# FLUTE: Figurative Language Understanding through Textual Explanations

Tuhin Chakrabarty1 Arkady Saakyan1 Debanjan Ghosh2 Smaranda Muresan 1
1Department of Computer Science, Columbia University
2Educational Testing Service
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Figurative language understanding has been recently framed as a recognizing
textual entailment (RTE) task (a.k.a. natural language inference, or NLI).
However, similar to classical RTE/NLI datasets, the current benchmarks suffer
from spurious correlations and annotation artifacts. To tackle this problem,
work on NLI has built explanation-based datasets such as e-SNLI, allowing us
to probe whether language models are right for the right reasons. Yet no such
data exists for figurative language, making it harder to assess genuine
understanding of such expressions. To address this issue, we release flute, a
dataset of 9,000 figurative NLI instances with explanations, spanning four
categories: Sarcasm, Simile, Metaphor, and Idioms. We collect the data through
a model-in-the-loop framework based on GPT-3, crowd workers, and expert
annotators. We show how utilizing GPT-3 in conjunction with human annotators
(novices and experts) can aid in scaling up the creation of datasets even for
such complex linguistic phenomena as figurative language. The baseline
performance of the T5 model fine-tuned on flute shows that our dataset can
bring us a step closer to developing models that understand figurative
language through textual explanations.

##  1 Introduction

Figurative language such as metaphors, similes or sarcasm plays an important
role in enriching human communication, allowing us to express complex ideas
and emotions in an implicit way Roberts and Kreuz (1994); Fussell and Moss
(1998). However, understanding figurative language still remains a bottleneck
for natural language processing Shutova (2011). Recently Jhamtani et al.
(2021) show that when faced with dialog contexts consisting of figurative
language,some models show very large drops in performance compared to contexts
without figurative language. Despite the fact that Transformer-based pre-
trained language models (LMs) get even larger Brown et al. (2020); Raffel et
al. (2020), they are still unable to comprehend the physical world, cultural
knowledge, or social context in which figurative language is embedded Bisk et
al. (2020).

In recent years, there have been several benchmarks dedicated to figurative
language understanding, which generally frame “understanding” as a recognizing
textual entailment (a.k.a natural language inference (NLI)) task — deciding
whether one sentence (premise) entails/contradicts another (hypothesis)
Chakrabarty et al. (2021); Stowe et al. (2022); Srivastava et al. (2022).
However, similar to general NLI datasets, these benchmarks suffer from
spurious correlations and annotation artifacts McCoy et al. (2019); Poliak et
al. (2018b). These can allow large language models (LLMs) to achieve near
human-level performance on in-domain test sets, yet turn brittle when
evaluated against out-of-domain or adversarial examples Glockner et al.
(2018); Ribeiro et al. (2016, 2020). To tackle these problems, research in NLI
has argued that it is not enough to correctly predict the entail/contradict
labels, but also to explain the decision using natural language explanations
that are comprehensible to an end-user assessing model’s reliability Camburu
et al. (2018); Majumder et al. (2021); Wiegreffe et al. (2021a), leading to
novel datasets such as e-SNLI Camburu et al. (2018). Yet, there is no such
dataset for figurative language, hindering our ability to assess LLMs’ genuine
understanding of figurative language.

Table 1: flute examples of figurative text (hypothesis) and their respective
literal entailment(E) and contradiction (C) premises, along with the
associated explanations. * For simile, metaphor, and idiom, figurative
examples are the hypothesis whereas for sarcasm, we have both figurative and
literal hypotheses (see Section 2).

In this paper, we make several contributions towards the goal of building
models and assessing their ability to understand figurative language:

  * •

flute: a new benchmark for figurative language understanding through textual
explanations. FLUTE contains 9,000 high-quality <literal, figurative> sentence
pairs with entail/contradict labels and the associated explanations. The
benchmark spans four types of figurative language: sarcasm, simile, metaphor,
and idiom. Table 1 shows examples from our dataset. A noteworthy property of
flute is that both the entailment/contradiction labels and the explanations
are w.r.t the figurative language expression (i.e., metaphor, simile, idiom)
rather than other parts of the sentence.

  * •

A scalable model-in-the-loop approach for building flute. Model-in-the-loop
approaches (i.e., GPT-3 Brown et al. (2020) and crowdsourcing) have been
recently proposed to generate NLI datasets, as well as free-form textual
explanations (a.k.a natural language explanations Camburu et al. (2018)) for
model decisions Liu et al. (2022a); Wiegreffe et al. (2021a). For figurative
language, Ghosh et al. (2020) has shown that crowdworkers are mostly good at
performing minimum edits to generate a literal sentence from a sarcastic one
(e.g., using negation or antonyms), which can lead to trivial examples easily
classified by LLMs Chakrabarty et al. (2021). Thus, for building flute, we
leverage the power of GPT-3 to generate diverse and high quality literal text
(paraphrases/contradictions and/or explanations) using few-shot prompting,
coupled with minimal human involvement (e.g., crowdworkers to minimally edit a
literal sentence to make it sarcastic and experts for judging and minimally
editing GPT-3 output to ensure quality control) (Section 2).

  * •

Comprehensive set of experiments to assess flute’s usefulness towards building
models that understand figurative language. We propose a setup inspired by
instruction-based learning Mishra et al. (2021); Sanh et al. (2021); Wei et
al. (2021) and train a T5 Raffel et al. (2019) model to jointly predict the
label (entail/contradict) and explanation. We train two variants: T5 trained
on e-SNLI dataset Camburu et al. (2018) and T5 trained on flute. We evaluate
our model on the flute test set (Section 3.2). We propose extensive automatic
and human evaluation experiments to assess model understanding through
explanations (Section 3.2). We show that the model trained on flute produce
higher quality explanations compared to model trained on e-SNLI (Section 4).

Our code and experimental setup is available at
111https://github.com/tuhinjubcse/model-in-the-loop-fig-lang.Our data can be
accessed at 222https://huggingface.co/datasets/ColumbiaNLP/FLUTE

##  2 Model-in-the-loop for building flute

flute consists of pairs of premises (literal sentences) and hypotheses
(figurative sentences)333Given sarcasm is the opposite of the literal meaning,
we would only have contradictions in the dataset, thus we also generate a
literal hypothesis that entails the literal premise., with the corresponding
entailment or contradiction labels (NLI instances), along with explanations
for each instance (Table 1). We describe the model-in-the-loop methods for
creating premise-hypothesis pairs for each type of figurative language
(Section 2.1) and the associated explanations (Section 2.2).

###  2.1 flute: Premise-Hypothesis Pair Creation

(a) Model in the Loop for flute: Simile, Metaphor, Idiom Data

(b) Model in the Loop for flute: Sarcasm Data

####  2.1.1 Sarcasm

When asked to generate literal equivalents of sarcastic sentences crowdworkers
on Amazon Mechanical Turk (MTurk) usually perform trivial rephrasings at
word/phrase level Ghosh et al. (2020), which can lead to NLI datasets for
sarcasm understanding where LLMs can achieve near-human performance (95%) due
to simple lexical cues, such as negation or antonyms Chakrabarty et al.
(2021). Additionally, in many cases sarcasm data is collected from Twitter
using hashtags, e.g., #sarcasm, which can be noisy and not diverse.

To address these issues we take a model-in-the-loop approach: given a literal
sentence we first use GPT-3 with few-shot prompting to generate a literal
paraphrase, and then use crowdworkers to minimally edit this new literal
sentence to form a sarcastic one (Figure 1(b)).444Using GPT-3 to directly
generate sarcastic sentences led to low quality output (no semantic
consistency, social biases, stereotypes, or toxic content. Then we pair the
original literal sentence with generated literal paraphrase as entailment
pair, and with the sarcastic one as a contradiction pair. Below we describe
these two steps.

##### Entailment Pairs.

Jointly modeling emotion and sarcasm had been shown beneficial for sarcasm
detection Chauhan et al. (2020). Thus, we select the literal sentences from
the Empathetic Dialogue dataset Rashkin et al. (2019). Each conversation in
the dataset is grounded in a situation with emotion label provided. We select
literal sentences labeled with negative emotions such as _angry_ , _afraid_ ,
_embarrassed_.555Although positive emotions can be used to generate sarcasm,
we leave it for the future work. Including the emotion in the GPT-3 prompts
serves two purposes: (a) the generated paraphrases are complex and often more
creative than the original literal input, and (b) it is easier for
crowdworkers to transform paraphrases with emotional content into sarcastic
counterparts with minimal edits. To generate the literal paraphrases, we
provide the literal sentence and the associated emotion in the prompt and ask
GPT-3 to paraphrase the input (top part of Figure 1(b), see prompt in Appendix
A.1.2). Every paraphrase generated from GPT-3 is verified by 3 experts. If the
quality of the generated paraphrase is deemed insufficient, it is resampled
from the model. Any individual example undergoes at most three rounds of
sampling, with 15% of them judged as appropriate upon the first round.

##### Contradiction pairs.

We recruit crowd workers on MTurk to convert the manually checked
GPT-3-generated literal paraphrases into sarcastic sentences. The workers were
provided with the paraphrases and instructed to make minimal edits (e.g.,
through negations, antonyms) to generate sarcastic sentences. We conducted a
qualification test, and recruited 29 distinct workers from the original set of
85 workers.666with native English proficiency, location in the U.S. and a hit
success rate of 96%. We recruit two independent workers for every paraphrase
input. The resulting sarcastic outputs are verified by three experts. 25% of
instances were deemed insufficient quality and edited by the experts. Consider
the sarcasm example in Table 1. The literal input “My next door …” is the
premise. The first step generated the paraphrase hypothesis by adding the
implicitly stated emotion “annoyed” and paraphrasing. Next, Turkers modified
the paraphrase to its sarcastic counterpart - by replacing "annoying" (the
emotion word) to it antonym “pleasant”.

The final flute sarcasm benchmark consists of 2,678 sarcastic sentences
contradicted by the 1,339 seed literal sentences, as well as the 1,339
paraphrased sentences (entailment pairs), as seen in Table 2.

####  2.1.2 Simile

##### Entailment Pairs.

We start by extracting sentences containing similes from Chakrabarty et al.
(2022, 2021). To generate an entailment pair, we perform two steps (see Fig.
1(a)): 1) given the sentence that contains a simile, create an auxiliary
literal sentence by simply replacing the simile with the simile’s property
(e.g., replace _like a steam sauna_ with _hot and moist_); 2) given the
auxiliary literal sentence and the property, prompt GPT-3 to generate a
_literal paraphrase_ consistent with the property (see prompt in Appendix
A.2.3). Experts deemed 720 generated instances satisfactory. To illustrate the
pipeline, see the example for simile in Table 1: “In fact the assembly hall
was now like a steam sauna”. The object _steam sauna_ is replaced with its
property _hot and moist_ , and the resulting sentence is fed into GPT-3 to
generate the paraphrase “The assembly hall was now hot and moist, more so than
usual”.

