as-graph for the preposition, such as the lexical as-graph
for “in” in Fig. 4(e). This new lexical as-graph consists of only the ‘LOC’
edge, with the root node at its origin, and sources s and o. At the same time,
we remove that edge from the lexical as-graph that contained it before (here
“house”). We then update the AM dependency edges accordingly, for example in
the transformation from Fig. 4(b) to Fig. 4(e), we remove the
mod${}_{\text{{mod}}}$ edge from “playing” to “house” and instead connect the
new constant for “in” with an outgoing App${}_{\text{{S}}}$ edge to “house”
and an incoming mod${}_{\text{{O}}}$ edge from “playing”, creating the conn
pattern now in PSD as well.

To find the edge in PSD in the general case, we use a similar strategy as we
employed for copula: we look at the parent of the preposition in PAS
(“playing” in the example) as well as the App-child of the preposition
(“house”), and consider the PSD AM tree edge between them (here
mod${}_{\text{{mod}}}$). We then identify the graph edge (here ‘LOC’) via the
source of the AM tree operation: the ‘LOC’ edge here is the edge from “house”
to the mod-source. Such a matching PSD AM tree edge exists in 86% of the
cases; in the other cases PAS and PSD disagree on the heads of the preposition
arguments and we cannot apply an automatic transformation.

Note that this transformation also changes the mod pattern at “house” to the
more consistent basic pattern. In total, we have changed a [conn / conn /
empty] and a [basic / basic / mod] signature to a [conn / conn / conn] and a
[basic / basic / basic] signature. In the [empty / conn / empty] case, we
employ the same strategy to change the AM trees for both DM and PSD.
Further restrictions: IN or TO POS tag at the preposition (covers 99% of [conn
/ conn / empty] and 66% of [empty / conn / empty]).

#### Binary Coordination: [empty / app2 / app2], [empty / other / other].

[sdpgraph] [column sep=1em] Mice & run & and & hide
[edge unit distance=2ex]2ROOT [edge unit distance=1ex]21ARG1 [edge unit
distance=1ex]41ARG1 [edge unit distance=.5ex]24_and_c

(a) DM graph

[sdpgraph] [column sep=3em] Mice & run & and & hide
[edge unit distance=2ex]3ROOT [edge unit distance=1ex]21verb_ARG1 [edge unit
distance=1ex]41verb_ARG1 [edge unit distance=1ex]32coord_ARG1 [edge unit
distance=1ex]34coord_ARG2

(b) PAS graph

[amdep] [column sep=.5cm]Mice & run & and & hide
& & &
[edge unit distance=2ex]3root [edge unit distance=2ex]31App${}_{\text{{s}}}$
[edge unit distance=2ex]32App${}_{\text{{op1}}}$ [edge unit
distance=2ex]34App${}_{\text{{op2}}}$ (n1) [below of = 21] ; (n2) [below of =
22] ; (n3) [below of = 23] ; (n4) [below of = 24] ;

(c) Normalized AM dependency tree for DM

[amdep] [column sep=.5cm]Mice & run &[+6mm] and &[+6mm] hide
& & &
[edge unit distance=2ex]3root [edge unit distance=2ex]31App${}_{\text{{s}}}$
[edge unit distance=2ex]32App${}_{\text{{op1}}}$ [edge unit
distance=2ex]34App${}_{\text{{op2}}}$ (n1) [below of = 21] ; (n2) [below of =
22] ; (n3) [below of = 23] ; (n4) [below of = 24] ;

(d) L19 AM tree for PAS

(e) Partial result of (6(c)) after App${}_{\text{{op1}}}$ and
App${}_{\text{{op2}}}$ operations

(f) Partial result of (6(d)) after App${}_{\text{{op1}}}$ and
App${}_{\text{{op2}}}$ operations

Figure 6: Coordination of two verbs with a common subject.

