# LPAttack: A Feasible Annotation Scheme for Capturing
Logic Pattern of Attacks in Arguments

###### Abstract

In argumentative discourse, persuasion is often achieved by refuting or
attacking others arguments. Attacking is not always straightforward and often
comprise complex rhetorical moves such that arguers might agree with a logic
of an argument while attacking another logic. Moreover, arguer might neither
deny nor agree with any logics of an argument, instead ignore them and attack
the main stance of the argument by providing new logics and presupposing that
the new logics have more value or importance than the logics present in the
attacked argument. However, no existing studies in the computational
argumentation capture such complex rhetorical moves in attacks or the
presuppositions or value judgements in them. In order to address this gap, we
introduce LPAttack, a novel annotation scheme that captures the common modes
and complex rhetorical moves in attacks along with the implicit
presuppositions and value judgements in them. Our annotation study shows
moderate inter-annotator agreement, indicating that human annotation for the
proposed scheme is feasible. We publicly release our annotated corpus and the
annotation guidelines.

Keywords: argument, counterargument, attack, logic pattern, scheme, reasoning,
annotation, debate

languageresourceLanguage Resources

LPAttack: A Feasible Annotation Scheme for Capturing
Logic Pattern of Attacks in Arguments

Farjana Sultana Mim1, Naoya Inoue2,3, Shoichi Naito1,3,4, Keshav Singh1,
Kentaro Inui1,3
---
1Tohoku University, 2Stony Brook University 3RIKEN, 4Ricoh Company, Ltd.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
{mim.farjana.sultana.t3, naito.shoichi.t1<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Argumentation plays a central role in human communication where refuting or
attacking other’s arguments is a common strategy of persuasion [Walton et al.,
2010]. _Attacks_ in arguments can have different modes [Walton, 2009, Cramer
and Guillaume, 2018] and often comprise complex rhetorical moves, e.g., one
might attack the _conclusion_ (i.e., statement expressing position or belief
of the arguer) while agreeing with a _premise_ (i.e., statement providing
support or reason for the conclusion) of the argument [Afantenos and Asher,
2014].

Consider an example debate in Fig. 1, where two argumentative speeches are
given by each opposing team of the debate. In the debate, the counterargument
(CA) does not deny the _premise_ of the initial argument (IA), i.e., _death
penalty deprives the chance of rehabilitation of the criminals_ , instead she
implicitly agrees with it while she denies the _conclusion_ of the IA i.e.,
_death penalty should be abolished_ by giving more importance or value to the
_death penalty_ than the _rehabilitation of the criminals_. Although this
value judgement is implicit in the CA speech, CA explicitly provides a reason
behind her value judgement (bold text in CA). Automatically identifying such
internal logic patterns can help a wide range of natural language processing
(NLP) applications. For example, in an educational domain, this can help
machines diagnose learners’ arguments and provide feedback to the learners.

Figure 1: An example of logic pattern of attack of a debate captured by
the proposed LPAttack annotation scheme.

Prior studies in NLP that focused on _attacks_ in arguments mainly worked on
classification of argumentative relations (e.g., support, attack, neural),
identifying attackable points in arguments or counterargument generation [Stab
and Gurevych, 2014, Deguchi and Yamaguchi, 2019, Kobbe et al., 2019, Jo et
al., 2021a, Walton et al., 2008, Jo et al., 2020, Wachsmuth et al., 2018, Hua
et al., 2019, Reisert et al., 2019, Alshomary et al., 2021, Jo et al., 2021b].
Comparatively, less attention has been paid to identifying the logic pattern
of attacks in arguments.

Although some recent studies [Reisert et al., 2018, Jo et al., 2021a]
developed annotation schemes and logical mechanisms to capture the reasoning
process behind support and attack relations where they exploited implicit
causal links and sentiments, these studies didn’t capture other implicit
information e.g., presupposition or value judgements in arguments that also
contribute to the underlying logical structure of attacks. Furthermore, none
of these studies capture the modes of _attacks_ (e.g, whether the
counterargument denies the conclusion or the premise of the attacked argument)
and the complex rhetorical moves in them (e.g., agreeing with a premise while
attacking the conclusion).

In order to address these gaps, we introduce LPAttack (Logic Patterns of
Attack), a new annotation scheme that captures common modes of attacks and
complex rhetorical moves in them as well as the implicit information and value
judgements that contribute to the logical structure of attacks. Fig. 1 shows
an example annotation. The logic pattern of IA speech is represented by our
logic pattern, which can be interpreted as follows: _death penalty (=X) is
considered a negative thing because death penalty suppresses chance of
rehabilitation of the criminals (=Y), something good_. The logic pattern of CA
speech then represents their value judgement on _death penalty_ and _chance of
rehabilitation of the criminals_ , where more value is given to _death
penalty_. This value judgement then attacks the conclusion of IA. Given that
information, one can understand how and which part of the IA is attacked by
CA.

Our contributions can be summarized as follows:

  * •

We introduce LPAttack, a novel annotation scheme that captures the common
modes and complex rhetorical moves in attacks along with the implicit
information, presuppositions or value judgements in them (§3).

  * •

We conduct an annotation study using the proposed scheme that yields moderate
agreement between two annotators indicating the feasibility of the human
annotation for the scheme (§4).

  * •

We provide the annotated corpus consisting of logic patterns of attacks of 250
debates and the annotation guidelines as a publicly available resource to
encourage future research111Our annotated corpus and annotation guidelines are
publicly available at http://..

