# Towards LLM-based Fact Verification on News Claims with a Hierarchical Step-
by-Step Prompting Method

Xuan Zhang  Wei Gao
School of Computing and Information Systems
Singapore Management University
80 Stamford Rd, Singapore 178902
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

While large pre-trained language models (LLMs) have shown their impressive
capabilities in various NLP tasks, they are still under-explored in the
misinformation domain. In this paper, we examine LLMs with in-context learning
(ICL) for news claim verification, and find that only with 4-shot
demonstration examples, the performance of several prompting methods can be
comparable with previous supervised models. To further boost performance, we
introduce a Hierarchical Step-by-Step (HiSS) prompting method which directs
LLMs to separate a claim into several subclaims and then verify each of them
via multiple questions-answering steps progressively. Experiment results on
two public misinformation datasets show that HiSS prompting outperforms state-
of-the-art fully-supervised approach and strong few-shot ICL-enabled
baselines.

##  1 Introduction

Misinformation such as fake news often causes confusion or wrong belief
because they contain claims that are factually false or inaccurate Lazer et
al. (2018). To combat misinformation in news claims, stakeholders rely on
fact-checking practices for claim verification. Fact-checking services online,
such as PolitiFact111https://www.politifact.com/. and
Snopes222https://www.snopes.com/.) require laborious manual efforts, making it
challenging to match the rapid pace of misinformation being produced.

In recent years, deep neural networks-based misinformation detection and fact-
checking methods have been studied extensively Wang (2017); Rashkin et al.
(2017); Popat et al. (2018); Ma et al. (2019); Kotonya and Toni (2020);
Atanasova et al. (2020); Yang et al. (2022). In particular, pre-trained
language models (PLMs) like BERT Kenton and Toutanova (2019) have demonstrated
superior results and surpassed traditional methods in fake news related
benchmarks Soleimani et al. (2020); Atanasova et al. (2020); Kruengkrai et al.
(2021), thanks to their strong ability to understand nuanced context for more
accurate decision. Recently, large pre-trained language models (LLMs) with a
massive number of parameters, such as GPT-3.5, have shown impressive
performances in various downstream tasks Brown et al. (2020); Wei et al.
(2022); Zhou et al. (2022); Press et al. (2022). But it is basically unclear
how well LLMs can perform on fact verification task as this is not at the core
of LLM pre-training Brown et al. (2020); Anil et al. (2023).

While it is not practical to directly fine-tune most LLMs, in-context learning
(ICL) Brown et al. (2020) offers an alternative way to instruct LLMs to learn
new tasks via inference only, conditioning on demonstration examples without
any gradient updates. Properly prompted LLMs can carry out similar steps of
logical traces with that in demonstration examples, which is known as Chain-
of-Thought (CoT) reasoning Wei et al. (2022). This generative reasoning
process not only enhances the model’s performance on tasks such as arithmetic,
commonsense, and symbolic reasoning, but also facilitates the understanding of
the underlying rationale behind the results from LLMs.

Figure 1: An example of claim verification based on vanilla CoT prompting. The
claim (underlined) and CoT (in green) are given as a demonstration. The
generated CoT (in italics) leads to an incorrect judgment due to (1) omission
of necessary thoughts regarding “nukes”, and (2) fact hallucination about the
war-loving speeches without specific evidence in the generated CoT (in blue).