##### Contradiction pairs.

To generate the contradictions, we prompt GPT-3 to invert the meaning of the
_literal paraphrase_ (see above) w.r.t the property (see prompt in Appendix
A.2.4). Out of 720 generated contradictions, experts deemed 642 satisfactory.
We further selected 108 challenging $<$simile, entailment, contradiction$>$
instances from Liu et al. (2022b), defined by low RoBERTa Zhuang et al. (2021)
logit of the correct option (see Appendix A.2.2 for details), for a total of
750 entailment and 750 contradiction pairs (Table 2).

####  2.1.3 Metaphors

##### Entailment Pairs.

We follow a similar model-in-the-loop approach as the one for similes (Figure
1(a)). We manually select a total of 750 metaphors from the following
datasets: Chakrabarty et al. (2021); Srivastava et al. (2022); Stowe et al.
(2022). Next, we prompt GPT-3 to generate paraphrases given the metaphoric
sentences (see prompt in Appendix A.4.2). Although the original datasets
contain the literal equivalents of the metaphoric sentence, they are not fully
adequate for our purposes because: (a) not all metaphor examples have the
literal counterpart, and (b) often, the literal counterpart is a minimal
modification (one-word edit) of the metaphor Srivastava et al. (2022);
Chakrabarty et al. (2021), which can lead to trivial examples for LLMs.

##### Contradiction Pairs.

To generate contradictions pairs, we start with the GPT-3 generated literal
sentence that entails the metaphoric sentence. Consider the metaphor in the
Table 1, “He _absorbed_ the knowledge or beliefs of his tribe” (taken from
Stowe et al. (2022)). In the original dataset, the non-entailment counterpart
is “He _absorbed_ the beverages of his tribe”, which is created by using a
verb in a different sense (literal sense of “absorb”) to fit the context of
beverage drinking. On the contrary, since we are interested in generating
instances that contradict the _metaphor itself_ , a more appropriate
modification would be “He _utterly_ _decimated_ his tribe’s most deeply held
beliefs” (more examples are in Table 6 in the Appendix). We follow the same
method to generate contradiction examples (using GPT-3) as for similes (see
prompt in Appendix A.4.3).

Both paraphrases and contradictions are verified by three experts and edited
when required. Our flute benchmark contains 750 entailment and 750
contradictions pairs for metaphors (Table 2).

####  2.1.4 Idioms

Observing the successful generations of paraphrases and contradictions by
GPT-3 in the case of simile and metaphors, we jointly generate paraphrases
along with contradictions using GPT-3 (See Figure 1(a) blue dotted lines). We
provide the idiom and its meaning in the prompt (see prompt in Appendix A.6).
Three experts manually verified all the generated sentences and edited a total
of 23% of total generations. We found that jointly generating paraphrases and
contradictions greatly eased the data creation process and resulted in
relatively high quality of generations.

flute benchmark consists of 1000 entailment and 1000 contradiction pairs for
idioms (Table 2).

| Entails | Contradicts | Total
---|---|---|---
Paraphrase | 1339 | - | 1339
\+ Sarcasm | - | 2678 | 2678
Simile | 750 | 750 | 1500
Metaphor | 750 | 750 | 1500
Idiom | 1000 | 1000 | 2000
Table 2: Dataset statistics showing distribution of Figurative Language across
flute.

###  2.2 flute: Generating Textual Explanations

Our task prediction requires that the model not only correctly infer the
label, but also explain _why_ a given premise entails or contradicts the
hypothesis. Towards this goal, we generate textual explanations for every
$<$premise, hypothesis$>$ pair.

For simile, metaphor, and sarcasm we provide the premise, hypothesis, and
label (entailment or contradiction) and prompt GPT-3 to generate an
explanation. We provide a natural language instruction followed by several
examples. We generate entailment and contradiction explanations separately.

For idioms, the idiom meaning in the seed dataset already makes up for a great
explanation. Thus, we utilize the provided idiom meaning to jointly generate
the explanation for the entailing premise, as well as for the contradicting
premise using GPT-3. Hence, for idiom data in addition to premise, hypothesis,
and labels, we also provide the idiom itself and its meaning in the prompt.

Three experts manually verified all explanations to ensure their correctness
and ability to explain the essence of the entailment or contradiction in
reasonable detail rather then learning a simple template (see Table 1). In
cases where explanations were not accurate, experts edited them to ensure they
are coherent, logically consistent, and grammatical. For explanations
pertaining to sarcasm+paraphrase, experts edited a total of 21% of the
generated explanations, while for simile, metaphor and idiom it was 27%, 40%
and 10% respectively, which further demonstrates the potential of using GPT-3
to significantly reduce the human effort that goes into collecting textual
explanations datasets. See Appendix A for details on hyperparameters and
prompts.

##  3 Experimental Setup

###  3.1 Models

Prior works in explainability have trained two types of models. Pipeline
models map an input to a rationale (I $\rightarrow$ R), and then a rationale
to an output (R $\rightarrow$ O).777Rationales are “textual explanation” in
this work, sometime used interchangeably. Joint Self Rationalizing models map
an input to an output and rationale (I $\rightarrow$ OR). Recently Wiegreffe
et al. (2021b) have exposed the short-comings of free-text pipelines and have
empirically shown that joint model rationales are more indicative of labels.
Following this, we fine-tune a joint self-rationalizing T5 model. Taking
advantage of the text-to-text format of T5 Raffel et al. (2020) and the recent
success of instruction-based models Sanh et al. (2021); Wei et al. (2021), we
design the following instruction for a given literal premise (P) and a
figurative hypothesis (H):

Does the sentence "P" entail or contradict the sentence "H"? Please answer
between "Entails" or "Contradicts" and explain your decision in a sentence.

The above instruction is fed to the encoder of T5. The decoder outputs the
label followed by the rationale. We fine-tune T5 with the following setups: in
the first one, we fine-tune on e-SNLI Camburu et al. (2018), and in the
second, we fine-tune on flute.

T5${}_{\text{e-SNLI}}$: e-SNLI Camburu et al. (2018) dataset comes with
supervised ground-truth labels and rationales. We fine-tune the 3B version of
T5 on e-SNLI for one epoch with a batch size of 1024, and an AdamW Optimizer
with a learning rate of $1e-4$. We remove the Neutral examples from e-SNLI
because our test data does not have such a category. We take the longest
explanation per example in e-SNLI since our data has only one reference
explanation. In case the explanations are more than one sentence we join them
using ‘and’ since our data contains single-sentence explanations. This leaves
us with 366,603 training and 6,607 validation examples.

T5${}_{\textsc{flute}}$: We fine-tune the 3B version of T5 model for 10 epochs
with a batch size of 1024, and an AdamW Optimizer with a learning rate of
$1e-4$ in a multitask fashion with data from all the four types of figurative
languages combined. Our training data consists of 7,035 samples which is 50X
smaller than e-SNLI. For validation we use 500 examples which is used for
selecting best checkpoint based on loss.

###  3.2 Evaluation Setup

To evaluate the above models, we built a test set by randomly selecting 750
instances (i.e., $<$premise, hypothesis$>$ pairs with associated explanations)
from the sarcasm dataset, and 250 examples each from simile, metaphor and
idiom datasets, for a total of 1,500 instances.

Below we describe several automatic metrics and human evaluations we consider
to assess the models’ ability to understand figurative language.

##### Automatic Metrics

To judge the quality of the explanations we compute the average between
BERTScore Zhang et al. (2020) 888We use the deberta-mnli version that has
shown to have highest correlation with human judges. and BLEURT Sellam et al.
(2020), which we refer to as _explanation score_ (between 0 and 100). Instead
of reporting only label accuracy, we report label accuracy at three thresholds
of explanation score (0, 50, and 60). Accuracy@0 is equivalent to simply
computing label accuracy, while Accuracy@50 counts as correct only the
correctly predicted labels that achieve an explanation score greater than 50.

##### Rationale Quality

Human simulatability Doshi-Velez and Kim (2017) has a rich history in machine
learning interpretability research as a reliable measure of rationale quality
from the lens of utility to an end-user. Simulatability measures the
additional predictive ability a rationale R provides over the input I for a
given label O, computed as the difference between task performance when a
rationale is given as input vs. when it is not (IR $\rightarrow$ O minus I
$\rightarrow$ O).In prior work on Explanability using Natural Language
Wiegreffe et al. (2021b) pointed out that model predictions are often unable
to be simulated because they degenerate under high values of noise. Following
Wiegreffe et al. (2021b) we thus use a variant of this metric that relies on
predicting the gold labels as our measure of rationale quality: (IR
$\rightarrow$ Ô minus I $\rightarrow$ Ô).

To compute rationale quality, we first train IR $\rightarrow$ O and I
$\rightarrow$ O for both flute and e-SNLI data. We then compute the test
accuracy for flute for both the IR $\rightarrow$ O and I $\rightarrow$ O
models trained on e-SNLI and flute using predicted rationales from the
respective I $\rightarrow$ OR models. We also compute rationale quality with
gold rationales (R*).

| T5${}_{\text{e-SNLI}}$ | T5${}_{\textsc{flute}}$ |
---|---|---|---
|  | Acc
---
@0
| Acc
---
@50
| Acc
---
@60
H${}_{\text{score}}$ | Yes% | No% |  | Acc
---
@0
| Acc
---
@50
| Acc
---
@60
H${}_{\text{score}}$ | Yes% | No%
Sarcasm | 60.6 | 15.7 | 2.4 | 34.2 | 14.7 | 52.0 | 91.6 | 86.2 | 56.2 | 85.3 | 75.3 | 8.7
Simile | 61.2 | 22.8 | 3.6 | 43.6 | 22.0 | 40.7 | 62.8 | 57.2 | 30.4 | 84.9 | 74.7 | 8.0
Metaphor | 81.8 | 31.8 | 11.6 | 55.3 | 36.0 | 28.0 | 73.3 | 55.6 | 23.7 | 80.2 | 64.0 | 6.0
Idiom | 84.8 | 46.4 | 7.6 | 60.9 | 37.3 | 24.7 | 79.2 | 77.2 | 66.8 | 83.1 | 69.3 | 8.7
Table 3:  Accuracy scores across four figurative language types by varying
thresholds of explanation score, along with human evaluation scores
H${}_{\text{score}}$, Yes% (higher is better), and No% (lower is better) for
explanations generated by T5 fine-tuned on e-SNLI (T5${}_{\text{e-SNLI}}$) and
T5 fine-tuned on flute (T5${}_{\textsc{flute}}$). $p<0.001$ via Wilcoxon
signed-rank test for all bolded results.