## 2\. Related Work

Computational analysis of argumentation has gained considerable attention in
recent years due to its importance in many NLP application such as essay
scoring, argumentative writing support system, providing educational feedback
etc. Common lines of work in this area include argumentative units (e.g.,
claim, premise) identification [Levy et al., 2014, Rinott et al., 2015, Stab
and Gurevych, 2014], argumentative relations (e.g., support, attack, neural)
classification [Peldszus and Stede, 2015, Cocarascu and Toni, 2017, Niculae et
al., 2017, Stab and Gurevych, 2014, Deguchi and Yamaguchi, 2019, Kobbe et al.,
2019, Jo et al., 2021a], qualitative assessment of arguments [Persing et al.,
2010, Persing and Ng, 2013, Persing and Ng, 2014, Persing and Ng, 2015,
Persing and Ng, 2016, Rahimi et al., 2015, Wachsmuth et al., 2016, Habernal
and Gurevych, 2016, Wachsmuth et al., 2017, Mim et al., 2019, Mim et al.,
2021] and retrieval or generation of counterargument [Hua and Wang, 2018,
Wachsmuth et al., 2018, Hua et al., 2019, Reisert et al., 2019, Alshomary et
al., 2021, Jo et al., 2021b]

Lately, researchers have started focusing on one of the complex and
challenging facet of argument analysis, i.e., capturing or explicating the
encapsulated knowledge in arguments (e.g., causal knowledge, commonsense
knowledge, factual knowledge) which are often implicit [Habernal et al., 2017,
Hulpus et al., 2019, Becker et al., 2019, Becker et al., 2020, Al-Khatib et
al., 2020, Becker et al., 2021b, Becker et al., 2021a, Singh et al., 2021,
Saha et al., 2021]. Although for deeper understanding of argumentation, we
also need to comprehend the underlying reasoning patterns of arguments, less
attention has been paid to representing such underlying reasoning patterns and
explicating the implicit information that contribute to these patterns.

We focus on this gap and address the problem of explicating internal logic
pattern of attacks in arguments that comprise complex rhetorical moves and
implicit causal information, sentiments, presuppositions or value judgements.
Our inspiration for designing such annotation scheme comes from Walton’s
argumentation schemes [Walton et al., 2008] which represent the common
reasoning structures in arguments. To give an example, Walton’s scheme of
_Argument from Negative Consequences_ has the conclusion _A should not be
brought about_ which is supported by the premise _if A is brought about, then
bad consequences will occur_. Although Walton’s schemes explicate the unstated
assumptions or propositions as a form of reasoning pattern, they are not
intended for capturing the logic pattern of _attacks_ , i.e., how an argument
is attacked by a counterargument. It should be noted that each of the Walton’s
scheme has a set of critical questions (CQs) associated with it that are used
to judge if an argument fitting a scheme is good or fallacious. Some CQs for
the above scheme are _How strong is the likelihood that the cited consequences
will occur?_ , _Are there other opposite consequences that should be taken
into account?_. However, the critical questions in Walton’s schemes only
specify the attackable points in an argument, they do not represent the
reasoning pattern of attacks.

Some recent studies adopted Walton’s schemes to represent the logic behind
support and attack relations. One of these studies [Reisert et al., 2018]
developed an annotation scheme that uses argument templates to capture
reasoning patterns behind support and attack relations. Another study [Jo et
al., 2021a] composed a set of rules specifying logical mechanisms that signal
the support or attack relation. While these studies identified implicit causal
reasoning, sentiments or factual contradiction in attacks, they did not
capture other implicit information e.g., contradictory causal reasoning,
assumptions or value judgements that significantly contribute to the logical
structure of attacks.

One recent study [Saha et al., 2021] created commonsense explanation graphs
that explicates the commonsense reasoning process involved in inferring
support and attack relations. However, the focus of this study is commonsense
explanation, not the reasoning pattern of _attack_ (or support). Therefore,
although this study exploited implicit causal knowledge, the fine-grained
implicit knowledge explicated in this study is not effective in representing
the logical structure of _attacks_ which requires distinct coarse-grained
implicit information e.g., contradictions or value judgements in arguments.

Besides, there is still no work in computational argumentation that captures
the modes of attacks in arguments (e.g, whether the counterargument denies the
conclusion or the premise of the attacked argument) or the complex rhetorical
moves in them (e.g., agreeing with a premise while attacking the conclusion,
providing a contradictory premise that leads to denying the conclusion etc.).
Our work addresses these gaps by introducing an annotation scheme that can
capture common modes of attacks, complex rhetorical moves in them as well as
implicit causal reasoning, sentiments, presuppositions or value judgements
that contribute to the logic pattern of attacks.

Figure 2: Base logic patterns with examples.

## 3\. LPAttack Annotation Scheme

In our study, we hypothesize that the logic pattern of attacks in arguments
are not uniformly distributed rather highly skewed and following this
hypothesis, we develop our annotation scheme to capture the common logic
pattern of attacks in arguments.

### 3.1. Pre-Study and Scheme Design

In order to examine what sort of strategic moves, assumptions or value
judgements are common during attack, we first conduct a preliminary study
where we perform qualitative analysis of how one argument attack another (see
Appendix for further details). For our pre-study, we select 35 debates from
the TYPIC dataset222This dataset is publicly available [Naito et al., 2022]
which consists of multiple, diverse debate themes. Each debate comprise an
argument and a counterargument conveyed by two opposing team of the debates.

This analysis of the internal structure of attacks provide insights into how
we can represent the attacking logic so that human annotation is plausible.
Then based on these insights, we design our annotation scheme, define the
annotation guidelines and formulate the task of capturing logic pattern of
attacks in arguments. In the following subsections, we describe our annotation
scheme along with the findings in pre-study.

#### 3.1.1. Base Logic Patterns

Generally, when people argue _for_ some belief, they show positive sentiment
towards a concept of that belief. Conversely, when they argue _against_ some
belief, they show negative sentiment towards a concept. For example, for the
beliefs _‘homework should be abolished’_ and _‘death penalty should be
abolished’_ , the arguers have _against_ stance and in both cases they have
_negative_ sentiment towards the concepts _homework_ and _death penalty_. Now,
counterarguments generally have the opposite stance and sentiment of initial
argument e.g., the counterargument _homework should not be abolished_ have
_for_ stance as well as _non-negative or positive_ sentiment towards the
concept _homework_. No matter how diverse the topics of the arguments are, the
sentiment towards a certain concept is generally dependent on these _for_ or
_against_ stance.

Argumentation Schemes [Walton et al., 2008] highlighted this fact and
extensively utilized the positive or negative sentiment of the arguer towards
a certain concept or consequence. Motivated by that, we design two base
patterns (shown in Fig. 2) for our scheme where the sentiments towards the
main concept in the argument acts as the _conclusion_ of the argument. _Base
pattern 1_ represents _against_ stance of the initial argument and therefore
presents the logic: {_Initial Argument_ : $X$ is negative; _Counterargument_ :
$X$ is not negative} where $X$ is a slot for the concept. Conversely, _Base
pattern 2_ represents the case where initial argument has _for_ stance. Both
of the base patterns have two slots for premises, one in the initial argument
and one in the counterargument.