To connect two conjuncts in coordination, such as “cats” and “dogs” in “cats
and cute dogs” in Fig. 1, DM uses an edge (here ‘_and_c’; strategy (i)), while
PAS and PSD make the conjunction “and” the head of the phrase with outgoing
edges to the conjuncts (strategy (ii)). Thus, both PSD (as preprocessed by
L19) and PAS show the same app2 pattern in the L19 AM trees (Fig. 4(4(b),
4(c))), while DM exhibits the empty pattern. If both conjuncts have a common
argument, such as in Fig. 6(b), the source for the common argument is first
passed up through the coordination before being filled with the argument,
resulting in an additional App child for the coordination (see Fig. 6(6(d),
6(f))). Thus, in such cases PAS and PSD L19 AM trees exhibit the other
pattern. In conclusion, binary coordination can have the [empty / app2 / app2]
or [empty / other / other] signatures.

Binary coordination makes up approximately 86% of all coordinations; we leave
the more challenging case of three or more conjuncts to future work.
Our fix: Similar to the preposition fix, we add a lexical as-graph for the
conjunction token (e.g. “and”) in DM containing the dedicated conjunction edge
that was previously part of one of the conjuncts lexical as-graph (compare
Fig. 4(a) and 4(d)). The incident nodes use op1\- and op2-sources to indicate
the open positions for the conjuncts, and we connect the conjuncts as
App${}_{\text{{op1}}}$ and App${}_{\text{{op2}}}$ children accordingly. One of
the conjuncts was previously the head of the coordination in DM (here “cats”),
we shift its incoming edge (here App${}_{\text{{S}}}$) to the conjunction
instead. The root of the conjunction’s lexical as-graph is placed at the
op1-source where “cats” will be placed, such that all outside graph edges
attach correctly. If the conjuncts share a common argument like the subject
“mice” in Fig. 6(a), we add respective source annotations to the op1\- and
op2-sources (Fig. 6(c)). Fig. 6(e) shows the partial result after evaluating
the App${}_{\text{{op1}}}$ and App${}_{\text{{op2}}}$ operations in this
example.
Further restrictions: CC POS tag, exactly two outgoing ‘*.member’ edges in PSD
(these are the coordination edges, exactly two ensures binary coordination).
This covers 89% of the two patterns.

##  5 Evaluation

We evaluate the quality of the normalized AM dependency trees by measuring
cross-formalism similarity as well as their suitability for parsing on the SDP
corpora of the 2015 shared task [Oepen et al., 2015]. We apply L19’s
preprocessing of coordination for PSD (see also the discussion in Section 6).
Due to the similarities of the graphbanks, it is very appealing to use multi-
task learning (MTL), which shares a subset of the model weights across
graphbanks in order to let the training data for one graphbank influence the
prediction on another. We expect this to help particularly in scenarios where
there is limited data for one graphbank but large amounts for others.

###  5.1 Structural Similarity

Tables 1 and 3 summarize the structural comparison of the graphs and AM
dependency trees, respectively. Note that the F-scores on the graphs don’t
take the edge labels into account. We additionally calculate the percentage of
lexical as-graphs that change by graph formalism as a result of our
normalization: 13.53% for DM, .35% for PAS, and 33.22% for PSD.

There is a consistent increase in similarity by at least 12.8 points F-score
of the normalized AM dependency trees in comparison to the L19 AM trees over
all metrics. The labeled F-score that disregards the source names is the upper
bound on how much similarity can be achieved by systematically making the
source names more uniform as well.

