problem[2][]

#2
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# Context-Preserving Text Simplification

Christina Niklaus13, Matthias Cetto1, André Freitas2 Siegfried Handschuh13
1 University of St.Gallen
{christina.niklaus, matthias.cetto<EMAIL_ADDRESS>
2 University of Manchester
<EMAIL_ADDRESS>
3 University of Passau
{christina.niklaus<EMAIL_ADDRESS>

###### Abstract

We present a context-preserving text simplification (TS) approach that
recursively splits and rephrases complex English sentences into a semantic
hierarchy of simplified sentences. Using a set of linguistically principled
transformation patterns, input sentences are converted into a hierarchical
representation in the form of core sentences and accompanying contexts that
are linked via rhetorical relations. Hence, as opposed to previously proposed
sentence splitting approaches, which commonly do not take into account
discourse-level aspects, our TS approach preserves the semantic relationship
of the decomposed constituents in the output. A comparative analysis with the
annotations contained in the RST-DT shows that we are able to capture the
contextual hierarchy between the split sentences with a precision of 89% and
reach an average precision of 69% for the classification of the rhetorical
relations that hold between them.

##  1 Introduction

Sentences that present a complex linguistic structure can be hard to
comprehend by human readers, as well as difficult to analyze by semantic
applications. Identifying grammatical complexities in a sentence and
transforming them into simpler structures is the goal of syntactic TS. One of
the major types of operations that are used to perform this rewriting step is
sentence splitting: it divides a sentence into several shorter components,
with each of them presenting a simpler and more regular structure that is
easier to process by both humans Siddharthan and Mandya (2014); Saggion et al.
(2015); Ferrés et al. (2016) and machines Štajner and Popovic (2016); Štajner
and Popović (2018); Saha and Mausam (2018).

A fluoroscopic study which is known as an upper gastrointestinal series is
typically the next step in management, although if volvulus is suspected,
caution with non water soluble contrast is mandatory as the usage of barium
can impede surgical revision and lead to increased post operative
complications. Input: Complex Sentence

$\downarrow$

ROOTContrast(a)Elaboration(b)A fluoroscopic
study is
typically
the next step
in management.A fluoroscopic
study is known
as an upper
gastrointestinal
series.Condition(c)Volvulus is
suspected.Background(d)Caution
with non
water soluble
contrast
is mandatory.List(e)The usage of
barium can
impede
surgical revision.The usage of
barium can lead
to increased post
operative
complications.Transformation Stage Semantic Hierarchy of Minimal Propositions

Figure 1: A complex sentence is transformed into a semantic hierarchy of
simplified sentences in the form of minimal, self-contained propositions that
are linked to each other via rhetorical relations. The output presents a
regular, fine-grained structure that preserves the context of the input in the
form of hierarchically ordered and semantically interconnected sentences.

We propose a sentence splitting approach that can be used as a preprocessing
step to generate an intermediate representation. The objective is to
facilitate and improve the performance of downstream tasks whose predictive
quality deteriorates with sentence length and complexity. Our approach aims to
break down a complex sentence into a set of minimal propositions, i.e. a
sequence of sound, self-contained utterances with a simple and regular
structure. Each of them presents a minimal unit of coherent information and
cannot be further decomposed into meaningful propositions.

However, any sound and coherent text is not simply a loose arrangement of
self-contained units, but rather a logical structure of utterances that are
semantically connected Siddharthan (2014). Consequently, when carrying out
syntactic TS operations without considering discourse implications, the
rewriting may easily result in a disconnected sequence of simplified
sentences, making the text harder to interpret. The vast majority of existing
structural TS approaches though do not take into account discourse-level
aspects. Therefore, they are prone to producing a set of incoherent utterances
where important contextual information is lost. Thus, in order to preserve the
coherence structure of the input we propose a context-preserving TS approach.
It establishes a semantic hierarchy between the split components by (1)
setting up a contextual hierarchy and (2) classifying the semantic
relationship that holds between them (see Figure 1).

To the best of our knowledge, this is the first time that syntactically
complex sentences are split and rephrased within the semantic context in which
they occur. Our framework differs from previously proposed approaches by using
a linguistically grounded transformation stage that applies clausal and
phrasal disembedding mechanisms to transform sentences into shorter utterances
with a more regular structure. By using a recursive top-down approach, it
generates a novel hierarchical representation between those units, capturing
both their semantic context and relations to other units in the form of
rhetorical relations. By taking advantage of the resulting fine-grained
representation, the complexity of downstream tasks may be reduced, thus
improving their performance. In addition, by incorporating the semantic
context of the source sentences, our proposed representation preserves
important contextual information that is needed to maintain the coherence
structure of the input, allowing for a proper interpretation of complex
assertions.

In summary, we make the following contributions: (i) We conduct a systematic
analysis of syntactic phenomena involved in complex sentence constructions
whose findings are materialized in a small set of 35 hand-crafted
transformation rules. (ii) We propose a context-preserving syntactic TS
approach which transforms complex sentences into a semantic hierarchy of
minimal propositions. (iii) The proposed method is linguistically grounded and
does not require any training data. (iv) As a proof of concept, we develop a
reference implementation. (v) We perform a comprehensive empirical evaluation,
demonstrating that we reach state-of-the-art performance in the classification
of both the hierarchical order and the semantic relationship that hold between
the split sentences.

##  2 Context-Preserving Sentence Splitting

We present DisSim, a context-preserving TS approach that creates a semantic
hierarchy of simplified sentences.111The source code of our framework is
publicly available for download under the following link:
https://github.com/Lambda-3/DiscourseSimplification. It takes a sentence as
input and performs a recursive transformation stage that is based upon a small
set of 35 hand-crafted rules.

###  2.1 Transformation Patterns

ROOT $<<:$ (S $<$ (NP $\$..$ (VP $<+$(VP) (SBAR $<,$ (IN $\$+$ (S $<$ (NP
$\$..$ VP)))))))
---
Table 1: Example of a transformation pattern (for decomposing adverbial
clauses). They are specified in terms of Tregex patterns Levy and Andrew
(2006). A boxed pattern represents the part of a sentence that is extracted
from the input and transformed into a new stand-alone sentence. A pattern in
bold is deleted from the source. The underlined part is labelled as a context
sentence, while the remaining part represents core information. The italic
pattern is used as a cue phrase for identifying the rhetorical relation that
holds between the decomposed spans.