#### 3.1.2. Relations and Attributes

In order to capture the logic of _premise_ that will support the conclusion
(i.e., the sentiment towards the central concept), we design a set of
relations and attributes. See Table 1 in Appedix for the overview.

##### Causal relation

Previous studies that worked on the representation of implicit reasoning
behind support or attack mostly adopted Argumentation Schemes [Walton et al.,
2008] and have shown that a majority of the arguments can be represented by
the implicit causal links [Reisert et al., 2018, Al-Khatib et al., 2020, Jo et
al., 2021a, Singh et al., 2021]. In our pre-study, we see the similar
phenomena i.e., most of the logics in arguments that are attacked or
acknowledged by counterarguments can be represented by causality. For example,
in Fig. 1, the logic of IA _death penalty deprives the chance of
rehabilitation of the criminals_ can be represented with the “suppress”
causality i.e., _{ death penalty, suppress, the chance of rehabilitation of
the criminals}_. We thus design our annotation scheme around two causal
relations _“promote”_ and _“suppress”_ (henceforth, _“base relations”_).

##### Value judgement

We observe that one of the common reasoning during attack is based on value-
judgements i.e., comparing two factors by giving more value or importance to
one than the other. We have found two phenomena: (i) counterarguments give
more importance to a certain concept of a logic while implicitly acknowledging
the logic, as shown in Fig. 1; and (ii) counterarguments neither acknowledge
nor deny any logic of the initial argument, instead ignore it and deny the
conclusion of the initial argument by providing new reasons presupposing that
the new reasons have more value. Consider the following example:

Initial Argument (IA)
_…homework should be abolished_ (Conclusion)
_…if homework were to be abolished, we could have more free time. As a result,
we could do what we really wanted like club activities…_ (Premise)
Counterargument (CA)
_…if homework is abolished, a number of people who don’t study at all will
increase…..To decrease a number of people who repeat years, homework is
necessary…_

In Example (3.1.2), CA neither affirms nor denies IA’s logic, instead ignores
it and provide new reasons that deny the conclusion “homework should be
abolished”. CA presupposes that the value or importance of _{ if homework is
abolished, a number of people who don’t study at all will increase}_ is
greater than the value of _{ if homework were to be abolished, we could have
more free time}_ and this presupposition is implicit in CA’s argument. In
order to represent these two phenomena of value judgements, we create the
relation _“is more important or severe or has greater weight”_.

##### Contradiction

We see another common attacking strategy in counterarguments i.e., providing
contradictory logic that ultimately leads to denying the conclusion of the
initial argument. One example of such scenario is given below:

Initial Argument (IA)
_..death penalty should be abolished_ (Conclusion)
_…death penalty is causing brutalization of modern society……it validates the
notion that the taking of someone’s life is a valid choice….._ (Premise)
Counterargument (CA)
_…death penalty sends a message that taking an innocent life will not be
tolerated by a civilized society. Thus, it serves as an antidote to
brutality…_

In the above example, while IA says: _{ death penalty is causing brutalization
of modern society}_, CA says the opposite: _{ it serves as an antidote to
brutality}_. CA’s logic contradics IA’s which leads to denying the conclusion
of IA. In order to capture such contradictory logic, we invent the
_“contradiction”_ relation.

##### Logic denial/agreement

In order to explicitly represent the denial of a premise logic or conclusion,
we create two relations _“nullify”_ and _“limit”_. These relations are
considered as the _“attacking relations”_ in our scheme. Besides, we represent
agreeing with a logic by the relation “acknowledgement”.

##### Negation

We observe that it is pretty common for counterarguments to _negate_
(explicitly or implicitly) certain logic, especially causal reasoning, by
providing some rationales or conditions. Consider the following example:
Initial Argument (IA)
_…homework should be abolished_ (Conclusion)
_…if students are always given homework, they will always be waiting for
instructions…..homework should be abolished so that students can study on
their own initiative…._ (Premise)
Counterargument (CA)
_…students will hardly be able to study on their own without homework because
continuous instructions or guidelines are needed for children to study or
learn something new…_

In Example (3.1.2), CA negates IA’s logic _{ homework should be abolished so
that students can study on their own initiative}_ by saying _{ students will
hardly be able to study on their own without homework}_ and provides a reason
behind it i.e., _“continuous instructions or guidelines are needed for
children to study or learn something new”_. In order to represent such
negation attribution and reasoning behind a logic we develop “negation”
attribute and “rationale/condition” relation.

##### Mitigation

Instead of completely negating a logic, counterarguments often express that
the severity of such phenomena can be mitigated. Consider the following
example:

Initial Argument (IA)
_…death penalty should be abolished_ (Conclusion)
_…death penalty causes executioner’s suffering…..they feel that they are
responsible themselves for killing the suspect…._ (Premise)
Counterargument (CA)
_..executioner’s stress can be reduced by making sure that would-be
executioners are fully prepared for the job and have a good mental support
system_

In this example, CA doesn’t completely negate IA’s logic _{ death penalty
causes executioner’s suffering}_ instead partially negate it by saying _{
executioner’s stress can be reduced}_. We create “mitigation” attribute to
represent such partial negation attribution.

#### 3.1.3. Slot-filling

Towards computationalizing the task of capturing logic pattern of attacks, we
analyzed if it is possible to represent the logic patterns using only the
information present in the argumentative texts (i.e., by not having any
external commonsense concepts). Our analysis suggested that it is fairly
possible to represent the logic behind attack without writing any external
commonsense concepts. We thus decided to formulate the task of slot filling as
a text-span selection task: annotators choose slot fillers in the base
patterns from the given arguments.

### 3.2. Task Setting

The task of representing logic pattern of attack for a given argument and
counterargument consists of the following steps:

  1. 1.

_Selection of base logic pattern and slot-filling:_ At first, a base logic
pattern is selected based on the central stance of initial argument and then
the slots of the pattern is filled with the central concept.

  2. 2.

_Selection of relations and attributes along with text-spans:_ In this step,
relations and attributes are chosen along with the text-spans from the given
arguments to complete the base pattern by representing the logic of the
premises.