##### Human Evaluation.

Finally, we measure the quality of the generated textual explanations from
T5${}_{\text{e-SNLI}}$ and T5${}_{\textsc{flute}}$ models via the MTurk
platform. We recruit 79 crowd workers with at least 98% HIT approval rate. We
compute human judgement scores (H${}_{\text{score}}$), identical to the e-ViL
score in Kayser et al. (2021). For each NLI instance (a total of 200 random
samples, 50 per figurative language type), we present two textual explanations
generated by the two models (T5${}_{\text{e-SNLI}}$ and
T5${}_{\textsc{flute}}$) and ask three workers the following question: Given
the two sentences, does the explanation justify the answer above? We provide
four options: _Yes_ ($1$), _Weak Yes_ ($\frac{2}{3}$), _Weak No_
($\frac{1}{3}$), and _No_ ($0$). For each explanation, we average the scores
by the three annotators and report the sample average in Table 3 as
H${}_{\text{score}}$. If the answer is anything other than _Yes_ , we ask to
categorize the shortcomings of the explanation: _Insufficient Justification_ ,
_Too Trivial_ , _To Verbose_ , _Untrue to Input_ , _Violates Common Sense_
Majumder et al. (2021).

Three workers were recruited for each instance and the IAA using
Krippendorff’s $\alpha$ Krippendorff (2011) between the workers is $0.45$,
indicating moderate agreement. See Appendix B for details on the
H${}_{\text{score}}$ computation and Figure 4 for a screenshot of the MTurk
task interface.

##  4 Results and Discussion

Table 3 shows accuracy at varying _explanation score_ thresholds. A threshold
of 0% does not account for the quality of the textual explanation and is
equivalent to simply reporting label accuracy. With an increase in threshold
to greater than 50% we see accuracy scores dropping almost by half for
T5${}_{\text{e-SNLI}}$, showing most explanations generated from the model
trained on e-SNLI are of poor quality. By increasing the threshold to greater
than 60%, the accuracy scores further decrease, demonstrating that models like
T5 fine-tuned on e-SNLI still struggle generating correct explanations even
when the label predictions are correct. On the contrary, Table 3 shows that
the accuracy scores for T5${}_{\textsc{flute}}$ are significantly higher for
each type of figurative language, indicating higher quality of explanations
achieved by fine-tuning the model on our dataset.

| Ac (IR $\rightarrow$ O) | Ac (I $\rightarrow$ O) | RQ.
---|---|---|---
e-SNLI | 68.4 | 74.5 | -6.1
flute | 89.3 | 90.5 | -1.2
flute (R*) | 95.6 | 90.5 | 5.1
Table 4: Rationale Quality ($p<0.001$ via Wilcoxon signed-rank test) (higher
is better) on FLUTE test set using accuracy of IR $\rightarrow$ O and I
$\rightarrow$ O models trained on e-SNLI and flute respectively. We use the
predicted rationale R obtained from respective I $\rightarrow$ OR models
T5${}_{\text{e-SNLI}}$ and T5${}_{\textsc{flute}}$ as well as gold rationale
(R*). Rationale Quality and accuracy are abbreviated as RQ. and Ac.

In terms of the Rational Quality (Table 4), using predicted rationales from I
$\rightarrow$ OR we observe that model fine-tuned on e-SNLI generates
explanations that are less useful for gold label prediction compared to a
model trained on flute. Simulatability scores obtained using predicted
rationales are negative compared to gold rationales since rationales generated
by the model are still imperfect and may introduce noise that negatively
affects the prediction. The positive score of gold rationales indicates that
explanations provided in the flute dataset are indeed useful for the model to
make predictions.

Premise |  | The camp life really had energized him.
---
Hypothesis |  | Life in the camp wilted him.
---
Label |  | Contradiction.
---
Gold_Exp |  | To wilt means to lose vitality or strength,
---
while to energize means to give strength
or energy to.
T5${}_{\text{e-SNLI}}$ |  | If the camp life energized him, then
---
he is not wilted.
T5${}_{\textsc{flute}}$ |  | To wilt something means to lose its energy or
---
vitality, while to have energized something
mean to feel a sense of excitement or life.
Premise |  | No one came to my wedding.
---
Hypothesis |  | No big deal that there were no guests at my
---
wedding reception considering it’s the only
one I’ll ever have.
Label |  | Contradiction.
---
Gold_Exp |  | A wedding reception is supposed to be a
---
joyous event with many guests, so not
having any guests at it would make it a
very uneventful and pointless one.
T5${}_{\text{e-SNLI}}$ |  |  No one is same as No big deal.
---
T5${}_{\textsc{flute}}$ |  | A wedding reception is a celebration of the
---
couple’s union and is usually attended by
family and friends so not having guests
there would be seen as snub.
Table 5: Examples of T5${}_{\text{e-SNLI}}$ and T5${}_{\textsc{flute}}$ model
generated explanations vs. gold explanations for NLI involving metaphor (top)
and sarcasm (bottom). More examples in Table 7 in Appendix. Figure 2: Bar plot
of the number of crowd worker-identified shortcomings of explanations
generated by T5${}_{\text{e-SNLI}}$ and T5${}_{\textsc{flute}}$ by shortcoming
and model type as percent of the sample (lower means fewer shortcomings). An
extended plot by figurative language type is available in Appendix, Figure 5.

Table 3 also presents the H${}_{\text{score}}$ scores for explanations
collected from human evaluation (see Appendix B and prior work Kayser et al.
(2021); Majumder et al. (2021) for details on computation). We notice the
scores for T5${}_{\textsc{flute}}$ are 51.1, 41.3, 24.9, 22.2 points better
than for T5${}_{\text{e-SNLI}}$ for Sarcasm, Simile, Metaphor, Idiom (in that
order). Crowd workers answered with solid ‘Yes’ (that the explanation
justifies the label) in 43.4% more cases on average when presented
explanations from T5${}_{\textsc{flute}}$ compared to T5${}_{\text{e-SNLI}}$.
Likewise, they answered ‘No’ in 28.5% less cases on average when presented
explanations from T5${}_{\textsc{flute}}$ compared to T5${}_{\text{e-SNLI}}$.
These results demonstrate a notably higher quality of explanations generated
by the model fine-tuned on flute compared to the model fine-tuned on e-SNLI,
despite the significantly larger size of the latter dataset.

Figure 2 shows that for cases where crowd workers did not choose an absolute
’Yes’, the most common error category to be found is Insufficient
Justification. The percentage of explanations for which shortcomings were
identified is higher or the same for across all categories for
T5${}_{\text{e-SNLI}}$. The second example (“No one is same …”) in Table 5
shows an insufficient explanation generated by T5${}_{\text{e-SNLI}}$.
T5${}_{\text{e-SNLI}}$ generated explanations were also more frequently marked
as Too Trivial. Often, they do not explain the reasoning but rather follow a
standard template if A then not B or if A then B, such as the first example in
Table 5 for T5${}_{\text{e-SNLI}}$. We share more such examples of erroneous
explanations in the Appendix, Table 8.

##  5 Related Work

In recent years, evaluating how well RTE models can capture specific
linguistic phenomena such as figurative language has attracted many NLP
researchers. In earlier work, Agerri (2008) analyze metaphor examples in the
RTE-1 Dagan et al. (2006) and RTE-2 Bar-Haim et al. (2006) datasets, whereas
Poliak et al. (2018a)’s diverse collection of RTE examples contains one type
of figurative language - puns. Our research is closer to Chakrabarty et al.
(2021), however, flute has better diversity and also contains explanations for
each NLI instance, as well as multiple expert checks to ensure higher quality
(see Section 2). A portion of flute’s metaphors are based on the Big-bench
corpora Srivastava et al. (2022) as well as the IMPLI dataset Stowe et al.
(2022), which is inspired by Zhou et al. (2021)’s prior work on the paired
idiomatic and literal dataset. However, there are several distinctions between
these datasets and flute. First, not all the metaphors in Big-bench have
literal paraphrase and contradictions. Second, in case of both the Big-bench
and IMPLI dataset, the non-entailment examples are created via minimal edits
to the original metaphor, often resulting in neutral examples or contradiction
to the non-metaphoric part of the sentence. In flute, we ensure that the non-
entailment examples are in contradiction to the metaphor (Table 6).

One of the motives for having NLEs with <literal, figurative> sentence pairs
like flute does is to evaluate model’s ability to explain their decisions.
Recent datasets such as CoS-E Rajani et al. (2019), Movie Reviews Zaidan and
Eisner (2008), and e-SNLI Camburu et al. (2018) have been released in a
similar vein. Recent work has also leveraged large language models to explain
humor in image captions Hessel et al. (2022) or sarcasm in dialouges Kumar et
al. (2022). The e-SNLI dataset (i.e., NLE of the entailment relations in the
SNLI dataset) has been used in related work Narang et al. (2020); Yordanov et
al. (2021); Majumder et al. (2021); Feng et al. (2022) for explanation
generation. In contrast to e-SNLI, which was created via crowdsourcing, we
rely on a model-in-the-loop framework for flute influenced by Wiegreffe et al.
(2021a). We have utilized the e-SNLI dataset for explanation generation and
observed a T5 model trained on flute performs notably better.

##  6 Conclusion

We release flute, a dataset for figurative language understanding spanning
across Sarcasm, Similes, Metaphors, and Idioms collected via a model-in-the-
loop framework. To encourage genuine understanding of figurative language, our
data also contains free-form textual explanations. Upon conducting baseline
experiments with state-of-the-art benchmark models (i.e., models trained on
the the e-SNLI dataset), we notice those models perform poorly. In contrast,
performance of the T5 model fine-tuned on flute shows that our dataset can
bring us a step closer to developing models that understand figurative
language through textual explanations. We hope our research on explanation
generation for figurative language will be a fruitful future direction, and
our dataset will be a challenging testbed for experimentation.