In the development of the transformation patterns, we followed a principled
and systematic procedure, with the goal of eliciting a universal set of
transformation rules. They were heuristically determined in a rule-engineering
process that was carried out on the basis of an in-depth study of the
literature on syntactic sentence simplification, e.g. Siddharthan (2006, 2014,
2002); Siddharthan and Mandya (2014); Evans and Orǎsan (2019); Heilman and
Smith (2010); Mitkov and Saggion (2018); Mallinson and Lapata (2019); Ferrés
et al. (2016). Next, we performed a thorough linguistic analysis of the
syntactic phenomena that need to be tackled in the sentence splitting task.
Details on the underlying linguistic principles, supporting the systemacity
and universality of the developed transformation patterns, can be found in
Section A in the appendix.

The transformation patterns encode syntactic and lexical features that can be
derived from a sentence’s phrase structure. Each rule specifies (1) how to
split up and rephrase the input into structurally simplified sentences and (2)
how to set up a contextual hierarchy between the split components and how to
identify the semantic relationship that holds between those elements. An
example of a transformation rule is provided in Table 1.222For reproducibility
purposes, the full set of patterns can be found online:
https://github.com/Lambda-3/DiscourseSimplification/tree/master/supplemental_material.

###  2.2 Data Model: Linked Proposition Tree

The transformation algorithm takes a complex sentence as input and recursively
transforms it into a semantic hierarchy of minimal propositions. The output is
represented as a linked proposition tree. Its basic structure is depicted in
Figure 2. A linked proposition tree is a labeled binary tree $LPT=(V,E)$.

$prop\in PROP$$prop\in PROP$$rel\in REL$$c\in CL$$c\in CL$ Figure 2: Basic
structure of a linked proposition tree $LPT$. It represents the data model of
the semantic hierarchy of minimal propositions.

Let $V\in\\{REL,PROP\\}$ be the set of nodes, where $PROP$ is the set of leaf
nodes denoting the set of minimal propositions. A $prop\in PROP$ is a triple
$(s,v,o)\in CT$, where $CT=\\{$SV, SVA, SVC, SVO, SVOO, SVOA, SVOC$\\}$
represents the set of clause types Del Corro and Gemulla (2013). Hence, $s\in
S$ denotes a subject, $v\in V$ a verb and $o\in\\{O,A,C,OO,OA,OA,\emptyset\\}$
a direct or indirect object, adverbial or complement (or a combination
thereof). Accordingly, a minimal proposition $prop\in PROP$ is a simple
sentence333A simple sentence comprises exactly one clause. that is reduced to
its clause type.444In addition, a specified set of phrasal elements were
extracted. The interested reader may refer to Section A in the appendix for
more details. Thus, it represents a minimal unit of coherent information where
all optional constituents are discarded, resulting in an utterance that
expresses a single event consisting of a predicate and its core arguments.

Furthermore, let $REL=\\{$Contrast, List, Disjunction, Cause, Result,
Temporal, Background, Condition, Elaboration, Explanation, Spatial,
Attribution, Unknown$\\}$ be the set of rhetorical relations, comprising the
set of inner nodes. A $rel\in REL$ represents the semantic relationship that
holds between its child nodes. It reflects the semantic context of the
associated propositions $prop\in PROP$. In that way, the coherence structure
of the input is preserved.

Finally, let $E\in CL$, with $CL\in\\{$core, context$\\}$, be the set of
constituency labels. A $c\in CL$ represents a labeled edge that connects two
nodes $V\in LPT$. It enables the distinction between core information and less
relevant contextual information. In that way, hierarchical structures between
the split propositions $prop\in PROP$ are captured. Figure 1 shows a concrete
example of the linked proposition tree that is generated by our TS approach on
an example sentence.

###  2.3 Transformation Algorithm

The transformation algorithm of our approach is given in Algorithm 1. It takes
a sentence as input and applies the transformation patterns to recursively
transform it into a semantic hierarchy of minimal propositions, represented as
an $LPT$.

Algorithm 1 Transform into Semantic Hierarchy

1:complex source sentence $str$

2:linked proposition tree $tree$

3:

4:function Initialize($str$)

5: $new\\_leaves$ $\leftarrow$ source sentence $str$

6: $new\\_node\leftarrow$ create a new parent node for $new\\_leaves$

7: $new\\_node.labels$ $\leftarrow$ None

8: $new\\_node.rel$ $\leftarrow$ ROOT

9: linked proposition tree $tree$ $\leftarrow$ initialize with $new\\_node$

10: return $tree$

11:end function

12:

13:procedure TraverseTree($tree$)

14: $\triangleright$ Process leaves (i.e. propositions) from left to right

15: for $leaf$ in $tree.leaves$ do

16: $\triangleright$ Check transformation rules in fixed order

17: for $rule$ in $TRANSFORM\\_RULES$ do

18: if $match$ then

19: $\triangleright$ (a) Sentence splitting

20: $simplified\\_propositions\leftarrow$ decompose $leaf$ into a

21: set of simplified propositions

22: $new\\_leaves\leftarrow$ convert $simplified\\_propositions$

23: into leaf nodes

24: $\triangleright$ (b) Constituency Type Classification

25: $new\\_node\leftarrow$ create a new parent node for $new\\_leaves$

26: $new\\_node.labels\leftarrow$ link each leaf in $new\\_leaves$ to

27: $new\\_node$ and label each edge with the leaf’s constituency

28: type $c\in CL$

29: $\triangleright$ (c) Rhetorical Relation Identification

30: $cue\\_phrase\leftarrow$ extract cue phrase from $leaf.parse\\_tree$

31: $new\\_node.rel\in REL\leftarrow$ match $cue\\_phrase$ against a

32: predefined set of rhetorical cue words

33: $\triangleright$ Update Tree

34: $tree.replace(leaf,new\\_node)$

35: $\triangleright$ Recursion

36: TraverseTree($tree$)

37: end if

38: end for

39: end for

40: return $tree$

41:end procedure

##### Initialization

In the initialization step (1-7), the linked proposition tree $LPT$ is
instantiated. At this point, it consists of the source sentence, which is
represented as a single leaf node, and an unlabeled edge to the root node.

##### Tree Traversal

Next, the linked proposition tree is recursively traversed, splitting up the
input in a top-down approach (9). Starting from the root node, the leaves are
processed in depth-first order. For every leaf (11), we check if its phrasal
parse tree matches one of the transformation patterns (13). The rules are
applied in a fixed order that was empirically determined.555See Section B in
the appendix. The first pattern that matches the proposition’s parse tree will
be employed (14). For instance, the first rule that matches the source
sentence from Figure 1 is the pattern depicted in Table 1.