Since there is no fixed template for representing the logic of the premises in
initial argument and counterargument of base pattern, one important question
is how many relations, attributes and text spans should be chosen for premise
representations and how to choose it. In this regard, we take a summarization
approach i.e., we create one or two line mental summary of the counterargument
(CA) considering its main points and find the logic in initial argument (IA)
that is attacked by the CA. If CA attacks IA’s conclusion instead of attacking
any logic behind the conclusion, then we also create a one line mental summary
of IA’s main points. Then, we choose suitable text spans, relations and
attributes to represent that one or two line mental summary of CA, the
attacked logic or mental summary of IA and how CA attacks IA (i.e., which part
of the IA logic is denied and if CA agrees with any of the IA logic). We refer
to these representations as _CA-pattern_ , _IA-pattern_ and _attack-pattern_.

We set some constraints on how many relations and attributes can be selected
for each of these three representations. For IA-pattern, at most two and for
CA-pattern, at most three relations or attributes can be selected where using
a base causal relation is mandatory for IA-pattern and using _good_ or _bad_
attributes are prioritized in both cases. For representing the attack-pattern,
choosing at least one attacking relation (i.e., nullify or limit) is
mandatory. It should be noted that attack-pattern basically represents the
relation between CA-pattern and IA-pattern or CA-pattern and the conclusion of
IA.

We also set a constraint on how long the text spans should be. We specify that
although text spans can be long up to two small sentences or one compound
sentence, we should try to choose smaller text span (e.g., short phrase) as
much as possible. We also specify that we should choose such text spans that
when we read the patterns as a standalone logic (i.e., without reading the
debates), they are understandable.

## 4\. Annotation Study

The key requirements for identifying logic pattern of attacks in arguments are
two-fold: (i) identify as much logic pattern of attacks as possible and (ii)
make human annotation feasible. In order to verify whether our LPAttack scheme
satisfies these requirements, we observe two metrics during annotation study:
(i) coverage of the scheme and (ii) inter-annotator agreement (IAA).

### 4.1. Setup

Two expert annotators participate in the annotation study and annotate logic
pattern of attacks independently using our annotation scheme333We use
diagrams.net for annotation..

We trained the annotators in a pilot annotation phase where both of the
annotators were asked to annotate 20 debates. After the pilot annotation, we
discussed the disagreements and, if needed, adjourned the annotation
guidelines. One issue that we observed in the pilot annotations is that when
we read the annotated patterns as a standalone logic (i.e., without reading
the debates), some of them didn’t make complete sense because the chosen text
spans had information gap. In order to reinforce that the logic patterns are
understandable independently, we ask the annotators to write the text form of
the logic patterns during our main annotation i.e., the way they read the
logic patterns. For example, the text form of the logic pattern in Fig. 1 is:

  * IA: _{ “death penalty” is negative}_ because _{ “death penalty” suppress (“chance of rehabilitation of the criminals” which is good)}_

  * CA: _{ “death penalty” is not negative}_ because _{ “death penalty” is more important/severe/has greater weight than “chance of rehabilitation of the criminals which is good” given the rationale/condition that “while executing prisoners is completely effective in ensuring….”}_

We expect that writing the text form would serve as a double checking for the
logic patterns and when the annotators would read it separately from the
debates, they will understand if there is an information gap or if the logic
pattern is self-sufficient.

In our main annotation study, 50 debates are annotated by two annotators and
145 debates are annotated by a single annotator. For coverage and IAA, we
report the results of dual annotations on 50 debates.

### 4.2. Coverage

We ask the annotators to mark an attacking strategy as “Not Applicable (NA)”
if it cannot be represented by our scheme. We obtain 90% (45/50) coverage for
the LPAttack scheme. This result validates our hypothesis that logic pattern
of attacks in arguments are not uniformly distributed rather highly skewed
i.e., logic pattern of a wide range of attacks can be captured with a limited
set of relations and attributes.

### 4.3. Inter-annotator agreement (IAA)

We measure the IAA for relations and attributes444We ignore calculating the
agreement for the selection of base pattern since all the debates used in the
annotation study has the same base pattern (i.e., base pattern 1). using
Cohen’s ($\kappa$) [Cohen, 1960]. For the calculation of IAA, we consider IA-
pattern, CA-pattern and attack-pattern (described in section 3.2) as our
markables. Since we want to know how much the annotators agree on each of
these logics as well as on the overall debate, we employ two strategies for
the calculation of IAA: (i) calculate IAA considering each markables, (ii)
concatenate these three markables to have a single representation of the whole
debate and calculate IAA.

We obtain Cohen’s $\kappa$ of 0.63 in case of (i) which indicates a
substantial agreement and in case of (ii), we obtain 0.49 which indicates a
moderate agreement.

Additionally, we evaluate whether the text spans are the same in cases where
relations and attributes are agreed. It should be noted that among the three
markables, only IA-pattern and CA-pattern have text spans and therefore we
consider these two markables for this matching calculation but follow the same
strategy as above (i.e., (i) and (ii)). In each of the markables, if all of
the text spans exactly match, we call it _exact-match_ , if all of the text
spans leniently (semi-exactly) match or some them have lenient matching while
others have exact matching, we call it _lenient-match_. We see that in case of
(i), 68% (47/69) text spans are similar _(43% (30/69) exact-match, 25% (17/69)
lenient-match)_. For (ii), we obtain 46% (12/26) match _(19% (5/26) exact-
match, 6% (7/26) lenient-match)_.

These results suggest that annotators can choose appropriate relations and
attributes as well as text spans with a certain degree of reliability.

Figure 3: Example of debate where two annotators have different
interpretation.

### 4.4. Analysis of Annotations

We perform a manual analysis of the annotations in order to examine the
correctness of the logic patterns, disagreements between the annotators and
common attacking strategies captured by these annotations.

##### Correctness of the logic patterns

We check how many annotated logic patterns are correct i.e., the logic
patterns capture essence of the attacks and are understandable enough when
read independently without reading the debates. We adopt the following
strategy in this regard:

  * •

Exact logic pattern match between annotators ➾ mark it as a correct logic
pattern

  * •

Non-exact match between annotators ➾ manually check the logic patterns and
discuss with the annotators ➾ mark it as correct or incorrect based on the
decision in the discussion

Following the above strategy, we find that 90% (45/50) annotations of one
annotator is correct where 86% (43/50) annotations of the another annotator is
correct. Both the annotators had incorrect patterns for two of the debates.
For four of the debates, one annotator chose “NA” while other had correct
patterns and for one of the debates one annotator chose “NA” while other had
incorrect pattern. This result indicates that if we have at least two
annotations for a single debate, then the possibility of having a correct
annotation from one of the annotators is pretty high.