Comparing the similarities of the normalized AM dependency trees to the
similarities of the graphs, there is an even larger gain in the unlabeled,
directed F-score of up to 52.7 points. Importantly, we only compute F-scores
on the edges of the AM dependency trees, not on the as-graph constants. The
higher similarity in the tree structure is achieved primarily by moving
discrepancies to the lexicon. In particular, this is true for edge directions
and edge labels, which are specified in the lexicon, not in the compositional
structure.

|  | L19 |  $\Delta$Det |  $\Delta$Aux |  $\Delta$Prep |  $\Delta$Coord |  $\Delta$Copula |  $\Delta$Neg |  $\Delta$ Punct | All changes
---|---|---|---|---|---|---|---|---|---|---
|  | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD | PAS | PSD
DM | UF | 63.5 | 55.7 | 0.0 | 6.8 | 1.3 | 0.8 | 3.3 | 9.9 | 2.8 | 3.2 | 0.6 | 0.6 | 0.5 | 0.5 | 4.3 | 1.3 | 76.3 | 78.8
A/M F | 57.8 | 46.7 | 0.0 | 7.2 | 1.3 | 1.0 | 3.4 | 11.3 | 2.8 | 3.2 | 0.6 | 0.7 | 0.5 | 0.5 | 4.5 | 1.8 | 70.9 | 72.4
LF | 29.7 | 26.2 | 0.0 | 8.1 | 1.6 | 1.4 | 3.6 | 1.3 | 2.6 | 2.9 | 0.6 | 0.7 | 0.5 | 0.4 | 5.3 | 3.9 | 43.9 | 44.9
PAS | UF |  | 57.0 |  | 5.9 |  | 1.3 |  | 13.4 |  | 0.0 |  | 0.0 |  | 0.0 |  | 4.4 |  | 82.0
A/M F |  | 50.1 |  | 6.1 |  | 1.4 |  | 13.7 |  | 0.0 |  | 0.0 |  | 0.0 |  | 4.6 |  | 75.9
LF |  | 36.8 |  | 6.7 |  | 1.5 |  | 14.2 |  | 1.1 |  | 0.0 |  | 0.0 |  | 4.9 |  | 65.2

Table 3: Directed F-scores between AM dependency trees at different stages of
normalization. UF is unlabeled F-score, A/M F is labeled F-score disregarding
source names and LF is the usual labeled F-score.

###  5.2 Suitability for Parsing

An important question to ask is if the normalization of the AM dependency
trees has an impact on how accurate parsing is. In particular, we expect the
high similarity to be beneficial for projecting from a large corpus to a
smaller one. Therefore, we conduct parsing experiments in the usual high-
resource scenario and also in a simulated low-resource scenario, for which we
sampled 3 subsets of the training data with 100 instances each and used those
subsets throughout all low-resource experiments.

We use the parser of L19 but, in contrast, we use only those training
instances for which we have AM dependency trees in _all_ formalisms, resulting
in 29,200 training instances. Moreover, we don’t use embeddings for lemmas
because of data sparsity in the low-resource condition. We follow their way of
performing MTL, that is, we share an LSTM over all tasks in addition to using
task-specific ones. Scores for edges and lexical as-graphs are produced from
single-layer feedforward networks that take the concatenation of the output of
the shared and task-specific LSTM as input. MTL experiments use the full
training data of the non-target formalism and the specified amount for the
target task; no data beyond the SDP corpora was used for the MTL experiments.
For full details of hyperparameters, see appendix B.

Table 4 summarizes the results. MTL in the low-resource scenario is hugely
beneficial and works better with the normalized AM dependency trees. We think
that the slight to moderate decrease in performance of the normalized trees in
the full data condition comes from the noise the normalization creates because
not all phenomena are detected perfectly and sometimes we restrict fixes to
specific instances only.