##### (a) Sentence Splitting

In a first step, the current proposition is decomposed into a set of shorter
utterances that present a more regular structure (15, 17). This is achieved
through disembedding clausal or phrasal components and converting them into
stand-alone sentences. Accordingly, the transformation rule encodes both the
split point and the rephrasing procedure for reconstructing grammatically
sound sentences. Table 2 provides an overview of the linguistic constructs
that are tackled by our approach.666Note that this subtask is presented in
detail in Niklaus et al. (2019b) and Niklaus et al. (2019a). Therefore, we
focus on subtasks b and c in this work. Each split will create two sentences
with a simpler syntax. They are represented as leaf nodes in the linked
proposition tree (18, 19) (see subtask a in Fig. 3). To establish a semantic
hierarchy between the split spans, two further subtasks are carried out.

| Clausal/Phrasal type | Hierarchy | # rules
---|---|---|---
Clausal disembedding
1 | Coordinate clauses | coordinate | 1
2 | Adverbial clauses | subordinate | 6
3a | Relative clauses (non-restrictive) | subordinate | 5
3b | Relative clauses (restrictive) | subordinate | 4
4 | Reported speech | subordinate | 4
Phrasal disembedding
5 | Coordinate verb phrases | coordinate | 1
6 | Coordinate noun phrases | coordinate | 2
6 | Participial phrases | subordinate | 4
8a | Appositions (non-restrictive) | subordinate | 1
8b | Appositions (restrictive) | subordinate | 1
9 | Prepositional phrases | subordinate | 3
10 | Adjectival and adverbial phrases | subordinate | 2
11 | Lead NPs | subordinate | 1
| Total |  | 35
Table 2: Linguistic constructs addressed by our proposed context-preserving TS
approach DisSim.

##### (b) Constituency Type Classification

To set up a contextual hierarchy between the split sentences, the
transformation rule determines the constituency type $c\in CL$ of the leaf
nodes that were created in the previous step (22-24). To differentiate between
core sentences that contain the key message of the input and contextual
sentences that provide additional information about it, the transformation
pattern encodes a simple syntax-based method. Based on the assumption that
subordinations commonly express background information, simplified
propositions resulting from subordinate clausal or phrasal elements are
classified as context sentences, while those emerging from their superordinate
counterparts are labelled as core sentences. Coordinations, too, are flagged
as core sentences, since they are of equal status and typically depict the
main information of the input (see subtask b in Fig. 3).777This approach
relates to the concept of nuclearity in RST. For details, see Section C in the
appendix.

##### (c) Rhetorical Relation Identification

To preserve the semantic relationship between the simplified propositions, we
classify the rhetorical relation $rel\in REL$ that holds between them. For
this purpose, we utilize a predefined list of rhetorical cue words adapted
from the work of Taboada and Das (2013).888The full list of cue phrases that
serve as lexical features for the identification of rhetorical relations is
provided in the appendix in Section D. To infer the type of rhetorical
relation, the transformation pattern first extracts the cue phrase of the
given sentence (26). It is then used as a lexical feature for classifying the
type of rhetorical relation that connects the split propositions (27, 28). For
example, the rule in Table 1 specifies that the phrase “although” is the cue
word in the source sentence of Fig. 1, which is mapped to a “Contrast”
relationship according to the findings in Taboada and Das (2013) (see subtask
c in Figure 3).

(c) “although” $\rightarrow$ Contrast Figure 3: Semantic hierarchy after the
first transformation pass. (Subtask a) The source sentence is split up and
rephrased into a set of syntactically simplified sentences. (Subtask b) Then,
the split sentences are connected with information about their constituency
type to establish a contextual hierarchy between them. (Subtask c) Finally, by
identifying and classifying the rhetorical relation that holds between the
simplified sentences, their semantic relationship is preserved.

##### Recursion

Next, the linked proposition tree is updated by replacing the leaf node that
was processed in this run with the newly generated subtree (30). It is
composed of the simplified propositions, their semantic relationship $rel\in
REL$ and constituency labels $c\in CL$. Fig. 3 depicts the result of the first
transformation pass on the example sentence from Fig. 1. The resulting leaf
nodes are then recursively simplified in a top-down fashion (32).

##### Termination

The algorithm terminates when no more rule matches the set of simplified
propositions $prop\in PROP$ in the leaf nodes. It outputs the source
sentence’s linked proposition tree $LPT$ (36), representing its semantic
hierarchy of minimal semantic units. In that way, the input is transformed
into a set of hierarchically ordered and semantically interconnected sentences
that present a simplified syntax. Figure 1 shows the final linked proposition
tree of our example sentence.

##  3 Evaluation

###  3.1 Experimental Setup999To enable reproducible research, all code /
datasets used in the experiments are provided online:
https://github.com/Lambda-3/DiscourseSimplification/tree/master/supplemental_material.

####  3.1.1 Automatic Metrics

We evaluate the constituency type classification and rhetorical relation
identification steps by mapping the simplified sentences that were generated
in the sentence splitting subtask to the Elementary Discourse Units (EDUs) of
the RST-DT corpus.111111https://t1p.de/n6t9 This dataset is a collection of
385 Wall Street Journal articles annotated with rhetorical relations based on
the RST framework Mann and Thompson (1988). For matching simplified sentences
generated by our TS approach to the annotations of the RST-DT corpus, we
compare each split sentence to all the EDUs of the corresponding input
sentence. For each pair, we search for the longest contiguous matching
subsequence. Next, based on the size of the matched sequences, a similarity
score between the two input strings is calculated. Each pair whose similarity
score surpasses an empirically determined threshold of 0.65 is considered a
match.

##### Constituency Type Classification

To determine whether the hierarchical relationship that is assigned by our TS
framework between a pair of simplified sentences is correct, we check if the
hierarchy of its contextual layers corresponds to the nuclearity of the
aligned text fragments of the RST-DT. For this purpose, we make use of the
nuclearity status encoded in the annotations of this dataset. In addition, we
compare the performance of our TS approach with that of a set of widely used
sentence-level discourse parsers on this task.

##### Rhetorical Relation Identification

To assess the performance of the rhetorical relation identification step, we
determine the distribution of the relation types allocated by our TS approach
when operating on the 7,284 input sentences of the RST-DT and compare it to
the distribution of the manually annotated rhetorical relations of this
corpus. Moreover, we examine for each matching sentence pair whether the
rhetorical relation assigned by our TS framework equates the relation that
connects the corresponding EDUs in the RST-DT dataset. For this purpose, we
apply the more coarse-grained classification scheme from Taboada and Das
(2013), who group the full set of 78 rhetorical relations that are used in the
RST-DT corpus into 19 classes of relations that share rhetorical meaning.
Finally, we analyze the performance of our framework on the relation labeling
task in comparison to a number of discourse parser baselines.