## Limitations

While we focused on four types of figurative language and generated a diverse
dataset, we believe it is just a first step towards capturing figurative NLI
instances and their explanations, since figurative language is able to draw on
a wide variety of cultural knowledge and contexts. Although the sarcasm
portion captures the most common type of incongruity between sarcastic context
and sentiment, sarcasm can manifest in many different forms - situational,
underplayed, or dramatic, for which examples and explanations will differ.
Finally this study doesn’t explicitly focus on faithfulness of model generated
Natural Language Explanations, however we hope to evaluate the faithfulness of
these using methods described in contemporaneous literature on faithfulness of
NLE’s Sia et al. (2022); Chan et al. (2022)

## Ethics Statement

We use a model-in-the-loop framework for content generation. Although we use
language models trained on data collected from the Web, which have been shown
to have issues with gender bias and abusive language, we have verified
carefully that our FLUTE data does not contain any toxic text and it underwent
manual inspection by the authors and experts.We pay Amazon Mechanical Turkers
at the rate of 15 $/hr which is compliant with the minimum hourly wage in
United States.

## Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments.
Finally we are also grateful to Esin Durmus and Faisal Ladakh for initial
brainstorming on model in the loop strategies for dataset creation. Tuhin is
funded by Columbia Center of Artifical Intelligence & Technology (CAIT) and
the Amazon Science Ph.D. Fellowship).

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##  Appendix A Appendix

In this section we report the details of the experiments (e.g.,
hyperparameters used for GPT-3 based generations, examples, as well as the
prompts for the figurative language.

###  A.1 Sarcasm dataset

####  A.1.1 Hyperparameters for the sarcasm dataset

We use GPT-3-Davinci-001 model for auxillary paraphrase generations from which
crowd workers create sarcasm. To generate paraphrase, we use the following
hyperparameters: temperature=1, max tokens=100, top p=0.9, best of=1,
frequency penalty=0.5, presence penalty=0.5.

To generate explanations, we use the following hyperparameters:
temperature=1.0, max tokens=100, top p=0.9, frequency penalty=0.5, presence
penalty=0.5, stop=["."].

####  A.1.2 Prompts for generation of Paraphrase from which Sarcasm is created

You will be presented with examples of some literal input sentences and their
creative paraphrases. For each example, we also provide the associated
emotion. Your task is then to generate a creative paraphrase for a given
literal sentence where the creative paraphrase should reflect the associated
emotion without changing its meaning. Make sure to use some sort of humor and
commonsense about everyday events and concepts

  * 1) Literal: A lot of people have got engaged recently.

  * Emotion: surprised

  * Creative Paraphrase: The way all the couples are pairing off lately and naming the big day, I think Cupid’s really busy.

  * 2) Literal: We have enough candles mom

  * Emotion annoyed

  * Creative Paraphrase: I think the Catholic church is going to have to canonize a whole new generation of saints to justify our candle use mom
…

’

####  A.1.3 Prompts for generation of Explanation for Paraphrase from which
Sarcasm is created (Entailment)

You will be presented with examples of two sentences typically a premise along
with an entailing paraphrase of the premise called the hypothesis. Your task
is to generate natural language explanations to justify the Entailment between
the premise and the hypothesis.

  * 1) Premise: Awful seeing a naked man run through my neighborhood.

  * Hypothesis: The sight of a man running through my neighborhood sans clothes was pretty disgusting.

  * Explanation: It is socially unacceptable to not wear clothes and step out of one’s house so seeing a man who is running naked in the neighborhood is pretty shameful and disgusting.

  * 2) Premise: My mother didn’t cook her chicken all the way through at dinner the other night.

  * Hypothesis: The fact that my mother didn’t cook her chicken all the way through at dinner makes me feel like I’m going to vomit.

  * Explanation: Eating undercooked chicken can cause food poisoning and so finding out that the chicken at dinner wasn’t cooked all the way through often makes people throw up …

####  A.1.4 Prompts for generation of Explanation for Sarcasm (Contradiction)

You will be presented with examples of some literal and sarcastic sentences.
Your task is then to write explanations to justify why it is sarcastic w.r.t
the literal

  * 1) Literal: When I moved into my apartment it was full of bugs

  * Sarcasm: I absolutely loved when I moved into my apartment and found it crawling with bugs.

  * Explanation: Bugs are usually disgusting and most people are terrified of them therefore it is unlikely to love seeing someone’s apartment infested by them.

  * 2) Literal: I’ve been hearing some strange noises around the house at night.

  * Sarcasm: I am completely comforted by the weird noises I keep hearing around the house at night.

  * Explanation: Hearing weird noises around the house at night could invoke a potential danger such as a robbery or someone breaking in with malicious intent which makes someone scared rather than comforted. …

###  A.2 Simile dataset

####  A.2.1 Hyperparameters for the simile dataset

We use GPT-3-Davinci-002 model for simile data generations. To generate
paraphrase, we use the following hyperparameters: temperature=1, max
tokens=256, top p=0.5, best of=1, frequency penalty=0.5, presence penalty=0.1.

To generate contradictions, we use the following hyperparameters:
temperature=0, max tokens=100, top p=1, frequency penalty=0, presence
penalty=0.

To generate explanations, we use the following hyperparameters:
temperature=0.7, max tokens=86, top p=1, frequency penalty=0, presence
penalty=0, stop=["."].

Figure 3: Model-in-the-loop to generate a Simile NLI dataset.

####  A.2.2 Challenging instances for the simile dataset

To select challenging instances from FigQA dataset Liu et al. (2022b), we use
RoBERTa fine-tuned on SNLI, MNLI, FEVER-NLI, and ANLI (R1, R2, R3). We choose
<simile, literal, contradiction> instances by taking the average between
RoBERTa logit for entailment given <simile, literal> as input and the logit
for contradiction given the <simile, contradiction> input. We then choose
instances with the lowest such score. In this way, we are able to select
instances for which RoBERTa is essentially confusing entailment and
contradiction. From our observation, these are usually ironic similes, e.g.
for the simile _I’m sharp as a pillow_ and the literal sentence _I’m not
sharp_ , RoBERTa would have a low logit for entailment, while for the sentence
_I’m sharp_ it would have a low logit for contradiction.

####  A.2.3 Prompts for Simile Paraphrase Generation

You will be presented with examples of some literal input sentences and their
creative paraphrases. You will also be presented words that need to be
preserved. Your task is to generate a creative paraphrase for a given literal
sentence consistent in meaning. DO NOT CHANGE words after "Preserve:" keyword.

  * 1\. Sentence: overwhelmingly , it began to draw him in.
Preserve: overwhelmingly
Creative Paraphrase: He was overwhelmingly obsessed with it.
. . .

####  A.2.4 Prompts for Simile Contradiction Generation

You will be presented with a Sentence and a Property. Your goal is to invert
the meaning of the Sentence with respect to the Property via a minimal edit.

  * 1\. Sentence: The place looked impenetrable and inescapable.
Property: impenetrable and inescapable
Inversion: This place looked easy to walk into and exit from.
. . .

####  A.2.5 Prompts for Simile Contradiction Explanation Generation

You will be provided with a Simile and a contradictory sentence after the word
"Contradiction". Your task is to explain why the contradictory sentence
contradicts the Simile.

  * 1\. Simile: like a psychic whirlpool , it began to draw him in.
Contradiction: Mildly, it began to draw him in
Explanation: A whirlpool is a strong current, so a psychic whirlpool drawing
in indicates that it was drawing him in intensely, rather than mildly.
. . .

###  A.3 Prompts for Simile Entailment Explanation Generation

You will be presented with a sentence containing a simile (Simile Sentence)
and an entailing sentence (Entail Sentence). Please provide an explanation for
why Simile Sentence is implied by the Entail Sentence.

  * 1) Simile Sentence: The place looked like a fortress
Entail Sentence: The place looked impenetrable and inescapable
Explanation: A fortress is a military stronghold, hence it would be very hard
to walk into, or in other words impenetrable.
. . .

###  A.4 Metaphor dataset

####  A.4.1 Hyperparameters for the metaphor dataset

We use GPT-3-Davinci-002 model for metaphor data generations. To generate
paraphrase, we use the following hyperparameters: temperature=1, max
tokens=100, top p=0.8, best of=1, frequency penalty=0.5, presence penalty=0.1.

To generate contradictions, we use the following hyperparameters:
temperature=0, max tokens=100, top p=0.8, frequency penalty=0.5, presence
penalty=0.1.

To generate explanations, we use the following hyperparameters:
temperature=0.8, max tokens=100, top p=0.9, frequency penalty=0, presence
penalty=0, stop=["."].

####  A.4.2 Prompts for Metaphor Paraphrase Generation

You will be presented with examples of some metaphor input sentences and their
creative paraphrases. Your task is to generate a creative paraphrase for a
given literal sentence consistent in meaning.

  * 1\. Sentence: A golden sun shines high in the sky.
Creative Paraphrase: a very bright sun shines high in the sky.
. . .

####  A.4.3 Prompts for Metaphor Contradiction Generation

You will be presented with examples of some literal input sentences and their
contradictions. Your task is then to generate contradiction of the new
sentences via a minimal edit.

  * 1\. Sentence: The company released him after many years of service.
Contradiction: The company hired him after many years of service.
. . .

####  A.4.4 Prompts for Metaphor Entailment Explanation Generation

You will be presented with examples of sentences containing a metaphor along
with an entailing paraphrase of the sentence. Your task is to generate natural
language explanations to justify the Entailment with the input.

  * 1\. Metaphor: Krishna is an early bird.
Contradiction: Krishna wakes up early everyday.
Explanation: Early bird means the person who wakes up early in the morning.
. . .

####  A.4.5 Prompts for Metaphor Contradiction Explanation Generation

You will be provided with a sentence containing a Metaphor and a contradictory
sentence after the word “Contradiction". Your task is to explain why the
contradictory sentence contradicts the Metaphor.

  * 1\. Metaphor: Joseph has the heart of a lion.
Contradiction: Joseph has the calm demeanor of a lamb.
Explanation: A lamb is typically seen as a gentle and timid creature while a
lion is seen as a brave and fierce creature.
. . .

Metaphor |  A weather vane _crowns_ the building.
---|---
Original Entailment |  The king crowned the prince.
GPT-3 Contradiction |  A weather vane mars the building.
Metaphor |  A _golden_ sun shines high in the sky.
Original Entailment |  A very expensive sun shines high in the sky A sunset is setting in the west.
GPT-3 Contradiction |  A sunset is setting in the west.
Metaphor |  Fear had changed him to a _shaken jelly_.
Original Entailment |  He was afraid of shaking jelly.
GPT-3 Contradiction |  He had conquered his fear and now he is a strong and capable person.
Table 6: Example of Metaphors and their contradictions (from prior work and
generated for this paper.). Note, examples from prior work replace the
metaphor sentence with adding words that fits into their context whereas in
this work we generate examples that _contradicts_ the metaphor.