|  | L19 AM tree |  | Normalized AM dependency trees
---|---|---|---|---
Training data | 100 | full |  | 100 | full
|  | Single | MTL | $\Delta$ | Single | MTL |  | Single | MTL | $\Delta$ | Single | MTL
DM | id F | 65.6  $\pm 1.4$ | 72.2  $\pm 0.9$ | 6.6 | 92.8  $\pm 0.1$ | 92.9  $\pm 0.1$ |  | 66.1  $\pm 1.2$ | 72.9  $\pm 0.2$ | 6.8 | 92.3  $\pm 0.2$ | 92.2  $\pm 0.1$
ood F | 63.3  $\pm 1.3$ | 69.1  $\pm 1.4$ | 5.8 | 88.9  $\pm 0.1$ | 88.9  $\pm 0.1$ |  | 63.0  $\pm 1.2$ | 69.5  $\pm 0.5$ | 6.5 | 88.0  $\pm 0.3$ | 87.7  $\pm 0.3$
PAS | id F | 72.6  $\pm 1.2$ | 81.5  $\pm 0.2$ | 8.9 | 94.7  $\pm 0.1$ | 94.6  $\pm 0.1$ |  | 72.5  $\pm 1.4$ | 84.1  $\pm 0.8$ | 11.5 | 94.6  $\pm 0.1$ | 94.5  $\pm 0.1$
ood F | 69.2  $\pm 2.0$ | 78.9  $\pm 0.7$ | 9.7 | 92.6  $\pm 0.1$ | 92.6  $\pm 0.1$ |  | 69.0  $\pm 1.6$ | 80.7  $\pm 1.0$ | 11.7 | 92.5  $\pm 0.1$ | 92.6  $\pm 0.1$
PSD | id F | 51.1  $\pm 0.7$ | 58.9  $\pm 1.2$ | 7.8 | 81.2  $\pm 0.1$ | 80.9  $\pm 0.3$ |  | 52.9  $\pm 1.8$ | 60.2  $\pm 0.9$ | 7.4 | 80.5  $\pm 0.1$ | 80.3  $\pm 0.1$
ood F | 50.3  $\pm 2.2$ | 58.2  $\pm 1.8$ | 7.9 | 80.3  $\pm 0.1$ | 80.1  $\pm 0.2$ |  | 51.6  $\pm 1.2$ | 59.1  $\pm 0.8$ | 7.5 | 79.7  $\pm 0.1$ | 79.5  $\pm 0.1$

Table 4: Means and standard deviations of labeled F-scores on test sets for
varying amounts of training data and use of MTL. Computed over 3 data samples
with replacement and different random initializations.

##  6 Discussion

The methodology we detail in Sections 3 and 4 allows us to normalize roughly
60$\%$ of differences across compositional structures of the SDP graphbanks.
Our work establishes that a small set of linguistically-grounded
transformations are quite powerful towards creating uniform compositional
structure. In future work, we plan to automate our methodology to broaden the
range of phenomena we are able to normalize.

The individual phenomena that create inconsistencies across graph
representations and their corresponding L19 AM trees predictably adhere to a
Zipfian distribution (Appendix A). Currently, we do not account for the
interaction of these phenomena, such as negation combined with auxiliary verbs
(“Mary will/might/should not sleep”) or adjectival copula with coordination
(“the tree is tall and green”). We find that the order of implementing our
fixes as shown in Table 3 minimally impacts F-scores and plan to explore this
in more depth. Automating our methodology will enable us to better address
these complex interactions and shed further light on how they impact
compositional structure.

Although the AM+ algebra we present to handle discrepancies in compositional
structure is more flexible than the L19 AM algebra, there are a few phenomena
that still pose challenges. In particular, this applies to the way
coordination is originally handled in PSD. The coordinate structure in Fig.
1(c) represents the PSD graph after the (reversible) preprocessing of L19. In
a PSD graph without preprocessing, there would be two ‘ACT-arg’ edges,
pointing to “cats” and “dogs”. The AM+ algebra cannot handle this original
structure unless we allow the lexical as-graph for “playing” to have two ‘ACT-
arg’ edges, an analysis that is questionable from a syntactic point of view.