Previous research has suggested the importance of reasoning in improving the
accuracy and explainability of fake news detection Jin et al. (2022). However,
leveraging LLM reasoning in the context of fake news related tasks remains
under-explored. In this work, we first evaluate three classical ICL methods,
including standard prompting and CoT-based methods for news claim
verification. The standard prompting takes in a news claim for LLM to return
its factuality judgment on the claim, while CoT additionally generates a
series of intermediate verbal reasoning steps in the result. On two fake news
benchmark datasets RAWFC Yang et al. (2022) and LIAR Wang (2017), we find that
the standard prompting performs comparably well as strong supervised
baselines, but the vanilla CoT is worse than the standard prompting, which is
counter-intuitive. We found that there are two main issues causing the failure
of vanilla CoT, as illustrated in Figure 1: (1) Omission of necessary thoughts
– vanilla CoT tends to ignore some noteworthy parts in the claim, resulting in
inaccurate decisions; (2) Fact hallucination333This type of hallucination is
also referred to as the extrinsic hallucination Bang et al. (2023) that cannot
be verified with the given source, and the fact-conflicting hallucination
Zhang et al. (2023) that, more broadly, are not faithful to established world
knowledge. – When necessary information is not available, the model tends to
generate relevant but unreliable “facts” on its own, which misleads the final
prediction.

To address the issues, we instruct LLMs to decompose a complex claim into
smaller subclaims, so that the reasoning follows up with the fine-grained
decomposition. This aims to enable a much more thorough examination of the
claim, reducing the risk of overlooking necessary details in the claim and
enhancing the reasoning effect based on different reasoning chains. This is
analogous to breaking down complex questions into subquestions Press et al.
(2022) for QA and devising a plan for solving complex tasks into multiple
steps Wang et al. (2023). Additionally, we instruct LLM to employ a search
engine for providing up-to-date external information, aiding the model in
reasoning and mitigating the hallucination problem. In light of this, we
propose a Hierarchical Step-by-Step (HiSS) prompting method, which is composed
of two main processes: (1) Claim Decomposition, which prompts the LLM to split
a complex claim into smaller subclaims. (2) Subclaim Verification, which
prompts LLM to verify the subclaim step-by-step employing a search engine to
obtain relevant evidence. Our contributions are three-fold:

  * •

We investigate the ability of LLMs with ICL for news claim verification. And
we find that with only four-shot demonstration examples, LLMs can outperform
most of the supervised methods, which indicates LLM is a promising tool to
combat misinformation.

  * •

We propose a HiSS prompting method to prompt LLM to do fine-grained checking
of news claims. Experiments on two public datasets show that HiSS-prompted
LLMs outperform traditionally strong fully-supervised models with an
improvement of 4.95% on average in macro-average F1 and set a new state-of-
the-art for few-shot news claim verification444Code and prompts data is
available at https://github.com/jadeCurl/HiSS..

  * •

Compared with previous methods, our HiSS-prompted LLMs provide superior
explanations, which are more fine-grained and easier to follow based on
automatic and human evaluation.

##  2 Related Work

###  2.1 Explainable Fake News Detection

Existing research on explainable fake news detection is mainly focused on
generating explanations from input evidence. These approaches include
generating human-comprehensible explanations for candidate facts based on
background knowledge encoded in the form of Horn clauses Gad-Elrab et al.
(2019), as well as using attention-based models to highlight relevant factual
words Popat et al. (2018), news attributes Yang et al. (2019) and suspicious
users Lu and Li (2020). Such an approach is based on general deep neural
networks and knowledge base instead of language models.

Later, Atanasova et al. (2020) and Kotonya and Toni (2020) propose directly
producing veracity explanations based on extractive and abstractive
summarization. However, these methods predominantly generate explanations by
summarizing fact-checking articles. While such an approach can somewhat
explain fact-checking decisions following human thoughts written in the
articles, it does not reason based on raw evidence to form the thoughts for
drawing conclusions, which should be the core of fact verification.

###  2.2 Fact Verification with Language Models

Previous research has utilized PLMs (e.g., BERT and BART) in fake news related
tasks. For example, Lee et al. (2020) directly uses the internal knowledge
implicitly stored as PLMs’ parameters for fact verification. Lewis et al.
(2020) proposes a retrieval-augmented approach to endow language models with
document retrieval capability, which was applied for selecting relevant
evidence in fact extraction and verification. Instead of using language models
to provide evidence only, Lee et al. (2021) utilizes LLMs such as GPT-2
Radford et al. (2019) and their few-shot capability to assess the claim’s
factuality based on the perplexity of evidence-conditioned claim generation.