###  A.5 Idiom dataset

####  A.5.1 Hyperparameters for the idiom dataset

We use GPT-3-Davinci-002 model for idiom data generations. To jointly generate
paraphrase and contradiction, we use the following hyperparameters:
temperature=1, max tokens=200, top p=0.9, best of=1, frequency penalty=0.5,
presence penalty=0.5, stop=[".."].

To jointly generate explanations for entailment and contradiction, we use the
following hyperparameters:  temperature=0.7, max tokens=256, top p=0.9,
frequency penalty=0, presence penalty=0, stop=[".."].

###  A.6 Prompts for joint paraphrase and contradiction generation for idioms

You will be presented with examples of some input sentences containing an
idiom. You will be provided with the meaning of the idiom. Your task is to
first generate a paraphrase that complies with the meaning of the idiom and
then generate a negation of the paraphrase that contradicts the meaning of the
idiom. Please look at the span within bold tags when performing paraphrase and
negation.

  * 1) Sentence: He looked great, and he was smiling <b>to beat the band</b>.

  * Idiom: to beat the band

  * Meaning: To a huge or the greatest possible extent or degree.

  * Paraphrase: He looked awesome and was smiling <b>hilariously in an uncontrollable manner</b>

  * Negation: He looked awesome and was smiling <b> in a very coy and restrained manner</b>.
…

####  A.6.1 Prompts for joint generation of Explanation for idioms

You will be presented with examples of sentences containing an idiom along
with an entailing and contradictory paraphrase of the sentence. Your task to
generate natural language explanations to justify the Entailment or
Contradiction with the input.

  * 1) Sentence: Not to share the bank with the table, or to take some minor part of it, but to go the whole hog.

  * Idiom: go the whole hog

  * Meaning: To do something as thoroughly as possible or without restraint.

  * Entailment: Not to share the bank with the table, or to take some minor part of it, but to take it all for themselves without any restraint.

  * Contradiction: Not to share the bank with the table, or to take some minor part of it, but to show some restraint and not go overboard.

  * Entail_Explanation: Usually to go the whole hog refers to do something as thoroughly as possible , taking it all for oneself or without any restraint.

  * Contra_Explanation: Usually to go the whole hog refers to do something as thoroughly as possible, without any sort of restraint and is often characterized by being extreme or overboard..

  * 2) Sentence: I told her almost everything, including my reticence about seeing this work of literature go through the mill which was vanity publishing. Idiom: go through the mill

  * Idiom: go through the mill

  * Meaning: To be abused or treated very harshly; to suffer intense anguish, stress, or grief.

  * Entailment: I told her almost everything, including my reticence about seeing this work of literature be abused and treated in an extremely harsh manner which was vanity publishing.

  * Contradiction: I told her almost everything, including my reticence about seeing this work of literature be celebrated and treated in an extremely proper manner which was vanity publishing.

  * Entail_Explanation: To go through the mill in the context here refers that vanity publishing will abuse and treat the work of literature being very poorly.

  * Contra_Explanation: Usually when we say go through the mill it does not mean something being celebrated and treated well but instead being abused and treated poorly which is being said here in the context of vanity publishing doing to the work of literature..
…

Premise |  | My father continuously gambles all his earnings away even though he has two minors to
---
support.
Hypothesis |  | My father is the best dad in the world who gambles all his earnings away and leaves me
---
and my sister struggling.
Gold |  | A great parent is someone who provides for their children and protects them but gambling
---
away all the money earned leaves the children in a difficult position and
hence the father
cannot be considered the best dad.
FLUTE |  | Gambling away one’s earnings is not a good thing and it can lead to the father not being
---
able to support his children who are dependent on him.
e-SNLI | Gambling all his earnings away is not the best dad in the world.
Premise | He ran slowly.
Hypothesis | He ran like an olympic sprinter.
Gold |  | An Olympic sprinter is someone who runs extremely fast, so saying someone ran like
---
one would imply they ran quickly, not slowly
FLUTE |  | Olympic sprinters are known to run very fast, so saying someone ran like an olympic
---
sprinter would mean they ran very quickly
e-SNLI | Slowly and sprinter are not the same
Premise |  | He’s totally trustworthy as an executive secretary, but he’s relatively new and
---
inexperienced and does not know what to do on occasions like this
Hypothesis |  | He’s totally trustworthy as an executive secretary, and has been around long enough
---
to know the ropes on occasions like this.
Gold |  | To know the ropes means to understand or be familiar with the details, but the person is
---
relatively new and inexperienced
FLUTE |  | To know the ropes means to have experience in the job, but in this context the person is
---
relatively new and inexperienced
e-SNLI |  | He’s not been around long enough to know the ropes and he is relatively new
---
and inexperienced.
Table 7: Example of Sarcasm, Simile and Idiom NLI with respective gold and
generated explanations from T5 by finetuning on FLUTE and e-SNLI

##  Appendix B Details of Human evaluation

We follow Kayser et al. (2021) in all the below human evaluation procedures.
Refer to Figure 4 for the example of the interface for crowdworkers. We first
ask the crowdworkers to identify the relationship between the literal and
figurative sentence (whether it is a contradiction or entailment). Using in-
browser checks, we ensure that the crowdworkers understood the NLI pair by
only accepting the submission if the relationship was identified correctly.

Then, we provide 2 explanations: one generated by T5${}_{\textsc{flute}}$ and
one generated by T5${}_{\text{e-SNLI}}$. The crowdworkers do not know which
one is which. For each NLE, we ask: Given the two sentences, does the
explanation justify the answer above?, and provide four options: Yes, Weak
Yes, Weak No, and No. We also ask to provide the shortcomings of the
explanations if the worker selected a score lower than Yes. The workers have
the following options to choose from, following prior work by Majumder et al.
(2021); Kayser et al. (2021): _Violates Common Sense_ , _Insufficient
Justification_ , _Untrue to Input_ , _Too Trivial_ , _To Verbose_. Some
examples of these shortcoming are provided in Table 8. In addition to the
Figure 2, we provide the bar plot of the number of shortcomings as percentage
of the sample by figurative language type in Figure 5.

We map the answers to $1,\frac{2}{3},\frac{1}{3},0$ respectively. Then, we
compute the average score across 3 workers per entry, and the sample average
per figurative language type for the corresponding model.

Figure 4:  Amazon Mechanical Turk interface, borrowed from Kayser et al.
(2021), asking to first select the relationship between the sentences, and
then evaluate the explanation.

Shortcoming | Premise | Hypothesis | Label | Explanation
---|---|---|---|---
T5${}_{\text{e-SNLI}}$ | T5${}_{\textsc{flute}}$
| Violates
---
Common
Sense
The gesture was awkward. |  | The gesture was
---
like a ballet move
C |  | A gesture is not
---
a ballet move.
| A ballet move is
---
graceful and fluid,
while an awkward
gesture is not
graceful or fluid.
| Insufficient
---
Justification
| I stayed in a house with
---
a rat problem for a while,
it was not fun.
| I simply cannot
---
wait to stay in
a house again
that had a rat
problem because
it was not only
a lot of fun but also
super enjoyable
C |  | It was either not
---
fun or not fun.
| Rats are often
---
associated with
being filthy and
unhygienic and
so staying in a house
with rat problems
would not be fun at all.
| Untrue to
---
Input
| Even though he was
---
strong in his magic,
it felt weak compared
to the inferno I’d faced
when I killed Mab.
| He was strong
---
in his magic but,
compared to the
blazing inferno that
i’d faced when i’d
killed mab, his
power felt like
a candle flame.
E |  | The candle flame
---
is the inferno.
| A candle flame is a
---
very small flame,
which is weak compared
to blazing inferno.
Too Trivial | He ran fast. |  | He ran like
---
an olympic sprinter.
E |  | He ran fast like
---
an olympic sprinter.
| An olympic sprinter
---
is a very fast person,
so if he ran like an
sprinter, it would mean
that he ran fast.
Too Verbose |  | She asked the witnesses
---
for information.
| She petitioned the
---
witnesses for
information.
E |  | She asked the
---
witnesses for
information and
petitioned the witness.
| Petitioning someone
---
means asking them for
information, which is
what she did
in this sentence.

Table 8:  Examples of shortcomings of T5${}_{\text{e-SNLI}}$ explanations. For
this table, T5${}_{\text{e-SNLI}}$ explanations were sorted by most crowd
workers votes for a respective shortcoming. Figure 5: Bar plot of the number
of crowd worker-identified shortcomings of explanations generated by
T5${}_{\text{e-SNLI}}$ and T5${}_{\textsc{flute}}$ by type of shortcoming,
figurative language type, and by type of model as percent of the sample (lower
means fewer shortcomings).


# Steady gradient Ricci solitons with nonnegative curvature operator away from
a compact set

Ziyi $\text{Zhao}^{{\dagger}}$ and Xiaohua $\text{Zhu}^{{\ddagger}}$ BICMR and
SMS, Peking University, Beijing 100871, China<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

Let $(M^{n},g)$ $(n\geq 4)$ be a complete noncompact $\kappa$-noncollapsed
steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}>0$ away from a compact
set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact
split limit Ricci flow of type I arising from the blow-down of $(M,g)$, if
there is an $(n-1)$-dimensional noncompact split limit Ricci flow.
Consequently, the compact split limit ancient flows of type I and type II
cannot occur simultaneously from the blow-down. As an application, we prove
that $(M^{n},g)$ with $\rm{Rm}\geq 0$ must be isometric the Bryant Ricci
soliton up to scaling, if there exists a sequence of rescaled Ricci flows
$(M,g_{p_{i}}(t);p_{i})$ of $(M,g)$ converges subsequently to a family of
shrinking quotient cylinders.

###### Key words and phrases:

Steady gradient Ricci soliton, Ricci flow, ancient $\kappa$-solution, Bryant
Ricci soliton

###### 2000 Mathematics Subject Classification:

Primary: 53E20; Secondary: 53C20, 53C25, 58J05

${\ddagger}$ partially supported by National Key R&D Program of China
2020YFA0712800, 2023YFA1009900 and NSFC 12271009.