A final challenge we leave to future work involves better integrating our
methodology with cross-framework MR parsing and MTL in general. Once we
normalize representations across AM dependency trees, our target annotation
has to account for all distinctions the different formalisms make. This makes
parsing more difficult but we also believe this is where MTL can be of most
use. The results obtained in the simulated resource scenario (Section 5) are
promising first steps towards MR parsing in the low-resource scenario that is
often faced when developing new MRs. We expect that, by automating our
methodology to capture a broader range of phenomena, this will not only have
further positive effects for the similarity of the compositional structures of
MRs but will also make it more accessible to the community and help in the
annotation process of future graphbanks.

##  7 Conclusion

We have shown how annotations from different graphbanks (specifically, DM,
PAS, and PSD) can be normalized at the level of their compositional structure.
We achieve this by updating the AM algebra to the AM+ Algebra, which allows
more flexibility in adapting compositional structures across MRs. Our new
methodology includes a focused study of design differences between formalisms
based on specific linguistic phenomena and more general pattern-based graph
analysis. By working at the compositional level of AM dependency trees, we are
able to quantify mismatches between different graphbanks, systematically
reshape these mismatches to make them more uniform across graphbanks, and
subsequently increase the match between the compositional structures for the
three graphbanks. Such work contributes to a broader effort to increase
parallel MR parsing accuracy.

As it is difficult to isolate and manually address all linguistic and
structural discrepancies between graphbanks, we plan to extend the methodology
we present in this paper to a system that is able to automatically detect and
normalize mismatches. One possibility for this work includes detecting
compositional mismatches during the algorithmic decomposition process itself
[Groschwitz et al., 2018]. We also want continue exploring how normalization
across graphbanks can bolster results with multi-task-learning, particularly
in low-resource scenarios. As part of this work, we plan to extend our work
normalizing compositional representations of MRs to formalisms with design
architectures quite different from those discussed here.

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##  Appendix A Additional Pattern Signatures and Frequencies

See Table 5.

Rank |  Pattern signature ([DM / PAS / PSD]) |  Most commonly associated phenomena | Count | Percentage |  Cumulative percentage |  We fix
---|---|---|---|---|---|---
1 | [mod / mod / empty] | Determiners | $49\,965$ |  $13.38$% |  $13.38$% | ✓
2 | [empty / mod / empty] | Punctuation | $45\,544$ |  $12.20$% |  $25.58$% | ✓
3 | [conn / conn / empty] | Prepositions | $42\,395$ |  $11.35$% |  $36.93$% | ✓
4 | [basic / basic / mod] | NPs with prep.* | $30\,202$ |  $8.09$% |  $45.02$% | ✓
5 | [empty / conn / empty] | Prepositions | $22\,324$ |  $5.98$% |  $51.00$% | ✓
6 | [mod / basic / basic] | Named Entities* | $15\,835$ |  $4.24$% |  $55.24$% | –
7 | [mod / basic / mod] | Named Entities* | $14\,234$ |  $3.81$% |  $59.05$% | –
8 | [empty / cmod / empty] | Temporal auxiliaries | $14\,074$ |  $3.77$% |  $62.82$% | ✓
9 | [mod / mod / basic] | Named Entities* | $7844$ |  $2.10$% |  $64.93$% | –
10 | [app1 / mod / mod] | Verbal Negation* | $7215$ |  $1.93$% |  $66.86$% | ✓
11 | [basic / basic / app1] | Named Entities* | $6670$ |  $1.79$% |  $68.64$% | –
12 | [app1 / basic / basic] | Named Entities* | $6595$ |  $1.77$% |  $70.41$% | –
13 | [basic / mod / mod] | Named Entities* | $5714$ |  $1.53$% |  $71.94$% | –
14 | [empty / app2 / app2] | Coordination | $5709$ |  $1.53$% |  $73.47$% | ✓
15 | [basic / mod / basic] | $ signs | $5008$ |  $1.34$% |  $74.81$% | –
16 | [empty / mod / basic] | Relative pronouns | $3703$ |  $0.99$% |  $75.80$% | –
17 | [app2 / conn / empty] | Subordinating conjs. | $3609$ |  $0.97$% |  $76.77$% | –
18 | [app1 / cmod / empty] | Modal auxiliaries | $3431$ |  $0.92$% |  $77.69$% | –
19 | [empty / conn / app2] | Punctuation | $2952$ |  $0.79$% |  $78.48$% | –
20 | [empty / capp2 / app2] | Copula (adjectival) | $2915$ |  $0.78$% |  $79.26$% | ✓
21 | [empty / app2 / other] | Coordination | $2848$ |  $0.76$% |  $80.02$% | ✓
22 | [empty / other / other] | Coordination | $2828$ |  $0.76$% |  $80.78$% | ✓
23 | [app1 / basic / app1] | NCP | $2712$ |  $0.73$% |  $81.51$% | –
24 | [app2 / app2 / other] | Particle verbs | $2612$ |  $0.70$% |  $82.21$% | –
25 | [conn / mod / empty] | Determiners* | $2234$ |  $0.60$% |  $82.80$% | –
26 | [empty / mod / mod] | Adverbs* | $2119$ |  $0.57$% |  $83.37$% | –
27 | [basic / basic / conn] | Named Entities* | $1992$ |  $0.53$% |  $83.90$% | –
28 | [conn / conn / other] | NCP | $1851$ |  $0.50$% |  $84.40$% | –
29 | [mod / conn / empty] | Subordinating conjs. | $1845$ |  $0.49$% |  $84.89$% | –
30 | [empty / app2 / empty] | Punctuation | $1842$ |  $0.49$% |  $85.39$% | –
Table 5: Most frequent pattern signature differences across the three
graphbanks. Asterisks indicate that additional phenomena comprise a
significant portion of the pattern. NCP denotes ‘no common pattern’ or
patterns that lack a single most common corresponding phenomenon.