##### Disagreements between the annotators

There are generally two types of disagreements between the annotators: (i)
same interpretation of the debate but different logic patterns and (ii)
different interpretation of the debate.

One example of case (i) is given below:

  * Annotation 1: _{ “homework” is more important than ”free time” given the condition/rationale that “homework can establish basic foundation of studying”}_

  * Annotation 2: _{{ ”homework” promote “establish basic foundation of studying”} which is more important/severe/has greater weight than {”homework” suppress ”free time”}}_.

In this example, both annotations have the same interpretation i.e., _homework
is more important than free time because it establish basic foundation of
studying_ but the interpretation is represented differently.

Figure 4: Distribution of logic patterns

One important factor that we noticed is that in all of the cases of (ii) where
annotators have different interpretation of the debates, one of the
annotations has been found incorrect. One example of such case is shown in
Fig. 3. In this example, the interpretation of CA-pattern is different in two
annotations i.e., _{ death penalty suppress life imprisonment system which is
a bad thing}_ and _{ no death penalty doesn’t promote rehabilitation of the
criminals}_. Although both of the annotated patterns are understandable
without reading the debates, _Annotation 1_ is has been marked as incorrect
pattern because in this debate CA doesn’t exactly express that life
imprisonment is bad, instead it expresses that the reason behind abolishing
death penalty is rehabilitation of the criminals but even if we abolish death
penalty, it doesn’t results in rehabilitation in life imprisonment and
_Annotation 1_ has failed to capture that notion.

We also see that many disagreement happens regarding the choice of text spans.
When we manually checked the debates, we have noticed that even when some
interpretations are quite the same (case (i)), text spans are different. This
is because sometimes two or more sentences express the same meaning and
annotators choose text spans from these different sentences. Consider the
following example:

  * Annotation 1: _{ “homework” promote “learns that the way to succeed is by making schedule”} is more important/severe/has greater weight than {“homework” suppress ”do more of what we really wanted”}_

  * Annotation 2: _{ “homework” promote “learns the importance of scheduling”} is more important/severe/has greater weight than {“homework” suppress “free time”}_.

In both cases, the text spans _“learns that the way to succeed is by making
schedule”_ and _“learns the importance of scheduling”_ basically have the same
interpretation but were chosen from different sentences and therefore are
considered as mismatched text spans.

Besides, when CA attacks IA’s conclusion, annotators often choose different
main points to represent the premise of IA and in such cases text spans don’t
match. In the above example we see that the representation of the main points
of IA is different i.e., _‘do more of what we really wanted’_ and _‘free
time’_ since in these annotation CA doesn’t attack these premises.

##### Common Attacking Strategies Captured by the Annotations

In order to examine what sort of common rhetorical moves, assumptions or value
judgements in attacks are captured by the annotations of two annotators, we
look into the distribution of relations and attributes used to annotate the
debates. Fig. 4 shows this distribution. From the distribution we observe that
the most common logic patterns are ‘attacking a premise by negating it’,
‘value judgements between two concepts of a premise that leads to agreeing
with the premise but denying the conclusion’, ‘providing a way for mitigating
the consequence of a premise that leads to agreeing with the premise and
nullifying it at the same time’ and ‘providing a contradictory premise that
leads to denying the conclusion’. Moreover, we observe that ‘value judgement
between two causal relations’ also happens quite often.

## 5\. Discussion and Future Work

In our annotation study, we have observed that although the initial arguments
were causal arguments, some logic in the arguments were evaluative judgement
e.g., “death penalty is cruel” or “A truly just society can do without the
death penalty” and the counterarguments focused on those logics. In such cases
annotators failed to annotate the attacking strategy. In future, we would like
to enrich our scheme so that these sort of logics in attack can be captured.
Besides, we plan to have a second annotation for the 145 debates in our corpus
that currently have single annotation. We also intend to perform a voting
between the two annotations so that we can select a single representation for
each attacking strategy based on majority voting. Furthermore, we plan to
formulate the task of automatic identification of logic pattern of attacks
from given arguments and counterarguments.

We acknowledge the fact that capturing logic pattern of attacks is quite a
challenging task, specially when the arguments are long and there is still
many room for improvements.

## 6\. Conclusions

We proposed LPAttack, a feasible annotation scheme for capturing underlying
logic pattern of attacks in arguments. LPAttack is designed to capture the
common strategic moves, assumptions or value judgements during attacks in
arguments. Our annotation study showed that even with a limited set of
relations and attributes, we can capture the logic pattern of a wide range of
attacks (90%) in a debate corpus consisting multiple, diverse debate themes.
The results also showed a moderate inter-annotator agreement (Cohen’s
$\kappa$=0.49) between two annotators, verifying the feasibility of the
proposed scheme.

## 7\. Bibliographical References

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## Appendix

Relations |  Description |  Example
---|---|---
promote |  represents something causing/ encouraging another thing |  no homework _promote_ free time
suppress |  represents something hindering/ preventing another thing. |  homework suppress free time
rationale/condition |  represents writer’s reasoning or justification behind a relation or attribution. |  homework is more important than free time given the _rationale/condition_ that homework is part of education
($Y$) is more important/severe/ has greater weight than ($X$) ($X$) is more important/severe/ has greater weight than ($Y$) |  (i) represents some relation has higher value than another or (ii) some concept or element has higher value than another |  (i){no homework promote people fail in exam} which _is more important/ severe/ has greater weight_ than {no homework promote free time}, (ii) Example in Fig. 1
contradiction |  represents opposing logics |  {homework promote “problems in family”} _contradicts_ {homework promote good family relation}
acknowledgement |  represents agreement between relations |  Example in Fig. 1
nullify (attacking relation) |  represents denying a relation or logic |  Example in Fig. 1
limit (attacking relation) |  represents agreeing with and denying a relation at the same time |  {death penalty promote executioner’s suffering can be mitigated given the condition that executioners have a good mental support system} which _limit_ {death penalty” promote executioner’s suffering}
function |  represents joining of two or more relations |  joining the two relations {homework suppress free time} and {free time promote unproductive activities} would produce the relation {homework suppress unproductive activities}
Attributes |  Description |  Example
negation |  represents negation form of a relation or concept |  {homework _doesn’t_ promote free time} or {_no_ homework promote free time}
mitigation |  represents mitigated form of a relation |  {death penalty promote executioner’s suffering can be _mitigated_ given the condition that executioners have a good mental support system}
good |  represents positive feeling of the arguer towards a concept |  {homework” should be abolished because homework suppress free time}. Here “free time” is a _good_ thing according to the arguer
bad |  represents positive feeling of the arguer towards a concept |  {death penalty should be abolished because death penalty promote executioner’s suffering }. Here, “executioner’s suffering” is a _bad_ thing according to the arguer
Table 1: Relations and attributes in LPAttack scheme

##### Pre-Study and LPAttack Scheme Design

The initial design of the annotation schemes, guidelines and the task were
developed by the authors of this paper. Then, two non-trained annotators
explored the initial designs and helped improving the overall design and the
guidelines. The primary feedback from the annotators were having a detailed
description for each of the relations and attributes, creating categories for
them as well as having a prioritization map for these relations and
attributes.