####  3.1.2 Manual Analysis

To get a deeper insight into the accuracy of the semantic hierarchy
established between the split components, the automatic evaluation described
above is complemented by a manual analysis. Three human judges independently
of each other assessed each decomposed sentence according to the following
four criteria: (i) Limitation to core information: Is the simplified output
limited to core information of the input sentence? (yes - no - malformed);
(ii) Soundness of the contextual proposition: Does the simplified sentence
express a meaningful context fact? (yes - no); (iii) Correctness of the
context allocation: Is the contextual sentence assigned to the parent sentence
to which it refers? (yes - no); and (iv) Properness of the identified semantic
relationship: Is the contextual sentence linked to its parent sentence via the
correct semantic relation? (yes - no - unspecified).

The first three categories of our analysis address the correctness of the
constituency type classification task, while the latter targets the rhetorical
relation identification step. The annotation task was carried out on a random
sample of 100 sentences from the OIE2016 Open IE benchmark Stanovsky and Dagan
(2016).

###  3.2 Results

####  3.2.1 Automatic Metrics

Using the matching function described in Section 3.1.1, we obtained 1,827
matched sentence pairs, i.e. 11.74% of the pairs of simplified sentences were
successfully mapped to a counterpart of EDUs from the RST-DT. The relatively
low number of matches can be attributed to the fact that the text spans we
compare have very different features.121212For details, see Section F in the
appendix. As we are primarily interested in determining whether the
constituency and relation labels that are assigned by our approach are
correct, we will focus on precision in the following.131313 The fraction of
labels that are successfully retrieved (i.e. recall) is of minor importance in
our setting. In addition, this score might be biased, since a large proportion
of EDUs from RST-DT is not mapped to a counterpart of simplified propositions
in our experiments. Therefore, we refrain from reporting recall scores.

##### Constituency Type Classification

In 88.89% of the matched sentence pairs, the hierarchical relationship that is
allocated between a pair of simplified sentences by our reference TS
implementation DisSim corresponds to the nuclearity status of the aligned EDUs
from RST-DT, i.e. in case of a nucleus-nucleus relationship in RST-DT, both
output sentences from DisSim are assigned to the same context layer, while in
case of a nucleus-satellite relationship the sentence mapped to the nucleus
EDU is allocated to the context layer cl, whereas the sentence mapped to the
satellite span is assigned to the subordinate context layer cl+1. The majority
of the cases where our TS approach assigns a hierarchical relationship that
differs from the nuclearity in the RST-DT corpus can be attributed to relative
clauses (see Table 3).

Source |  Mr. Volk, 55 years old, succeeds Duncan Dwight, who retired in September.
---|---
|  Mapped Span 1 |  Mapped Span 2
DisSim |  Volk succeeds Duncan Dwight. |  Duncan Dwight retired in September.
Nuclearity |  context layer 0 |  context layer 1
Relation | Elaboration
RST-DT |  Mr. Volk, 55 years old, succeeds Duncan Dwight, |  who retired in September.
Nuclearity |  nucleus |  nucleus
Relation | Elaboration
Table 3: Example. | nuclearity | relation
---|---|---
DPLP Ji and Eisenstein (2014) | 71.1 | 61.8
Feng and Hirst (2014) | 71.0 | 58.2
2-Stage Parser Wang et al. (2017) | 72.4 | 59.7
Lin et al. (2019) | 91.3 | 81.7
SPADE Soricut and Marcu (2003) | 56.1 | 44.9
HILDA Hernault et al. (2010) | 59.7 | 48.2
PAR-s Joty et al. (2015) | 75.2 | 66.1
Lin et al. (2019) | (86.4)* | (77.5)*
DisSim | 88.9 | 69.5
Table 4: Precision of DisSim and the discourse parser baselines, as reported
by their authors. (*) In case of automatic discourse segmentation, for Lin et
al. (2019) the F1-score is available only.

Table 4 displays the precision that the discourse parser baselines achieve on
the 991 sentences of the RST-DT test set in distinguishing between nucleus and
satellite spans (“nuclearity”). For the approaches in the upper part of the
table, the authors report the systems’ performance when using gold EDU
segmentation, while for those in the lower part the performance is indicated
based on automatic segmentation, i.e. when they are fed the output of their
respective discourse segmenter. Since our framework makes use of the
simplified sentences that were generated in the previous step when setting up
the semantic hierarchy, it is better comparable to the latter group. The
figures show that in this case our approach outperforms all other systems in
the constituency type classification task by a large margin of 13.7% at a
minimum.141414A very recent approach to intra-sentential sentence parsing was
proposed in Lin et al. (2019), achieving an F1-score of 86.4%. However, the
authors do not report its precision.

##### Rhetorical Relation Identification

Table 5 displays the frequency distribution of the 19 classes of rhetorical
relations that were specified in Taboada and Das (2013). The ten most
frequently occurring classes make up for 89.45% of the relations that are
present in the dataset. We decided to limit ourselves to these classes in the
evaluation of the rhetorical relation identification step, with two
exceptions. First, we did not take into account the “Topic-change” and “Same-
unit” classes. Second, we merged the two highly related classes of “Cause” and
“Explanation” into a single category.

Rhet. Relation | Count | Percent. | Precision
---|---|---|---
Elaboration | 7,675 | 25.65% | 0.5550
Joint | 7,116 | 23.78% | 0.6673
Attribution | 2,984 | 9.97% | 0.9601
Same-unit | 2,788 | 9.32% | —
Contrast | 1,522 | 5.09% | 0.7421
Topic-change | 1,315 | 4.39% | —
Explanation | 966 | 3.21% |
Cause | 754 | 2.52% | 0.7037
Temporal | 964 | 3.22% | 0.7895
Background | 897 | 2.30% | 0.4459
|  |  | avg.: 0.6948
Evaluation (2.0%), Enablement (1.8%), Comparison (1.5%), Textual organization
(1.2%), Condition (1.1%), Topic-comment (0.9%), Manner-means (0.7%), Summary
(0.7%), Span (0.0%)
Table 5: Frequency distribution of the 19 classes of relations from Taboada
and Das (2013) and the precision of DisSim’s rhetorical relation
identification step.

The right column in Table 5 displays the precision of our TS approach for each
class of rhetorical relation when run over the sentences from RST-DT. The
“Attribution” relation reaches by far the highest precision. The remaining
relations, too, show decent scores, with a precision of around 70%. The only
exception is “Background”. The difficulty with this type of relationship is
that it signifies a very broad category that is not signalled by discourse
markers and therefore hard to detect by our approach Taboada and Das (2013).
With an average precision of 69.5% in the relation labeling task (see Table
4), our framework again surpasses all the discourse parser baselines under
consideration when using automatic discourse segmentation.151515with the
exception of Lin et al. (2019)’s parser, for which only the F1-score is
reported by the authors, though. Hence, it is not directly comparable to the
other approaches whose performance is analyzed based on their precision.