##  1\. Introduction

Let $(M^{n},g;f)$ $(n\geq 4)$ be a complete noncompact $\kappa$-noncollapsed
steady gradient Ricci soliton with curvature operator $\rm{Rm}\geq 0$ away
from a compact set $K$ of $M$. Let $g(\cdot,t)=\phi^{*}_{t}(g)$
$(t\in(-\infty,\infty))$ be an induced ancient Ricci flow of $(M,g)$, where
$\phi_{t}$ is a family of transformations generated by the gradient vector
field $-\nabla f$. For any sequence of $p_{i}\in M$ $(\to\infty)$, we consider
the rescaled Ricci flows $(M,g_{p_{i}}(t);p_{i})$, where

(1.1) $\displaystyle g_{p_{i}}(t)=r_{i}^{-1}g(\cdot,r_{i}t),$

$r_{i}R(p_{i})=1$. By a version of Perelman’s compactness theorem for ancient
$\kappa$-solutions [14, Proposition 1.3], we know that
$(M,g_{p_{i}}(t);p_{i})$ converge subsequently to a splitting flow
$(N\times\mathbb{R},\bar{g}(t);p_{\infty})$ in the Cheeger-Gromov sense, where

(1.2) $\displaystyle\bar{g}(t)=h(t)+ds^{2},~{}{\rm on}~{}N\times\mathbb{R},$

and $h(t)$ ($t\in(-\infty,0]$) is an ancient $\kappa$-solution on an
$(n-1)$-dimensional $N$. For simplicity, we call $(N,h(t))$ a compact split
limit flow (by the blow-down) if $N$ is compact, otherwise, $(N,h(t))$ is a
noncompact split limit flow if $N$ is noncompact.

In this paper, we extend our previous result [14, Corollary 0.3] from the
dimension $4$ to any dimension. Namely, we prove

######  Theorem 1.1.

Let $(M^{n},g)$ $(n\geq 4)$ be a noncompact $\kappa$-noncollapsed steady
gradient Ricci soliton with nonnegative curvature operator. Suppose that there
exists a sequence of $p_{i}\in M$ $(\to\infty)$ such that the rescaled Ricci
flows $(M,g_{p_{i}}(t);p_{i})$ of $(M,g)$ converge subsequently to a family of
shrinking quotient cylinders. Then $(M,g)$ is isometric to the $n$-dimensional
Bryant Ricci soliton up to scaling.

Theorem 1.1 is also an improvement of Brendle [2, Theorem 1.2], and Deng-Zhu
[10, Theorem 1.3] and [9, Lemma 6.5]. In the above papers, one shall assume
that for any sequence of $p_{i}\in M\rightarrow\infty$ the rescaled flows
$(M,g_{p_{i}}(t);p_{i})$ of $(M,g)$ converge subsequently to a family of
shrinking cylinders. We note that the nonnegativity condition of curvature
operator can be weakened as the nonnegativity of sectional curvature when
$n=4$ [14, Corollary 0.3].

Our proof of Theorem 1.1 depends on the following classification of split
ancient $\kappa$-solutions.

######  Theorem 1.2.

Let $(M^{n},g)$ $(n\geq 4)$ be a noncompact $\kappa$-noncollapsed steady Ricci
soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}>0$ on $M\setminus K$. Suppose that
there exists a sequence of rescaled Ricci flows $(M,g_{p_{i}}(t);p_{i})$,
which converges subsequently to a splitting Ricci flow
$(N\times\mathbb{R},\bar{h}(t);p_{\infty})$ as in (1.2) for some noncompact
ancient $\kappa$-solution $(N,h(t))$. Then there is no any compact split limit
Ricci flow of type I arising from the blow-down of $(M,g)$.

Recall that a compact ancient solution $(N,h(t))$ $(t\in(-\infty,0])$ of type
I means that it satisfies

$\displaystyle\sup_{N\times(-\infty,0]}(-t)|R(x,t)|<\infty.$

Otherwise, it is called type II, i.e., it satisfies

$\displaystyle\sup_{M\times(-\infty,0]}(-t)|R(x,t)|=\infty.$

By Theorem 1.2, we also prove

######  Corollary 1.3.

Let $(M^{n},g)$ $(n\geq 4)$ be a noncompact $\kappa$-noncollapsed steady Ricci
soliton as in Theorem 1.2. Then the compact split limit ancient flows of type
I and type II cannot occur simultaneously from the blow-down of $(M,g)$.

By Theorem 1.2 and Corollary 1.3, we conclude that for any noncompact
$\kappa$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and
$\rm{Ric}>0$ on $M\setminus K$, either all split limit flows $(N,h(t))$ as in
(1.2) are compact and of type I, or there exists at least one noncompact split
limit flow. The conclusion can be regarded as a generalization of Chow-Deng-
Ma’s result [6, Theorem 1.3, Claim 6.4] from dimension $4$ to any dimension.

Compared to the proof of [14, Theorem 0.2] for the $4d$ steady Ricci soliton,
we shall modify the argument for one of Theorem 1.2 in the case of higher
dimensions, since we have no classification result for codimemsional one
compact ancient $\kappa$-solutions of type II as dimension $3$ [3, 1, 5]. We
will get a distance estimate between two compact level sets, each of which has
a large diameter, to see (3.8). All main results will be proved in Section 3.

##  2\. Preliminaries

In this section, we review some results proved in our previous article [14],
which will be also used in this paper. As in [14], we will always assume that
$(M^{n},g;f)$ $(n\geq 4)$ is a noncompact $\kappa$-noncollapsed steady
gradient Ricci soliton with curvature operator $\rm{Rm}\geq 0$ on $M\setminus
K$.

The following result can be regarded as a Harnack type estimate for steady
Ricci solitons.

######  Lemma 2.1.

([14, Lemma 1.3]) Let $(M^{n},g)$ be a complete noncompact
$\kappa$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ on $M\setminus
K$. Let $\\{p_{i}\\}\to\infty$ be a sequence in $(M,g)$. Then, for any
$q_{i}\in B_{g_{p_{i}}}(p_{i},D)$, there exists $C_{0}(D)>0$ such that

(2.1) $\displaystyle C_{0}^{-1}R(p_{i})\leq R(q_{i})\leq C_{0}R(p_{i}).$

###  2.1. A decay estimate of curvature

By [14, Proposition 1.3], for any sequence $\\{p_{i}\\}\to\infty$, rescaled
Ricci flows $(M,g_{p_{i}}(t);p_{i})$ converge subsequently to a splitting
Ricci flow $(N\times\mathbb{R},\bar{g}(t);p_{\infty})$ in the Cheeger-Gromov
sense, where $\bar{g}(t)=h(t)+dr^{2}$ as in (1.2). We assume that the ancient
$\kappa$-solution $(N,h(t))$ is compact. Namely, there is a constant $C$ such
that

(2.2) $\displaystyle\text{Diam}(N,h(0))\leq C.$

Then we have the following curvature decay estimate.

######  Lemma 2.2.

([14, Lemma 2.2]) Let $(M^{n},g)$ be a complete noncompact steady Ricci
soliton with $\text{Ric}>0$ and $\rm{Rm}\geq 0$ on $M\setminus K$. Suppose
that there exists a sequence of $p_{i}\rightarrow\infty$ such that the split
$(n-1)$-dimensional ancient $\kappa$-solution $(N,{h}(t))$ satisfies (2.2).
Then the scalar curvature of $(M,g)$ decays to zero uniformly. Namely,

(2.3) $\displaystyle\lim_{x\rightarrow\infty}R(x)=0.$

By Lemma 2.2 and the normalization identity

(2.4) $\displaystyle R+|\nabla f|^{2}=1,$

we have

(2.5) $\displaystyle|\nabla f(x)|\to 1~{}{\rm as}~{}\rho(x)\to\infty.$

Moreover, by [9, Lemma 2.2] (or [6, Theorem 2.1]), $f$ satisfies

(2.6) $\displaystyle c_{1}\rho(x)\leq f(x)\leq c_{2}\rho(x)$

for two constants $c_{1}$ and $c_{2}$. Hence, the integral curve $\gamma(s)$
generated by $X=\nabla f$ extends to the infinity as $s\rightarrow\infty$.

Based on Lemma 2.1 and Lemma 2.2, by studying the level set geometry of
$(M,g)$, we prove

######  Proposition 2.3.

([14, Proposition 2.7]) Let $(M,g)$ be the steady Ricci soliton as in Lemma
2.2 and $(N\times\mathbb{R},h(t)+ds^{2};p_{\infty})$ the splitting limit flow
of $(M,g_{p_{i}}(t);p_{i})$, which satisfies (2.2). Then there exists
$C_{0}(C)>0$ such that for any sequence of $q_{i}\in f^{-1}(f(p_{i}))$ the
splitting limit flow
$(h^{\prime}(t)+ds^{2},N^{\prime}\times\mathbb{R};q_{\infty})$ of rescaled
flows $(M,g_{q_{i}}(t);q_{i})$ satisfies

(2.7) $\displaystyle{\rm Diam}(h^{\prime}(0))\leq C_{0}.$

###  2.2. A classification of split compact ancient solutions

In case that all split ancient $\kappa$-solution $(N,h(t))$ satisfies
$(\ref{bound-h})$, we can classify $(N,h(t))$.

######  Proposition 2.4.

([14, Proposition 4.1]) Let $(M,g)$ be a steady Ricci soliton as in Lemma 2.2.
Suppose that there is a uniform constant $C$ such that all split ancient
$\kappa$-solution $(N,h(t))$ satisfies $(\ref{bound-h})$. Then every $h(t)$
must be an ancient $\kappa$-solution of type I.

Compact ancient $\kappa$-solution of type I has been classified as follows
(cf. [7, Theorem 7.34], [4], [12]).

######  Lemma 2.5.

Suppose that $(N^{n-1},h(t))$ of $(n-1)$-dimension is a compact ancient
$\kappa$-solution of type I with ${\rm R_{m}}\geq 0$. Then

$\displaystyle(N^{n-1},h(t))$ (2.8)
$\displaystyle\cong(N_{1},h_{1}(t))\times\cdots\times(N_{k},h_{k}(t))\times(\hat{N}_{1},\hat{h}_{1}(t))\times\cdots\times(\hat{N}_{\ell},\hat{h}_{\ell}(t)),$

where each $(N_{i},h_{i}(t))$ is a family of shrinking quotients of a closed
symmetric space with nonnegative curvature operator, and each
$(\hat{N}_{j},\hat{h}_{j}(t))$ is a family of shrinking round quotient
spheres.

To prove Proposition 2.4, we shall exclude the existence of ancient
$\kappa$-solutions of type II. Actually, we prove the following diameter
estimate for such ancient solutions.