##  Appendix B Hyperparameters

We use the implementation of L19: https://github.com/coli-saar/am-parser. The
hyperparameters common to all experiment are collected in Table 6. We train
the parser for 100 epochs and pick the model with the highest performance on
the development set after epoch 25. We perform early stopping with patience of
10 epochs.

We use the ”large-uncased” BERT model as available through AllenNLP and don’t
fine-tune.

Activation function in all MLPs | tanh
---|---
Optimizer | Adam
Learning rate | 0.001
Epochs | 100
Dim of lemma embeddings | 0
Dim of POS embeddings | 32
Dim of NE embeddings | 16
Hidden layers in all MLPs | 1
Hidden units in LSTM (per direction) | 256
Hidden units in edge existence MLP | 256
Hidden units in edge label MLP | 256
Hidden units in supertagger MLP | 1024
Hidden units in lexical label tagger MLP | 1024
Layer dropout in LSTMs | 0.3
Recurrent dropout in LSTMs | 0.4
Input dropout | 0.3
Dropout in edge existence MLP | 0.0
Dropout in edge label MLP | 0.0
Dropout in supertagger MLP | 0.4
Dropout in lexical label tagger MLP | 0.4
Table 6: Common hyperparameters used in all experiments. Deviations from L’19
are highlighted with boldface.

#### Single Task

We use a batch size of 16 sentences.

#### MTL

In our MTL experiments, we have one LSTM per graphbank and one that is shared
between the graphbanks. When we compute scores for a sentence, we run it
through its graphbank-specific LSTM and the shared one. We concatenate the
outputs and feed it to graphbank-specific MLPs. We use a separate LSTM for the
edge model (input to edge existence and edge label MLP) and the supertagging
model.

In effect, we have two LSTMs that are shared over the graphbanks: one for the
edge model and one for the supertagging model. All LSTMs have the
hyperparameters detailed in Table 6. We use a batch size of 16 for the target
formalism and a batch size of 64 for all non-target formalisms. Each batch
consists of instances from only one formalism.