When comparing the distribution of the rhetorical relations that were
identified by our TS approach on the source sentences from the RST-DT (see
Figure 5 in Section F in the appendix) to that of the manually annotated gold
relations displayed in Table 5, it turns out that there is a very high
similarity between the two of them. However, it must be noted that in about a
fifth of the cases, our TS approach is not able to identify a rhetorical
relation between a pair of decomposed sentences (“Unknown”). For the most
part, this can be attributed to sentence pairs whose relation is not
explicitly stated in the underlying source sentence. Since our approach is
based on cue phrases, searching for discourse markers that explicitly signal
rhetorical relations, it has difficulties in identifying relations that can
merely be implied.

####  3.2.2 Manual Analysis

The results of the human evaluation are displayed in Table 6. The inter-
annotator agreement was calculated using Fleiss’ $\kappa$ Fleiss (1971). The
figures indicate fair to substantial agreement between the three annotators,
suggesting that the evaluation scores present a reliable result.

Category | Yes | No | Malf. | Unspec. | $\kappa$
---|---|---|---|---|---
Limitation to core information | 68.2% | 20.0% | 11.9% | — | 0.39
Soundness of the contextual proposition | 83.1% | 16.9% | — | — | 0.51
Correctness of the context allocation | 93.2% | 6.8% | — | — | 0.41
Properness of the semantic relationship | 69.8% | 7.0% | — | 23.2% | 0.69
Table 6: Results of the manual analysis.

In more than two out of three cases, the annotators marked the propositions
that were classified as core sentences by our TS approach as correct, thus
approving that they have a meaningful interpretation and that their content is
truly restricted to core information of the underlying source sentence. Only
about 12% of the simplified sentences are malformed according to our
annotations. The remaining fifth of output core sentences was judged as being
misclassified, i.e. they rather contribute less relevant background data than
key information of the input. Regarding the soundness of the context
propositions, only about 17% of the output propositions that were classified
as context sentences were labelled as being inaccurate, while as many as 83%
present proper contextual propositions, expressing a meaningful context fact
that is asserted by the input and can be properly interpreted. Furthermore,
93% of the context sentences are assigned to their respective parent sentence,
whereas only 6% of them are misallocated, according to the annotators’ labels.
Finally, our evaluation revealed that our TS approach shows a decent
performance for the rhetorical relation identification step, too. More than
two-thirds of the sentence pairs are classified with the correct rhetorical
relation, according to our manual analysis. Only 7% of them are assigned an
improper relation. However, in nearly a quarter of the cases, our TS approach
was not able to identify a semantic relationship between the given pair of
sentences. This can be explained by the fact that for this subtask, our
framework follows a rather simplistic approach that is primarily based on cue
phrases. Therefore, it fails to identify a semantic relationship whenever none
of the specified keywords appears in the underlying input sentence. As a
result, our approach provides very precise results. Covering only a small
subset of rhetorical relations it lacks in completeness, though.

###  3.3 Extrinsic Evaluation

The fine-grained representation of complex sentences in the form of
hierarchically ordered and semantically interconnected propositions may serve
as an intermediate representation for downstream tasks. An application area
that may benefit greatly from our approach as a preprocessing step is the task
of Open IE Banko et al. (2007).

##### Coverage and Accuracy

We demonstrated that relational tuples in the form of predicate-argument
structures can be extracted more easily from the simplified sentences Cetto et
al. (2018b, a). In fact, a comparative analysis with state-of-the-art Open IE
systems Fader et al. (2011); Mausam et al. (2012); Del Corro and Gemulla
(2013); Angeli et al. (2015); Stanovsky et al. (2016); Mausam (2016);
Gashteovski et al. (2017); Saha et al. (2017); Stanovsky et al. (2018) on a
large benchmark corpus Stanovsky and Dagan (2016) revealed that their
performance in terms of precision and recall is improved by up to 25% and 27%,
respectively, when taking advantage of the split propositions instead of
dealing with the complex source sentences (see Table 7).

System | Precision | Recall | F1 | AUC
---|---|---|---|---
ReVerb | -7.2% | +19.1% | +8.4% | +39.2%
Ollie | +1.1% | -1.5% | -0.3% | -1.1%
ClausIE | +17.0% | -3.5% | +8.1% | +13.0%
Stanford Open IE | +25.0% | +27.2% | +25.5% | +35.5%
PropS | -6.1% | +16.9% | +4.5% | +12.4%
OpenIE-4 | +10.0% | +8.6% | +9.4% | +19.6%
MinIE | +23.7% | -1.8% | +12.5% | +21.3%
OpenIE-5 | +5.0% | +4.2% | +4.6% | +9.0%
RnnOIE | -15.0% | +0.9% | -8.3% | -14.1%
Table 7: Improvements in precision, recall, F1 and AUC on the OIE2016
benchmark when using our reference implementation DisSim as a pre-processing
step.

##### Coherence

Moreover, our TS approach enables existing Open IE systems to enrich their
output with semantic information. By capturing intra-sentential rhetorical
structures and hierarchical relationships between the extracted tuples, the
shallow semantic representation of Open IE tuples in the form of isolated
predicate-argument structures is extended. In that way, the semantic context
of the individual tuples is preserved.161616For details, see Section E in the
appendix.

##  4 Related Work

##### Discourse-level TS

The vast majority of structural TS approaches do not take into account
discourse-level aspects. However, two notable exceptions have to be mentioned.
Siddharthan (2006) was the first to use discourse-aware cues in the
simplification process. As opposed to our approach, though, where a semantic
relationship is established for each simplified output sentence, only a
comparatively low number of sentences is linked by such cue words. Another
approach that operates on the level of discourse was proposed by Štajner and
Glavaš (2017). It performs a semantically motivated content reduction by
maintaining only those parts of a sentence that belong to factual event
mentions. Our approach, on the contrary, aims to preserve all the information
contained in the source.

##### Discourse Parsing

The challenge of uncovering coherence structures in texts is pursued in the
field of Discourse Parsing. It aims to identify discourse relations that hold
between textual units in a document Marcu (1997). A well-established theory of
text structure used in this area is RST. Here, textual coherence is explained
by the existence of rhetorical relations that hold between adjacent text spans
in a hierarchical structure. Approaches to detect rhetorical structure
arrangements in texts range from early rule-based approaches Marcu (2000) to
supervised data-driven models that were trained on annotated corpora such as
the RST-DT Feng and Hirst (2014); Li et al. (2014); Lin et al.
(2019).171717Section G in the appendix elaborates on why it is not possible to
simply use an RST parser for establishing the semantic hierarchy between the
decomposed spans.