######  Lemma 2.6.

([14, Lemma 4.3]) Let $(N^{n-1},h(t))$ be an $(n-1)$-dimensional compact
ancient $\kappa$-solution of type II. Then for any sequence
$t_{k}\rightarrow-\infty$, it holds

$\displaystyle R_{min}(t_{k}){\rm Diam}(h(t_{k}))^{2}\rightarrow\infty,$

where $R_{min}(t)=\min\\{R(h(\cdot,t)\\}$. Consequently,

(2.9) $\displaystyle\lim_{t\rightarrow-\infty}R_{min}(t){\rm
Diam}(h(t))^{2}\rightarrow\infty.$

Lemma 2.6 can be regarded as a higher dimensional version of [1, Lemma 2.2].

##  3\. Proofs of main results

In this section, we prove Theorem 1.2 as well as and Corollary 1.3 and Theorem
1.1. First, we recall the following definition introduced by Perelman (cf.
[13]).

######  Definition 3.1.

For any $\epsilon>0$, we say a pointed Ricci flow
$\left(M_{1},g_{1}(t);p_{1}\right),t\in$ $[-T,0]$, is $\epsilon$-close to
another pointed Ricci flow $\left(M_{2},g_{2}(t);p_{2}\right),t\in[-T,0]$, if
there is a diffeomorphism onto its image
$\bar{\phi}:B_{g_{2}(0)}\left(p_{2},\epsilon^{-1}\right)\rightarrow M_{1}$,
such that $\bar{\phi}\left(p_{2}\right)=p_{1}$ and
$\left\|\bar{\phi}^{*}g_{1}(t)-g_{2}(t)\right\|_{C^{\left[\epsilon^{-1}\right]}}<\epsilon$
for all $t\in\left[-\min\left\\{T,\epsilon^{-1}\right\\},0\right]$, where the
norms and derivatives are taken with respect to $g_{2}(0)$.

By the compactness of rescaled Ricci flows [14, Proposition 1.3], we know that
for any $\epsilon>0$, there exists a compact set $D(\epsilon)>0$, such that
for any $p\in M\setminus D$, $(M,g_{p}(t);p)$ is $\epsilon$-close to a
splitting flow $(h_{p}(t)+ds^{2};p)$, where $h_{p}(t)$ is an
$(n-1)$-dimensional ancient $\kappa$-solution. Since the $\epsilon$-close
splitting flow $(h_{p}(t)+ds^{2};p)$ may not be unique for a point $p$, we may
introduce a function on $M$ for each $\epsilon$ as in [11],

(3.1) $\displaystyle
F_{\epsilon}(p)=\inf_{h_{p}}\\{{\rm{Diam}}(h_{p}(0))\in(0,\infty)\\}.$

For simplicity, we always omit the subscribe $\epsilon$ in the function
$F_{\epsilon}(p)$ below.

###  3.1. Proof of Theorem 1.2

We use the argument by contradiction. On the contrary, we suppose that there
exists a sequence of rescaled Ricci flows $(M,g_{q_{i}}(t);q_{i})$ (
$q_{i}\to\infty$), which converges to a limit Ricci flow
$(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$, where
$(N^{\prime},h^{\prime}(t))$ is a compact ancient $\kappa$-solution of type I.
Then by Lemma 2.5, there exists a constant $C_{0}>0$ such that for any small
$\epsilon>0$ it holds,

$\displaystyle F(q_{i})=F_{\epsilon}(q_{i})\leq C_{0}.$

By Proposition 2.3, it follows

(3.2) $\displaystyle F(q)\leq C,$

for all $q\in f^{-1}(f(q_{i}))$. We note that the constant $C$ is uniform by
the classification result, Lemma 2.5, i.e., it is independent of the sequence
of rescaled Ricci flows with a limit Ricci flow, which is a compact ancient
$\kappa$-solution of type I.

On the other hand, for the sequence of $(M,g_{p_{i}}(t);p_{i})$ in Theorem
1.2, we can choose a point $p_{i_{0}}\in\\{p_{i}\\}$ such that

(3.3) $\displaystyle F(p_{i})>100C,$

for all $i>i_{0}$. Let $\hat{X}=\frac{\nabla f}{|\nabla f|}$ and $\Gamma(s)$
be an integral curve of $\hat{X}$ starting from $p_{i_{0}}$, i.e.,
$\Gamma(0)=p_{i_{0}}$. We note that $\Gamma(s)$ tends to the infinity by (2.5)
and (2.6), since Lemma 2.2 holds. Thus by $\eqref{type1-bound}$ and
$\eqref{F_0}$, we can choose two sequences $\\{p^{1}_{i}\\}$ and
$\\{p^{2}_{i}\\}$ of points in $\Gamma(s)$ to the infinity, which satisfy the
following properties:

$\displaystyle 1)f(p^{1}_{i})<f(p^{2}_{i});$ $\displaystyle
2)f(p^{1}_{i})<f(p^{1}_{i+1}),f(p^{2}_{i})<f(p^{2}_{i+1});$ $\displaystyle
3)f(p^{1}_{i})<f(q_{i})<f(p^{2}_{i});$ $\displaystyle
4)F(p^{1}_{i})=F(p^{2}_{i})=10C$ (3.4) $\displaystyle 5)F(p)\leq
10C,~{}\forall~{}p\in\Gamma(s)~{}{\rm with}~{}f(p^{1}_{i})\leq f(p)\leq
f(p^{2}_{i}).$

Thus there are $s^{1}_{i},s^{2}_{i},s_{i}$ with $s^{1}_{i}<s_{i}<s^{2}_{i}$
such that

(3.5)
$\displaystyle\Gamma(s^{1}_{i})=p^{1}_{i},\Gamma(s^{2}_{i})=p^{2}_{i}~{}{\rm
and}~{}\Gamma(s_{i})=q^{\prime}_{i},$

where $q^{\prime}_{i}=\Gamma(s)\cap f^{-1}(f(q_{i}))$, see Figure 1.

Figure 1.

By [14, Proposition 1.3] and the relation 4) in (3.1),
$(M,g_{p^{1}_{i}}(t);p^{1}_{i})$ converges subsequently to
$(N_{1}\times\mathbb{R},h_{1}(t)+ds^{2};p^{1}_{\infty})$, where $h_{1}(t)$
satisfies

$\displaystyle{\rm Diam}(h_{1}(0))\in[10C-1,10C+1].$

Thus, by $\eqref{type1-bound}$, we see that $h_{1}(t)$ must be a compact
ancient $\kappa$-solution of type II. By Lemma 2.6, it follows

(3.6) $\displaystyle R(p^{1}_{\infty},t_{k}){\rm
Diam}(h_{1}(t_{k}))^{2}\rightarrow\infty$

for any $t_{k}\to-\infty$. Hence, there exists a $k_{0}$ such that

(3.7) $\displaystyle R(p^{1}_{\infty},t_{k_{0}})^{1/2}{\rm
Diam}(h_{1}(t_{k_{0}}))>100C.$

We fix $t_{k_{0}}$ so that $\epsilon^{-1}>-10t_{k_{0}}$ as long as
$\epsilon<<1$.

Next we claim there exists a constant $A>0$ such that

(3.8) $\displaystyle s^{2}_{i}-s^{1}_{i}<AR(p^{1}_{i})^{-1}$

for all $i\gg 1$.

Suppose that the above claim is not true. Then by taking a subsequence, we may
assume

(3.9) $\displaystyle R(p^{1}_{i})(s^{2}_{i}-s^{1}_{i})\to\infty.$

Recall that $\left\\{\phi_{t}\right\\}_{t\in(-\infty,\infty)}$ is the flow of
$-\nabla f$ with $\phi_{0}$ the identity and $(g(t),\Gamma(s))$ is isometric
to $\left(g,\phi_{t}(\Gamma(s))\right)$. Then

$\displaystyle\phi_{t}(\Gamma(s))=\Gamma\left(s-\int_{0}^{t}|\nabla
f|\left(\phi_{\mu}(\Gamma(s))\right)d\mu\right)$

Let
$T_{k_{0},i}=t_{k_{0}}R(p_{i}^{1})^{-1}=t_{k_{0}}R\left(\Gamma\left(s^{1}_{i}\right)\right)^{-1}$
and

$s^{\prime}_{i}=s^{1}_{i}-\int_{0}^{T_{k_{0},i}}|\nabla
f|\left(\phi_{\mu}\left(\Gamma\left(s^{1}_{i}\right)\right)\right)d\mu.$

It follows

(3.10) $\displaystyle s^{\prime}_{i}-s^{1}_{i}\leq-
T_{k_{0},i}=-t_{k_{0}}R\left(\Gamma\left(s^{1}_{i}\right)\right)^{-1}.$

By combining (3.9) and (3.10), we see that
$s^{\prime}_{i}\in[s^{1}_{i},s^{2}_{i}]$ for $i\gg 1$. Hence, by the relation
5) in (3.1), we obtain

(3.11) $\displaystyle F(\Gamma_{1}(s^{\prime}_{i}))\leq 10C.$

By the isometry, we have

$\displaystyle(M,R(\Gamma(s^{\prime}_{i}))g;\Gamma(s^{\prime}_{i}))$
$\displaystyle\cong(M,R(\Gamma(s^{1}_{i}),T_{k_{0},i})g(T_{k_{0},i});\Gamma(s^{1}_{i}))$
(3.12)
$\displaystyle\cong(M,\frac{R(\Gamma(s^{1}_{i}),T_{k_{0},i})}{R(\Gamma(s^{1}_{i}))}R(\Gamma(s^{1}_{i}))g(T_{k_{0},i});\Gamma(s^{1}_{i})).$

On the other hand, from the proof of [14, Proposition 3.6], we know that for
each $\Gamma(s^{1}_{i})$, there exists a $(n-1)$-dimensional compact ancient
$\kappa$-solution $h_{\Gamma(s^{1}_{i})}(t)$ such that

$\displaystyle(M,R(\Gamma(s^{1}_{i}))g(R(\Gamma(s^{1}_{i}))^{-1}t);\Gamma(s^{1}_{i}))$
(3.13)
$\displaystyle\overset{\epsilon-\text{close}}{\sim}(N\times\mathbb{R},h_{\Gamma(s^{1}_{i})}(t)+ds^{2};\Gamma(s^{1}_{i})).$

Since $R(\Gamma_{1}(s^{1}_{i}),T_{k_{0},i})\leq R(\Gamma(s^{1}_{i}))$ by the
monotonicity of scalar curvature along $\Gamma(s)$, by (3.1), we get