During our pre-study, we have noticed is that sometimes the counterargument
shows strong opposite sentiment towards an argument e.g., _{ Argument: X is
negative; Counterargument: X is positive}_ and sometimes the opposite
sentiment of the counterargument is not that strong e.g., _{ Argument: X is
negative; Counterargument: X is not negative}_. However, it usually depends on
human perception how “strong” the opposite sentiment is and might vary from
human to human. In order to reduce the complexity and confusion during human
annotation, we only keep the representation of the less strong opposite
sentiment of counterargument as shown in Fig. 2 of the paper.

|  Homework should be abolished
---|---
PM-1 |  Abolishing homework will give students more free time
PM-2 |  Forcing students to do homework will make them passive in character
PM-3 |  It is not good for students to be obliged to study by their teachers or parents
PM-4 |  Students have memorized the incorrect way to study with homework
PM-5 |  Schools should take the responsibility for children’s academic skills, not parents at home

|  Death penalty should be abolished
---|---
PM-1 |  Death penalty is inhumane punishment
PM-2 |  Abolishing death penalty will prevent the situation of ending the life of innocent people
PM-3 |  Because of the high stress on the executioner, death penalty should be abolished
PM-4 |  The death penalty deprives criminals of the opportunity for rehabilitation
PM-5 |  Society is being brutalized by the death penalty

Table 2: Main points of the initial arguments of the debates in the TYPIC
corpus
for which counterarguments are written Figure 5: Rules for calculating inter-
annotator agreement (IAA)

##### Source Data

For our pre-study and annotation study, we utilize the debates from TYPIC
dataset [Naito et al., 2022]. This dataset has 1,000 parliamentary style
debates where given a topic, two opposing teams i.e., Prime Minister (PM) and
the Leader of the Opposition (LO) argue by taking a position in favor and
against the topic respectively. In each debate, the PM speech acts as the
_initial argument_ and the LO speech acts as the _counterargument_. The corpus
consists 10 PM speeches which are written on two topics: _“Homework should be
abolished”_ and _“Death penalty should be abolished”_. Table 2 shows the main
points of these PM speeches. For each of the PM speech, there are 100 LO
speeches. The arguments of 8 PM speeches out of 10 are causal arguments
(underlined in the table).

Since our scheme is designed around two causal relations “promote” and
“suppress”, when we choose debates, we only choose these 8 PM speeches
(initial arguments) for annotations whose arguments are causal arguments and
then we randomly choose LO speeches (counterarguments) associated with these
PM speeches.

##### Rules for calculating IAA:

One factor to consider during the IAA calculation is that in our scheme, we
kept the flexibility of human representation i.e., the same interpretation can
be represented in a slightly different way e.g., “no homework promote free
time” has the same meaning as “homework suppress free time” but they are
different representations. In order to handle such different representations
that generally have the same meaning, we create some rules to consider these
different representation as same. Fig. 5 show these rules. As shown in the
figure, we consider the representations “no X promote Y” and “X suppress Y” as
same (marked as _Causal Type A_) since they have the same meaning. One of our
rules consider rationale/condition relation as auxiliary in certain cases
where having or not having it doesn’t affect the understanding of the logic
much. For example, in the case of the logic {{no homework promote people fail
in exam _given the rationale/condition that not doing homework will lead to
lack of preparation}_ which is more important or severe or has greater weight
than {no homework promote free time}}, even if we remove the
rationale/condition relation, the interpretation is understandable. We ignore
this relation in such cases during the calculation of IAA.

Figure 6: Annotation example of logic pattern of attack of a debate

##### Annotation examples of logic pattern of attacks

In order to provide a better understanding of what the annotated logic
patterns look like and what sort of text spans are chosen from the given
arguments, we provide some annotation examples of the logic pattern of attacks
in Fig. 6, 7 and 8.

Figure 7: Annotation example of logic pattern of attack of a debate Figure 8:
Annotation example of logic pattern of attack of a debate


# Some comments on the fracture of viscoelastic solids

B.N.J. Persson  Peter Grünberg Institute (PGI-1), Forschungszentrum Jülich,
52425, Jülich, Germany  MultiscaleConsulting, Wolfshovener str. 2, 52428
Jülich, Germany  G. Carbone  Department of Mechanics, Mathematics and
Management, Politecnico di Bari, Via Orabona 4, 70100, Bari, Italy  C. Creton
Soft Matter Sciences and Engineering (SIMM), ESPCI Paris, PSL University,
Sorbonne University, CNRS, 75005 Paris, France  G. Heinrich  Technische
Universität Dresden, Institut für Textilmaschinen und Textile
Hochleistungswerkstofftechnik, Hohe Straße 6, D-01069 Dresden, Germany  T.
Tada  Sumitomo Rubber Industries, Ltd., Material Research & Development HQ.
2-1-1, Kobe 651-0071, Japan

###### Abstract

Crack propagation in viscoelastic solids like rubber is of great practical
importance. Shrimali and Lopez-Pamies have proposed a new interesting approach
for the crack propagation in viscoelastic solids. We give comments on the
validity of the theory and point out some effects not included in the theory.

Subm. to Extreme Mechanics Letters, 1.09.2023

Introduction

Crack propagation in rubber-like materials has many applications, e.g. to tire
wear[1, 2], and has been studied for a long time[3, 4, 5, 6, 7, 8, 9, 10, 11,
12]. In a recent series of papers Shrimali and Lopez-Pamies have presented an
interesting new approach to crack propagation in viscoelastic solids[13, 14,
15]. The theory is based on the assumption that a slab of rubber fracture at a
critical stretch that is independent of the stretch rate. Here we argue that
the study is not as general as stated by the authors. In what follows we
assume for simplicity linear viscoelasticity and neglect inertia effects.