##  5 Conclusion

We presented a context-preserving TS approach that transforms structurally
complex sentences into a hierarchical representation in the form of core
sentences and accompanying contexts that are semantically linked by rhetorical
relations. In our experiments, we mapped the simplified sentences from our
reference implementation DisSim to the EDUs from RST-DT and showed that we
obtain a very high precision of 89% for the constituency type classification
and a decent score of 69% on average for the rhetorical relation
identification. In the future, we plan to improve the latter step by extending
our approach to also capture implicit relationships between the decomposed
sentences.

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##  Appendix A Process of Development of the Transformation Patterns

One of the fundamental goals of our context-preserving TS approach is to
decompose complex assertions into a set of self-contained minimal
propositions. Intuitively, the simplified propositions are “easy to read and
understand, and arguably easily processed by computers” Bach et al. (2011).
Below, we present a more formal specification of the minimality property we
aim for in the resulting simplified sentences, both on a syntactic and a
semantic level.

##### Minimality on the Syntactic Level

In syntax, four types of sentence structures are distinguished: simple
sentences, compound sentences, complex sentences and compound-complex
sentences Quirk et al. (1985). The goal of our approach is to transform a
given source sentence into a set of simple sentences. A simple sentence is a
sentence that comprises exactly one independent clause, i.e. a group of words
that has both a subject (S) and a verb (V), and optionally an indirect or
direct object (O), an adverbial (A) or a complement (C). It expresses a
complete thought, i.e. some coherent piece of information, as the following
example shows: “I admire her reasoning.”

Accordingly, the method we propose aims to split and rephrase complex multi-
clause sentences into sequences of simple sentences that each contain exactly
one independent clause. Fay (1990) distinguishes four central functional
categories of clauses:

  * •

coordinate clauses (1 rule),

  * •

adverbial clauses (6 rules addressing pre- and postposed adverbial clauses as
well as adverbial clauses of purpose introduced by the phrase “to do”),

  * •

relative clauses (9 rules targeting both restrictive and non-restrictive
relative clauses, lexicalized on the different types of relative pronouns) and

  * •

reported speech (4 rules addressing both direct and reported speech with pre-
and postposed attribution).

Thus, in order to divide complex assertions into single-clause elements, we
specified, in a first step, a set of transformation patterns that target
above-mentioned categories of clauses, converting complex and compound
sentences into a set of simple sentences.

However, simple sentences may still include optional constituents that render
it overly complex. For instance, the two adverbials “in Princeton” and “in
1955” in the simple sentence “[Albert Einstein]S [died]V [in Princeton]A [in
1955]A.” specify additional contextual information that can be left out
without producing ill-formed output. Rather, the remaining clause “Albert
Einstein died.” still carries semantically meaningful information. Hence, in
order to transform simple sentences into atomic semantic units, we need to
further reduce them to their corresponding clause type. According to Quirk et
al. (1985), clauses can be classified into seven different clause types based
on the grammatical function of their constituents, as illustrated in Table 8
Del Corro and Gemulla (2013).

|  Clause Type |  Example
---|---|---
T1 |  SV |  [Albert Einstein]S [died]V.
T2 |  SVA |  [Albert Einstein]S [remained]V [in Princeton]A.
T3 |  SVC |  [Albert Einstein]S [is]V [smart]C.
T4 |  SVO |  [Albert Einstein]S [has won]V [the Nobel Prize]O.
T5 |  SVOO |  [The Royal Swedish Academy of Sciences]S [gave]V [Albert Einstein]O [the Nobel Prize]O.
T6 |  SVOA |  [The doorman]S [showed]V [Albert Einstein]O [to his office]A.
T7 |  SVOC |  [Albert Einstein]S [declared]V [the meeting]O [open]C.
Table 8: The seven types of clauses. S: Subject, V: Verb, C: Complement, O:
Object, A: Adverbial.

The clause type conveys the minimal unit of coherent information in the
clause. Accordingly, if a constituent of a clause that is also part of its
type is removed, the resulting clause does not carry semantically meaningful
information any more Del Corro and Gemulla (2013). Hence, while constituents
that belong to the clause type are essential components of the corresponding
simple sentence, all other constituents are optional and can be discarded
without leading to an incoherent or semantically meaningless output.
Therefore, we do not only perform clausal disembedding. Instead, we go down to
the level of individual phrases, with the goal of extracting and converting
optional phrasal constituents into stand-alone sentences.

So far, a minimal proposition can be defined as a simple sentence that is
reduced to its clause type by omitting all optional constituents, i.e. all
elements that do not appear in the type of the underlying clause. However,
transforming complex source sentences into simple sentences and trimming them
to their clause types may still result in over-specified propositions due to
overly complex subclausal units. For instance, in the simple sentence “[Bell,
a telecommunication company,]S [makes and distributes]V [electrical goods,
computers and building products]O.” (of clause type SVO), the phrasal elements
in all three positions are unnecessarily complex. While the subject contains
an appositive phrase that further specifies the noun to which it refers
(“Bell”), both the verb and the object include coordinated conjunctions that
can be decomposed into separate elements, resulting in a set of clauses that
present a much simpler syntax. Consequently, we further simplify such
utterances by extracting phrasal elements from the input and transforming them
into stand-alone sentences. In total, following Heilman and Smith (2010), our
TS approach addresses seven types of phrasal constructs:

  * •

coordinate verb phrases (1 rule),

  * •

coordinate noun phrase lists (2 rules targeting lists of noun phrases in
subject and object position),

  * •

participial phrases (4 rules addressing both restrictive and non-restrictive
embedded, pre- and postposed participial phrases),

  * •

appositive phrases (2 rules addressing restrictive and non-restrictive
appositions),

  * •

prepositional phrases (3 rules addressing prepositional phrases that act as
complements of verb phrases and prepositional phrases that are offset by
commas),

  * •

adjectival and adverbial phrases (2 rules targeting pre- and postposed
adjectival and adverbial phrases) and

  * •

lead noun phrases (1 rule).

To sum up, by the notion of a minimal proposition we understand a simple
sentence that has been broken down to its essential constituents, i.e. the
elements that are part of its clause type, while also extracting a specified
set of phrasal units.