Research on utilizing the reasoning capabilities of LLMs, such as CoT-based
reasoning, in the misinformation domain is still limited. Recent works Press
et al. (2022); Yao et al. (2023); Jiang et al. (2023) find that combining
LLM’s reasoning capability with accessibility to external knowledge is helpful
to many reasoning-intensive NLP tasks including HotpotQA Yang et al. (2018)
and FEVER Thorne et al. (2018). In contrast to existing works, our research is
motivated by the counter-intuitive observation that CoT under-performs the
standard prompting in news claim verification, and explores how to better
elicit LLMs to mitigate two salient issues of LLMs in this task. We focus on
the verification of real-world news claims, which could be more temporally
dynamic and sensitive than FEVER type of claims, necessitating the model to
access up-to-date knowledge.

##  3 Our HiSS Prompting Method

Figure 2: Overview of the proposed HiSS model: Original human inputs are in
red background, LLM directly generated text is in white, and answers generated
based on search results are in green. We start by providing a few-shot
demonstration, followed by appending the claim to be checked (underlined).
HiSS prompts the LLM to (1) decompose the claim into subclaims; (2) verify
each subclaim step-by-step via raising and answering a series of questions.
For each question, we prompt LLM to assess if it is confident to answer it or
not, and if not, we input the question to a web search engine. The search
results are then inserted back into the ongoing prompt to continue the
verification process; (3) generate the final prediction. The detailed
demonstrations are omitted in this illustration for space which can be found
in Table 8 (a) and Table 8 (b) in Appendix B .

In this section, we address the two main issues of LLMs observed in the news
claim verification task, i.e., 1) Omission of necessary thoughts and 2) Fact
hallucination. We will first raise our specific research questions, and then
present our HiSS prompting method.

###  3.1 Research Questions

For the omission of necessary thoughts, the basic research question we need to
address would be:

  * •

_How to instruct LLMs not to overlook any crucial points of a claim in its
CoT?_

The context of real-world claims could be complex and deep. For example, the
seemingly easy claim _Donald Trump has said he loves war, “including with
nukes”_ is actually quite intricate, as it not only explicitly states Trump’s
declaration of love for both regular and nuclear wars, but also implies that
in order to verify the statement is factual or not, one has to examine whether
and in what circumstances he has expressed such passion on both types of wars.
Therefore, we propose to prompt LLMs to thoroughly generate all explicit and
implicit points that are check-worthy given a claim.

Hallucination is an intrinsic and fundamental problem of LLMs Ji et al.
(2023); Bang et al. (2023). We address it by providing relevant and up-to-date
contextual information to LLM as external knowledge, assuming that
hallucinations most likely result from the lack of knowledge on the necessary
context Bang et al. (2023). Our specific research question would be:

  * •

_How can we determine when external knowledge is needed during the
verification and assist LLM in acquiring the necessary knowledge to mitigate
fact hallucination?_

While the decomposition can prompt LLM to raise fine-grained questions, the
model may make up responses when background information is lacking. For
instance, if the model is unaware of the specific contexts of Trump’s wording
on “war”555The comment regarding Trump’s “love” of war comes from his speech
in Iowa on Nov. 12, 2015. In the speech, Trump theorized that former Iraqi
leader Saddam Hussein feigned having weapons of mass destruction to scare
Iran, before briefly sidetracking into his feelings on war generally: “This is
the Trump theory on war,” he said. “But I’m good at war. I’ve had a lot of
wars of my own. I’m really good at war. I love war in a certain way. But only
when we win.” and “nukes”666Trump made his comments about “nukes” in an April
3 interview with Fox News Sunday’s Chris Wallace. Wallace was asking Trump
about his suggestion that Japan might be better off with nuclear weapons.
Trump suggested that Japan might need to acquire nuclear weapons to defend
against neighboring North Korea. It’s worth noting that the comment wasn’t
about the United States using nuclear weapons, but about his belief that Japan
might be better off if it had nuclear weapons., it can lead to factually
inaccurate answer, such as “During his term as the 45th President of the US,
Donald Trump gave speeches proclaiming his love for war”.