Figure 1:  Crack in a rubber sheet with clamped upper and lower edges. The
height $h(t)=h_{0}+u(t)$ of the slab increases with time (undeformed height is
$h_{0}$). For an elastic solid (no viscoelasticity) the crack start to move
when the elongation strain $\epsilon=u(t)/h_{0}$ reach the critical value
$(2\gamma/Eh_{0})^{1/2}$ where $\gamma$ is the energy per unit surface area to
break the bonds at the crack tip.

A Dependency of the strain at fracture on the strain rate

The theory of Shrimali and Lopez-Pamies is based on the assumption that
fracture in “pure shear” (see Fig. 1) occur at a critical stretch that is
independent of the stretch rate. As support for this they cite several
experimental studies where this is indeed observed[16, 17]. However, the
experimental studies are for a limited range of applied stretch rates, and it
is easy to show that it cannot be true in general. To show this consider a
slab of elastomer (of undeformed height $h_{0}$) elongated at a rate
$\dot{h}$. We assume that the length $L$ of the slab is much bigger than the
height $h_{0}$ and that the crack tip is not close (on the scale of $h_{0}$)
to any end of the slab. For simplicity we assume linear viscoelasticity with
the frequency-dependent modulus $E(\omega)$. The modulus in the low-frequency
(rubbery) region $E(0)=E_{0}$ is much smaller than the modulus
$E_{1}=E(\infty)$ in the high frequency (glassy) region. (Note that the
rubbery region and the glassy region may be separated by 10 (or more) decades
in frequency.) If the elongation strain rate $\dot{\epsilon}=\dot{h}/h_{0}$
(or rather strain rate frequency $\omega=\dot{\epsilon}/\epsilon$) is very low
the rubber will effectively be in the rubbery state and the onset of crack
propagation will occur when the strain
$\epsilon_{0}=(2\gamma/h_{0}E_{0})^{1/2}$ where $\gamma$ (denoted by $G_{\rm
c}$ in Ref. [13]) is the energy per unit surface area to breaks the bonds at
the crack tip. However, if the strain rate is very high the fracture will
occur almost instantaneously and rubber will effectively be in the glassy
state everywhere both with respect to the deformations resulting from the
elongation and from the movement of the crack tip. Hence we can again apply
the standard Griffith fracture criteria but now using the high frequency
modulus so that the critical (fracture) strain
$\epsilon_{1}=(2\gamma/h_{0}E_{1})^{1/2}$ which is smaller than the low strain
rate fracture strain by a factor of $(E_{0}/E_{1})^{1/2}$ which is typically
in the range $0.1-0.01$ in practical cases. This conclusion is consistent with
the temperature-frequency relation valid for (simple) rubber materials[18]
where high frequencies are equivalent to low temperatures. At temperatures
below the glass transition temperature the time scale of deformation is
typically much higher than that of the (Brownian motion) of the molecular
chains of the rubber, and rubber compounds are hard and brittle like glass.

Experiments we have performed show that a critical stretch that is independent
of the stretch rate is a special case which is mostly encountered (in a finite
stretch rate range) in very tough viscoelastic materials[19, 20]. We have
performed few tests with the pure shear geometry but many tests in singe-edge
notch geometry (where the critical stretch depends on notch length) and the
stretch at break is markedly strain rate dependent, which may in part result
from a strain rate dependency of the bond breaking in the process zone. See
Ref. [19] and an earlier study on the fracture of very elastic
polyurethanes[20].

B Definition of the fracture energy

The fracture energy $G$ (denoted as $T_{\rm c}$ in Ref. [13]) is usually
measured under conditions where the strain far from the crack tip is in the
relaxed state. For example in some experiments the slab in Fig. 1 is first
stretched without the crack and then kept in the stretched state for
equilibrium to occur after which a crack is inserted using a razor blade. For
this situation $G(v)$ is the energy to break the bonds at the crack tip plus
the viscoelastic energy dissipation close to the crack tip due to its motion,
and will only depend on the crack tip speed $v$ (and the temperature, see
below). In this case $G(v)$ can be considered as a (useful) material parameter
which can be tabulated for different rubber compounds as done by Gent[4] for
three types of rubber [cis-polybutadiene (PB), styrene-butadiene copolymer
(SB) and ethylene-propylenecopolymer(EPR)].

For a general situation where the strain field far from the crack is time
dependent, as when the slab in Fig. 1 is elongated $h=h(t)$ in some arbitrary
complex way, Shrimali and Lopez-Pamies define the fracture energy as
$G=-dW/dA$ where $dW/dA$ is the change in the total (stored and dissipated)
deformation energy in the bulk with respect to added surface area $A$ of the
pre-existing crack. But using this definition $G$ will depend on the (time-
dependent) loading conditions and is no longer a material parameter but depend
on the loading history. Shrimali and Lopez-Pamies only studied the onset of
crack propagation where $v=0$ but in the most general situation where the
strain field far from the crack is time dependent, and the crack propagate
with the speed $v$, the crack propagation need to be studied theoretically (or
experimentally) for each such case separately using, e.g., numerical methods
(see Sec. F). The material input parameters for such studies are the
viscoelastic modulus and the energy to break the bonds per unit surface area
$G_{\rm c}=\gamma$. (In general $\gamma$ depend on complex processes occurring
in the crack tip process zone, which depend on the crack tip speed and the
temperature, i.e., $\gamma=\gamma(v,T)$ is a function of the crack tip speed
and the temperature.)

The theory proposed in Ref. [13] is stated to be valid only at the onset of
crack propagation. This case is not of very big interest in practical
applications since the (useful) toughness of rubber materials result from the
fact that the crack propagation energy increases very strongly with increasing
crack tip speed, e.g., by a factor of $E_{1}/E_{0}$ when the stress field far
from the crack is in the relaxed state.