##### Minimality on the Semantic Level

From a semantic point of view, a text can be seen as a collection of events
(also called “frames” or “scenes”) Abend and Rappoport (2013) that describe
some activity, state or property. From a semantic perspective, the goal of our
TS approach is to split sentences into separate frames, where each of them
presents a single event. An event typically consists of a predicate, a set of
arguments and optionally secondary relations. While the predicate (typically a
verb, but nominal or adjectival predicates are also possible) is the main
determinant of what the event is about, arguments describe its participants
(e.g., who and where). They represent core elements of a frame, i.e. essential
components that make it unique and different from other frames. Secondary
relations, in contrast, represent non-core elements that introduce additional
relations, describing further event properties (e.g., when and how). In order
to avoid over-specified propositions that are difficult to handle for
downstream applications, such non-core elements of a frame are extracted and
transformed into stand-alone sentences in our proposed TS approach.

To sum up, our TS approach aims to split sentences that present a complex
structure into its basic semantic building blocks in the form of events.
Complex sentences that contain several events are decomposed into separate
frames, while non-core elements of a frame are extracted and transformed into
stand-alone sentences. Hence, from a semantic point of view, a minimal
proposition can be seen as an utterance expressing a single event consisting
of a predicate and its core arguments.

##  Appendix B Execution Order of the Patterns

The execution order of the transformation patterns was determined by examining
which sequence achieved the best simplification results in a manual
qualitative analysis that was conducted on a random sample of 300 sentences
from Wikipedia.

The execution order is given in Table 9. The 35 rules could be grouped into 17
classes.

Order |  Rule Group
---|---
1 |  coordinate clauses
2 |  non-restrictive relative clauses
3 |  appositive phrases
4 |  preposed adverbial clauses and participial phrases
5 |  coordinate verb phrases
6 |  postposed adverbial clauses and participial phrases
7 |  reported speech with preposed attribution
8 |  postposed adverbial clauses
9 |  reported speech with postposed attribution
10 |  embedded participial phrases
11 |  restrictive relative clauses
12 |  prepositional phrases that act as complements of verb phrases
13 |  postposed participial phrases
14 |  adjectival/adverbial phrases
15 |  lead noun phrases
16 |  prepositional phrases that are offset by commas
17 |  lists of noun phrases
Table 9: Execution order of the patterns.

The execution order of the transformation patterns was balanced for various
criteria, including the frequency of the rules, their complexity and
specificity, as well as susceptibility to errors. In general, the least error-
prone rules are executed first (e.g., in a manual analysis, we determined that
99.1% of the coordinate clauses (1) were correctly split). During the
transformation process, we then work our way to the more error-prone ones,
such as the rules for decomposing lists of noun phrases (17), which can easily
be confused with appositive phrases. Beyond that, the patterns become more and
more complex towards the end of the list, since we need to check for special
cases and possibilities of confusion, e.g. (12) and (17). It also becomes
clear that we first operate on coarse-grained level, dividing up clauses
(1-11), before we go down to the phrasal level (10-17), resulting in much more
fine-grained splits. Moreover, specific rules are carried out early in the
process, such as the rules for disembedding relative clauses (lexicalized on
the relative pronouns, 2) and appositive phrases (based on named entities, 3).
More general rules, on the other hand, tend to be executed later. Finally,
there is a rough orientation on the frequency of the rules, starting with the
ones that are triggered more often, and working towards the ones that occur
less frequently in the transformation process.

To sum up, the execution order of the transformation patterns is based on the
following criteria, which have been balanced against each other:

  * (1)

susceptibility to errors: from less to more error-prone

  * (2)

granularity: from clausal to phrasal units

  * (3)

complexity: from simple to complex

  * (4)

specificity: from specific to general

  * (5)

frequency: from frequent to infrequent

##  Appendix C Constituency Type Classification

In RST, each text span is specified as either a nucleus or a satellite. The
nucleus span embodies the central piece of information and is comparable to
what we denote a core sentence, whereas the role of the satellite is to
further specify the nucleus, corresponding to a context sentence in our case.

##  Appendix D Mapping of Cue Phrases to Rhetorical Relations

Table 10 lists the full set of cue phrases that serve as lexical features for
the identification of rhetorical relations when establishing the semantic
hierarchy between a pair of split sentences. It further shows to which
rhetorical relation each of them is mapped.

Rhetorical Relation |  Cue Phrases
---|---
Contrast |  although, but, but now, despite, even though, even when, except when, however, instead, rather, still, though, thus, until recently, while, yet
List |  and, in addition, in addition to, moreover
Disjunction |  or
Cause |  largely because, because, since
Result |  as a result, as a result of
Temporal |  after, and after, next, then, before, previously
Background |  as, now, once, when, with, without
Condition |  if, in case, unless, until
Elaboration |  more provocatively, even before, for example, further, recently, since, since now, so, so far, where, whereby, whether
Explanation |  simply because, because of, indeed, so, so that
Table 10: Mapping of cue phrases to rhetorical relations.

In addition, Spatial relations are identified on the basis of named entities.
Attribution relations are detected using a pre-defined list of verbs of
reported speech and cognition Carlson and Marcu (2001).

##  Appendix E Merits for Downstream Applications

##### Coherence

Previous work in the area of Open IE has mainly focused on the extraction of
isolated relational tuples, ignoring the cohesive nature of texts where
important contextual information is spread across clauses or sentences.
Consequently, state-of-the-art Open IE approaches are prone to generating a
loose arrangement of tuples that lack the expressiveness needed to infer the
true meaning of complex assertions Niklaus et al. (2018).

Our approach allows existing Open IE systems to enrich their output with
semantic information. By leveraging the semantic hierarchy of minimal
propositions generated by our discourse-aware TS approach, they are able to
extract semantically typed relational tuples from complex source sentences.
Thus, the shallow semantic representation of state-of-the-art Open IE
approaches in the form of disconnected predicate-argument structures is
extended, capturing intra-sentential rhetorical structures and hierarchical
relationships between the relational tuples. In that way, the semantic context
of the extracted tuples is preserved, allowing for a proper interpretation of
complex assertions. See Figure
LABEL:fig:ComparativeAnalysisSystems_supervisedOIE for an illustrative
example.

##  Appendix F Evaluation

While the goal of our TS approach is to generate well-formed syntactically
simplified sentences, the EDUs in the RST-DT are copied verbatim from the
source, resulting in an output of varied length that is usually not
grammatically sound. Moreover, in many cases, the EDUs mix multiple semantic
units, whereas our approach aims to split the input into atomic components,
with each of them expressing a coherent and indivisible proposition.

Figure 5: Distribution of the rhetorical relations.

##  Appendix G Discourse Parsing

1-4: Condition1-2: ElaborationA fluoroscopic
studyknown as an
upper gastrointestinal
… the next step
in management,3-4: Listalthough if volvulus
is suspected,
caution … impede
surgical revisionand lead to
post operative
complications. Figure 6: Rhetorical structure tree of our example sentence,
generated using the RST parser proposed in Ji and Eisenstein (2014). The
leaves correspond to EDUs, while each node is characterized by its nuclearity
and a rhetorical relation between adjacent text spans.