In the following subsections, we will describe our Hierarchical Step-by-Step
(HiSS) prompting method. As shown in Figure 2, HiSS involves three processes:
(1) Claim decomposition, (2) Subclaim step-by-step verification, and (3) Final
prediction.

###  3.2 Claim Decomposition

At the first level of HiSS, we focus on instructing the model to capture all
the explicit points in the original claim and decompose them into subclaims.
This level aligns with previous studies Ousidhoum et al. (2022); Fan et al.
(2020) in fact-checking, which found that segmenting the original claim by
identifying entities or focal points can facilitate human fact-checkers in
making informed judgments. However, these models require the manual collection
of datasets for training, whereas our method prompts LLM to do the
decomposition guided by only a few demonstration examples.

Speficially, LLM is prompted with $K$-shot ($K$ is a hyperparameter)
demonstration examples (see Table 8 (a) and Table 8 (b) in Appendix B for
details) that serve to illustrate the entire verification process, followed by
the test claim to be checked, as shown in the Level 1 in Figure 2. The
demonstration examples exhibit to LLM how to break down a claim $c_{i}$ into a
series of subclaims $[s_{i1},s_{i2},\cdots,s_{iN_{i}}]$ that cover all check-
worthy points explicitly expressed. The demonstration examples vary in their
complexity, with some simple claims not undergoing deep decomposition and more
complex claims being decomposed into a few more subclaims. The LLM presumably
follows the demonstrated decomposition approach in accordance with the
complexity of the input claim $c_{i}$. Therefore, $N_{i}$ is determined by LLM
automatically. Figure 2 illustrates that LLM decomposes the test claim into
two subclaims.

###  3.3 Subclaim-level Step-by-Step Verification

In the second level, LLM individually verifies each subclaim obtained from
Level 1. Underlying the explicit points conveyed in each subclaim can be a few
implicit points that are not expressed but need further scrutinization in one
way or the other. For example, “Did Trump really say he loves war?”, “What is
his exact wording?”, “In what context did he express it?”, etc. for the first
subsclaim “Donald Trump has expressed a love for war”.

Specifically, we leverage the reasoning capability of LLM to delve deeper into
the underlying information needed to validate each subclaim $s_{ij}$ by
generating a series of probing questions $\\{q_{ij}^{m}\\}$, each $q_{ij}^{m}$
corresponding to an implicit point. Similarly, the number of probing questions
of each subclaim is determined by LLM automatically with reference to the
demonstration example. We adopt a progressive approach to generate the
questions. This allows us to adjust the subsequent question generation based
on the answers to previous questions and the acquired context information on
the chain. As a result, the generated questions become more targeted and in-
depth, facilitating a comprehensive analysis of the subclaims.

Once a question $q_{ij}^{m}$ is generated, the next step is to elicit the
corresponding answer $a_{ij}^{m}$ from LLM. Recent works have found that
providing LLMs with access to external knowledge can lead to notable
improvements Yao et al. (2023); Jiang et al. (2023). An important
consideration is how to prompt LLM to automatically decide when it needs to
consult with an external knowledge source (e.g., web search engine), to
mitigate fact hallucination. It is hypothesized that LLM can be prompted to
assess its own confidence in answering a question, so that we can acquire
relevant external information to aid it when it lacks confidence. We resort to
Google Search as an external source.