Consider the system in Fig. 1 exposed to time-dependent external forces. In
Ref. [13] it was proposed that the crack will start to move at the time $t$
when the condition

$-{dW^{\rm Eq}\over dA}=G_{\rm c}$ $None$

is obeyed. Here $W^{\rm Eq}$ is the elastic energy stored in the system at
equilibrium i.e. the elastic energy after keeping the system which prevail at
time $t$ for an infinite long time with fixed boundary conditions (as given by
the boundary conditions at time $t$) on the part of the boundary that are not
traction free. As support for this equation the author use the experimental
observation that the fracture for the system shown in Fig. 1 occur at a fixed
strain independent of the strain rate. However, we have argued above that this
may be true only for low enough strain rates, and will break down for strain
rates where the deformation frequency
$\omega=\dot{\epsilon}/\epsilon=\dot{u}/u$ (see Fig. 1) occur in the glassy
response region. This result in a violation of (1) as can be easily seen by
the following argument: Consider the system in Fig. 1 and assume first the
case of extremely slow stretch rate. In this case the material behaves as an
elastic body with low-frequency elastic modulus $E_{0}$ and the energy balance
would then require

$dW^{\rm Eq}=-G_{\rm c}dA$ $None$

where $dW^{\rm Eq}=dU$ is the change of elastic energy
$dU=E_{0}\epsilon_{0}^{2}h_{0}dA/2$ of the relaxed material. Next consider
extremely high stretch rate, i.e. a step change of the remote displacement. In
this case the material is in the glassy region and will behave elastically
with modulus $E_{1}$ and the energy balance will require that the total change
of the elastic energy must balance the fracture energy

$dW^{\rm Eq}+dW^{\rm NEq}=-G_{\rm c}dA$ $None$

Here the change in the total elastic energy equal $dU=dW^{\rm Eq}+dW^{\rm
NEq}=E_{1}\epsilon_{1}^{2}h_{0}dA/2$. The conditions (2) and (3) give the
Griffith fracture results for $\epsilon_{0}$ ans $\epsilon_{1}$ quoted in Sec.
A. It is evident that Eq. (6) in Ref. [13] cannot hold true in all cases as
the authors propose.

C Velocity dependency of the crack propagation energy $G(v)$

As we understand it the theory developed in Ref. [13, 14, 15] only address the
onset of crack propagation. Therefore, the dependency of the fracture energy
$G(v)=\gamma[1+f(v)]$ on the crack-tip speed cannot be deduced from the
theory. When the crack is propagating at constant speed one has to consider
the influence of non-conservative work of internal stresses to write down the
correct energy balance equation. This was done in Ref. [21] where Eqs. (6),
(8), (9) and (10) give the change of the elastic energy, and of the non-
conservative work of internal stresses, upon a small displacement of crack
front both for the opening and closing crack, thus extending the Griffith
criterion to viscoelastic cracks moving at constant speed.

Here we note that the classical experiments of Gent measured the $G(v)$
relation by peeling rubber sheets apart[4]. By performing measurements at
different peeling speeds and temperatures and using the temperature-velocity
shifting procedure he could map out the full $G(v)$ (master) curve.
Measurements of this type cannot be analyzed using the theory of Shrimali and
Lopez-Pamies.

Figure 2:  Schematic coupling between viscoelasticity (blue domain) and strand
breakage (red domain) at the crack tip. The enlarged region shows the
occurrence of bond scission (yellow stars) in the elastomer network. Bond
scission and viscoelastic dissipation are strongly coupled, with a joint
increase in bond scission and viscoelastic dissipation between the low
viscoelasticity (i) and large viscoelasticity regimes (ii). Adopted from Ref.
[19].

D Crack tip process zone

Shrimali and Lopez-Pamies claim that the advantage of their approach compared
to earlier theories is that they do not need to invoke a crack tip process
zone. However, a crack tip process zone will occur in all real materials and
it is important to study its influence on the crack propagation. If one assume
a priory that no crack tip process zone occur one cannot claim that it will
have no influence on the viscoelastic part of the crack propagation energy.
Here we note that for cohesive crack propagation the crack tip process zone
can be very complex involving cavitation, formation of filaments (stringing)
and a recent study has shown that breaking of chemical bonds may occur far
away from the crack tip in a region overlapping with the region where
viscoelastic energy dissipation occur, see Fig. 2 and Ref. [19]. Thus bond
scission, far from being restricted to a constant level near the crack plane,
can be delocalized over hundreds of micrometers and increase by a factor of
100, depending on the temperature and stretch rate, and the energy dissipated
by covalent bond scission accounts for a much larger fraction of the total
fracture energy than was previously believed. The situation may be less severe
for adhesive crack propagation (crack propagation at the interface between a
flat rigid solid and an elastomer film adhering to the substrate) but even in
these cases for weakly crosslinked elastomers (as in pressure sensitive
adhesives) the crack tip process zone is very complex and spatially large[22].

The existence of a (complex) process zone in rubber-like materials can be
(indirectly) seen through the roughness profiles of the surfaces formed by the
crack propagation[23, 24]. This is valid for all real materials, not only
rubber. The morphology of fracture surfaces reflect the complex processes
occurring on different length scales close to the crack tip, and depend on the
microstructure of the material.

E Delayed fracture

Shrimali and Lopez-Pamies studied the so-called delayed fracture test[14]. In
a typical delayed fracture test, a sheet of the elastomer of interest
containing a pre-existing crack is subjected to a load that is applied rapidly
over a very short time interval $[0,t_{0}]$ and then held constant. Nucleation
of fracture from the pre-existing crack occurs at a critical time $t_{\rm
c}>t_{0}$ , hence the name of the test. Delayed fracture sometimes occurs also
at fixed grips conditions (fixed stretch) for which $W^{\rm Eq}$ is time
independent. This delayed fracture at constant stretch cannot be explained by
the Shrimali and Lopez-Pamies theory.

F Numerical approaches to viscoelastic crack propagation

Shrimali and Lopez-Pamies performed a numerical study using the finite element
method applied to an elastomer with non-linear rheology. To determine when the
crack start to move they used the criteria (1) [see Eq. (6) in Ref. [13]]
involving only $G_{\rm c}=\gamma$ and the equilibrium part of the stored
elastic energy. In our opinion this approach may be valid only at the onset of
crack propagation and for low enough stretch rate. An alternative more general
approach, which can be used to obtain the full $G(v)$ curve, is to use a
discretized solids where the “atoms” are connected by (realistic) non-linear
springs or bonds, e.g using a non-linear Rouse-like model for the network
chains[25]. Such atomistic models have been used to study the $G(v)$ relation
for crack propagation in silicone[26] and recently also for adhesive cracks
involving viscoelastic solids in contact with a rigid flat substrate[27]. An
advantage with this approach is that it can also include inertia effects which
are crucial in some cases (see Ref. [26]).

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