The syntactic analysis we propose for establishing the semantic hierarchy
between the decomposed spans is bound to the RST discourse markers. However,
it is not possible to simply use an RST parser for this task. As illustrated
in Figure 6, such a parser does not return grammatically sound sentences.
Instead, it segments the input into basic textual units, so-called elementary
discourse units (EDUs), which are copied verbatim from the source. In order to
reconstruct proper sentences, rephrasing is required. For this purpose,
amongst others, referring expressions have to be identified, and phrases have
to be rearranged and inflected. Moreover, the textual units resulting from the
segmentation process are too coarse-grained for our purpose, since RST parsers
mostly operate on clausal level. The goal of our approach, though, is to split
the input into minimal semantic units, which requires to go down to the
phrasal level in order to produce a much more fine-grained output in the form
of minimal propositions.


$||\tilde{U}_{0}-\bar{U}_{0}||_{{\mathcal{E}}_{K}}<\nu.$ (4.15)

Using the definition (4.13), we decompose $\tilde{U}_{0}-\bar{U}$ as follows

$\tilde{U}_{0}-\bar{U}=\tilde{H}_{0}\Big{(}\tilde{u}_{0}(\tilde{\mathcal{M}}_{0}^{-1}z+\tilde{V}_{0})-u_{0}(\tilde{\mathcal{M}}_{0}^{-1}z+\tilde{V}_{0})\Big{)}+\Big{(}\tilde{H}_{0}u_{0}(\tilde{\mathcal{M}}_{0}^{-1}z+\tilde{V}_{0})-\bar{U}_{0}\Big{)}:=J_{1}+J_{2}.$
(4.16)

For $J_{1}$, using a change of variable, (4.12a), the assumption (1.17) for
$u_{0}-\tilde{u}_{0}$, and the embedding (3.13a), we obtain

$||J_{1}||_{{\mathcal{H}}^{K}}\lesssim||u_{0}-\tilde{u}_{0}||_{{\mathcal{H}}^{K}}\lesssim\epsilon_{0},\quad\lim_{\epsilon_{0}\to
0}\max_{|z|\leq 1}|\nabla^{i}J_{1}|=0,\quad\mathrm{for\ }i=0,1,2,3.$ (4.17)

Denote
$H_{1}=\tilde{H}_{0}H_{0}^{-1},\mathcal{M}_{1}=\mathcal{M}_{0}\mathcal{M}_{0}^{-1},V_{1}=\mathcal{M}_{0}(\tilde{V}_{0}-V_{0})$.
For $J_{2}$, using $u_{0}(z)=H_{0}^{-1}U_{0}(\mathcal{M}_{0}(z-V_{0}))$ (1.13)
and a change of variable, we obtain

$\displaystyle J_{2}$
$\displaystyle=\tilde{H}_{0}H_{0}^{-1}U_{0}(\mathcal{M}_{0}\mathcal{M}_{0}^{-1}z+\mathcal{M}_{0}(\tilde{V}_{0}-V_{0}))-\bar{U}_{0}=H_{1}U_{0}(\mathcal{M}_{1}z+V_{1})-\bar{U}$
$\displaystyle=H_{1}(U_{0}-\bar{U})(\mathcal{M}_{1}z+V_{1})+\Big{(}H_{1}\bar{U}(\mathcal{M}_{1}z+V_{1})-\bar{U}\Big{)}:=J_{21}+J_{22}.$

From (4.12b), we obtain that $\mathcal{M}_{1}-1=o(1),H_{1}-1=o(1),V_{1}=o(1)$.
Since $||U_{0}-\bar{U}_{0}||_{{\mathcal{E}}_{K}}<\nu$, by choosing
$\epsilon_{0}$ small enough, we yield

$||J_{21}||_{{\mathcal{E}}_{K}}<\nu.$ (4.18)

Using the smoothness of $\bar{U}$ and the embedding (3.13a), we obtain

$\lim_{\epsilon_{0}\to 0}||J_{22}||_{{\mathcal{H}}^{K}}+\max_{|z|\leq 1,i\leq
3}|\nabla^{i}J_{22}|=0.$ (4.19)

Next, we show that for $f$ with $\nabla^{j}f(0)=0,j\leq 2$ and $i\leq K$, we
have

$||\nabla^{i}f||_{\rho_{i}}\lesssim||f||_{{\mathcal{H}}^{K}}.$ (4.20)

Using the definition of (1.28) $\rho_{K}$, ${\mathcal{E}}_{K}$ (3.4),
$g_{k},{\mathcal{H}}^{K}$ (3.9), and embedding (3.13a), for $f$ with
$\nabla^{j}f(0)=0,j\leq 2$, we have

$\displaystyle||\nabla^{i}f||_{\rho_{i}}\lesssim||\nabla^{3}f||_{L^{\infty}}||\
|x|^{3-i}\mathbf{1}_{|x|\leq
1}||_{\rho_{i}}+||\nabla^{i}f||_{g_{i}}\lesssim||f||_{{\mathcal{H}}^{K}},\quad
i\leq 3,$
$\displaystyle||\nabla^{i}f||_{\rho_{i}}\lesssim||\nabla^{i}f||_{L^{\infty}}||\mathbf{1}_{|x|\leq
1}||_{\rho_{i}}+||\nabla^{i}f||_{g_{i}}\lesssim||f||_{{\mathcal{H}}^{K}},\quad
i\leq(d+5)/2.$

For $\frac{d+5}{2}<i\leq K$, $\rho_{i}$ and $g_{i}$ are equivalent and (4.20)
follows from (3.12c). From the definition of $\tilde{U}_{0}$ and $J_{i}$, for
$l\leq 2,i=1,2$, we obtain
$\nabla^{l}J_{21}(0)=0,\nabla^{l}(J_{1}+J_{22})(0)=0$. Applying (4.20), we
obtain

$||J_{1}+J_{22}||_{{\mathcal{E}}_{K}}\lesssim||J_{1}+J_{22}||_{{\mathcal{H}}^{K}},$

which goes to $0$ as $\epsilon_{0}\to 0$. Since the inequality (4.18) is
strict, for $\epsilon_{0}$ small enough, we prove (4.15). Condition (1.16)
follows from a similar argument, we conclude the proof of Theorem 2.

##### Acknowledgments:

The research of Y.Wang and T.Y.Hou was in part supported by NSF Grant
DMS-2205590 and the Choi Family Gift Fund. The work of J.Chen was in part
supported by the NSF Grant DMS-2408098 and by the Simons Foundation. V.T.
Nguyen is supported by the National Science and Technology Council of Taiwan.

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