$\displaystyle(M,R(\Gamma(s^{\prime}_{i}))g;\Gamma(s^{\prime}_{i}))$ (3.14)
$\displaystyle\overset{\epsilon-\text{close}}{\sim}(N\times\mathbb{R},\frac{R(\Gamma(s^{1}_{i}),T_{k_{0},i})}{R(\Gamma(s^{1}_{i}))}h_{\Gamma(s^{1}_{i})}(t_{k_{0}})+ds^{2};\Gamma(s^{1}_{i})).$

Note that there are also another $(n-1)$-dimensional compact ancient
$\kappa$-solutions $h_{\Gamma(s^{\prime}_{i})}(t)$ corresponding to the point
$\Gamma(s^{\prime}_{i})$ as in (3.1) such that

(3.15)
$\displaystyle(M,R(\Gamma(s^{\prime}_{i}))g;\Gamma(s^{\prime}_{i}))\overset{\epsilon-\text{close}}{\sim}(h_{\Gamma(s^{\prime}_{i})}(0)+ds^{2},\Gamma(s^{\prime}_{i})).$

Hence, combining $(\ref{s1-close1})$ and $(\ref{s1-close2})$, we derive

$\displaystyle h_{\Gamma(s^{\prime}_{i})}(0)$
$\displaystyle\overset{\epsilon-\text{close}}{\sim}\frac{R(\Gamma(s^{1}_{i}),T_{k_{0},i})}{R(\Gamma(s^{1}_{i}))}h_{\Gamma(s^{1}_{i})}(t_{k_{0}})$
(3.16)
$\displaystyle\overset{\epsilon-\text{close}}{\sim}R_{h}(\Gamma(s^{1}_{i}),t_{k_{0}})h_{\Gamma(s^{1}_{i})}(t_{k_{0}}).$

By the convergence, we have

(3.17) $\displaystyle
h_{\Gamma(s^{1}_{i})}(t_{k_{0}})\overset{\epsilon-\text{close}}{\sim}h_{1}(t_{k_{0}}),~{}{\rm
as}~{}i\gg 1.$

It follows

(3.18) $\displaystyle
R_{h}(\Gamma(s^{1}_{i}),t_{k_{0}})\overset{\epsilon^{\prime}(\epsilon)-\text{close}}{\sim}R(p^{1}_{\infty},t_{k_{0}}),$

where $R$ is the scalar curvature w.r.t. $h_{1}$. Thus by (3.1) and (3.7), we
estimate

$\displaystyle F(\Gamma(s^{\prime}_{i}))$ $\displaystyle\geq{\rm
Diam}(h_{\Gamma(s^{\prime}_{i})}(0))-\epsilon$ $\displaystyle\geq{\rm
Diam}(R_{h}(\Gamma(s^{1}_{i}),t_{k_{0}})h_{\Gamma(s^{1}_{i})}(t_{k_{0}}))-\epsilon(1+\epsilon^{\prime})$
(3.19) $\displaystyle>98C.$

But this is impossible by (3.11). Hence, (3.8) must be true.

Now we can finish the proof of Theorem 1.2. By (3.8) and the relation 3) in
(3.1), we have

(3.20) $\displaystyle s^{2}_{i}-s_{i}\leq s^{2}_{i}-s^{1}_{i}\leq
AR\left(\Gamma\left(s^{1}_{i}\right)\right)^{-1}\leq
AR\left(\Gamma\left(s_{i}\right)\right)^{-1},$

where the last inequality follows from the monotonicity of $R$ along
$\Gamma(s)$. Since $(M,g_{\Gamma(s_{i})}(t);\Gamma(s_{i}))$ converges
subsequently to the limit flow
$(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$, by the
shrinking property of type I solution $(N^{\prime},h^{\prime}(t))$ in Lemma
2.5, we know

(3.21)
$\displaystyle{\rm{Diam}}(R_{h^{\prime}}(q_{\infty},t)h^{\prime}(t)))\leq
2C,~{}\forall~{}t\leq 0.$

Then as in $(\ref{h-sequence-limit})$, we get

(3.22) $\displaystyle
h_{\Gamma(s^{2}_{i})}(0)\overset{\epsilon-\text{close}}{\sim}R_{h}(\Gamma(s_{i}),t^{\prime}_{i})h_{\Gamma(s_{i})}(t^{\prime}_{i})$

for some $t^{\prime}_{i}$, where $-t^{\prime}_{i}<2A$ by (3.20). Thus ,
combining $(\ref{type1-scaled-diameter})$ and (3.22) and by the convergence of
$(M,g_{\Gamma(s_{i})}(t);\Gamma(s_{i}))$, we obtain

(3.23) $\displaystyle F(\Gamma(s^{2}_{i}))\leq 3C.$

But this is impossible since by the relation 4) in (3.1) it holds

(3.24) $\displaystyle F(\Gamma(s^{2}_{i}))=F(p^{2}_{i})=10C.$

Therefore, we prove the theorem.

###  3.2. Proofs of Corollary 1.3 and Theorem 1.1

###### Proof of Corollary 1.3.

Suppose that there exist two splitting limit flows
$(N_{1}\times\mathbb{R},h_{1}(t)+ds^{2};p^{1}_{\infty})$ and
$(N_{2}\times\mathbb{R},h_{2}(t)+ds^{2};p^{2}_{\infty})$ of rescaled flows
$(M,g_{p^{1}_{i}}(t);p^{1}_{i})$ and $(M,g_{p^{2}_{i}}(t);p^{2}_{i}),$
respectively, such that $(N_{1},h_{1}(t))$ is a compact ancient
$\kappa$-solution of type I, and $(N_{2},h_{2}(t))$ is another compact ancient
$\kappa$-solution of type II. We claim that

(3.25) $\limsup_{\epsilon\to 0}\limsup_{p\to\infty}F_{\epsilon}(p)=\infty.$

On the contrary, there will be a uniform constant $C$ such that all split
ancient $\kappa$-solutions $(N^{n-1},h(t))$ of $(n-1)$-dimension satisfy
$(\ref{bound-h})$. Then by Proposition 2.4, it follows that every split limit
flow $h(t)$ must be an ancient $\kappa$-solution of type I. But this is
impossible since $h_{2}(t)$ is a compact ancient $\kappa$-solution of type II.
Thus the claim is true.

By (3.25), it is easy to see that there is a sequence of pointed flows
$(M,g_{q_{i}}(t);q_{i})$, which converges subsequently to a splitting Ricci
flow $(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$ for some
noncompact ancient $\kappa$-solution $(N^{\prime},h^{\prime}(t))$. Thus by
Theorem 1.2, there cannot exist a compact splitting limit flow of type I. But
this is impossible since $h_{1}(t)$ is a compact ancient $\kappa$-solution of
type I. This proves the corollary. ∎

###### Proof of Theorem 1.1.

By the assumption, the $(n-1)$-dimensional split ancient flow $(N,h(t))$ of
limit of $(M,g_{p_{i}}(t),p_{i})$ is a family of shrinking round quotient
spheres. Namely, $(N,h(0))$ is a quotient of round sphere, so it is of type I.
We first show that $(M,g)$ has positive Ricci curvature on $M$.

On the contrary, ${\rm Ric}(g)$ is not strictly positive. We note that the
scalar curvature $R(p_{i})$ decaying to zero is still true in the proof of
Lemma 2.2 without ${\rm Ric}(g)>0$ away from a compact set of $M$. Then as in
the proof of [8, Lemma 4.6], we see that $X_{i}=R(p_{i})^{-\frac{1}{2}}\nabla
f\to X_{\infty}$ w.r.t. $(M,g_{p_{i}}(t);p_{i})$, where $X_{\infty}$ is a non-
trivial parallel vector field. Thus according to the argument in the proof of
[8, Theorem 1.3], the universal cover of $(N,h(t))$ must split off a flat
factor $\mathbb{R}^{d}$ ($d\geq 1$). However, the universal cover of $N$ is
$S^{n-1}$. This is a contradiction! Hence, we conclude that ${\rm Ric}(g)>0$
on $M$.

Now we divide into two cases to prove the theorem.

Case 1:

$\limsup_{p\rightarrow\infty}F_{\epsilon}(p)<C.$

for any $\epsilon<1$. Then by Proposition 2.4 and Lemma 2.5, all
$(n-1)$-dimensional split ancient $\kappa$-solutions
$(N^{\prime},h^{\prime}(t))$ are of type I, and each of them is one described
in (2.5).

Case 2:

$\limsup_{\epsilon\to 0}\limsup_{p\rightarrow\infty}F_{\epsilon}(p)=\infty.$

In this case, there will exist a sequence of pointed flows
$(M,g_{q_{i}}(t);q_{i})$, which converges subsequently to a splitting Ricci
flow $(N^{\prime}\times\mathbb{R},h^{\prime}(t)+ds^{2};q_{\infty})$ for some
noncompact ancient $\kappa$-solution $(N^{\prime},h^{\prime}(t))$. But this is
impossible by Theorem 1.2 since we already have had a split ancient limit flow
$(N,h(t))$ of type I. Thus Case 2 can be excluded.

It remains to show that every split limit flow $(N^{\prime},h^{\prime}(t))$ in
Case 1 is in fact a family of shrinking round spheres.

By Lemma 2.2, the scalar curvature of $(M,g)$ decays to zero uniformly. Then
$(M,g)$ has unique equilibrium point $o$ by the fact ${\rm Ric}(g)>0$. Thus
the level set $\Sigma_{r}=\\{f(x)=r\\}$ is a closed manifold for any $r>0$,
and it is diffeomorphic to $S^{n-1}$ (cf. [8, Lemma 2.1]).

On the other hand, as in the proof of [14, Lemma 2.6], the level sets
$(\Sigma_{f(q_{i})},\bar{g}_{q_{i}};q_{i})$ converge subsequently to
$(N^{\prime},h^{\prime}(0);q_{\infty})$ w.r.t. the induced metric
$\bar{g}_{q_{i}}$ on $\Sigma_{f(q_{i})}$ by $g_{q_{i}}$. Since each
$\Sigma_{f(q_{i})}$ is diffeomorphic to $S^{n-1}$, $N^{\prime}$ is also
diffeomorphic to $S^{n-1}$. Thus $(N^{\prime},h^{\prime}(t))$ is a family of
shrinking round spheres.

By the above argument, we see that the condition (ii) in [14, Definition 0.1]
is satisfied. Thus by [9, Lemma 6.5], $(M,g)$ is asymptotically cylindrical.
It follows that $(M,g)$ is isometric to the Bryant Ricci soliton up to scaling
by [2]. Hence, the theorem is proved.

∎

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