Specifically, LLM follows the specific format of demonstration examples to
generate questions: it starts with the prefix “Question:” and presents the
generated question $q_{ij}^{m}$, followed by “Tell me if you are confident
...”. We control the model to pause at the end of $q_{ij}^{m}$ by setting the
phrase “Tell me if you are confident” as the stop sequence777The “stop
sequence” mechanism is a setting provided by the OpenAI API
(https://help.openai.com/en/articles/5072263-how-do-i-use-stop-sequences).
When a specific word or phrase is set as a “stop sequence”, the model will
halt its generation upon encountering that word or phrase, allowing users to
control the length or content of the generated output. . This aims to
facilitate 1) extracting the text of $q_{ij}^{m}$, and 2) probing the LLM to
assess its confidence in answering the question without additional
information. During its pause, we append the following instruction: Tell me if
you are confident to answer the question or not. Answer with ‘‘yes’’ or
‘‘no’’:, and set the stop sequence to ‘no’. This means that if the LLM
responds with ‘no’, the model will cease to further generate an answer for
$q^{m}_{ij}$, but wait for us to input $q_{ij}^{m}$ into Google Search
API888https://serpapi.com. to obtain top search results999Search results from
fact-checking websites are filtered to avoid ground-truth leakage.
Specifically, we remove the search results with URLs containing keywords such
as “fact check”, and “fact-checking” since the URL of fact-checking websites
and fact-check articles on mainstream media, e.g., NY Times
(https://www.nytimes.com/spotlight/fact-checks.), typically contain such
keywords. After filtering, we choose the top-one snippet from the search
result to feed into the LLM., so that we can feed them into the LLM for it to
generate the answer $a_{ij}^{m}$. However, if the LLM responds with “yes”, the
LLM does not halt and proceeds to generate the answer $a_{ij}^{m}$ to the
question. Following the specific format of the demonstration example, after a
prior question is addressed, the LLM continues to generate the subsequent
question until it ceases to produce any more questions, transitioning then to
the final prediction phase.

###  3.4 Final Prediction

Once all the subclaims have been verified, the LLM can make a final
prediction. At this point, it outputs “Among [label set], the claim is
classified as” before providing the final answer, where [label set] is
substituted with the actual label set for a specific dataset. This facilitates
the parsing of the final prediction, as the predicted class label will appear
after the word “as” in the last output line.

##  4 Experiments and Results

###  4.1 Experimental Setup

| RAWFC | LIAR
---|---|---
| Val. | Test | Val. | Test
Claim | 200 | 200 | 1274 | 1,251
# true | 67 | 67 | 169 | 205
# mostly-true | - | - | 251 | 238
# half-true | 66 | 66 | 244 | 263
# barely-true | - | - | 236 | 210
# false | 67 | 67 | 259 | 249
# pants-fire | - | - | 115 | 86
Table 1: Datasets statistics.

We conducted experiments on two standard English fake news datasets: 1) RAWFC
Yang et al. (2022) contains gold labels based on Snopes fact-check articles
and follows a three-class classification scheme (True/False/Half); 2) LIAR
Wang (2017) contains gold labels based on PolitiFact articles with six classes
(True/Mostly-true/Half-true/Barely-true/False/Pants-fire). Different from
FEVER Thorne et al. (2018) which uses manually synthesized claims from
Wikipedia articles, the claims in these two datasets are based on real-world
news. Table 1 displays the statistics of datasets. We use the provided valid-
test split of both datasets. The few-shot demonstration examples are randomly
selected from the training set.

Following Yang et al. (2022), we use macro-average precision ($P$), recall
($R$), and $F_{1}$ ($F_{1}=\frac{2RP}{R+P}$) scores as the metrics for
evaluation.

Model | RAWFC | LIAR
---|---|---
$P(\%)$ | $R(\%)$ | $F_{1}(\%)$ | $P(\%)$ | $R(\%)$ | $F_{1}(\%)$
Fully Supervised Models
CNN Wang (2017) | 38.8 | 38.5 | 38.6 | 22.6 | 22.4 | 22.5
RNN Rashkin et al. (2017) | 41.4 | 42.1 | 41.7 | 24.4 | 21.2 | 22.7
DeClarE† Popat et al. (2018) | 43.4 | 43.5 | 43.4 | 22.9 | 20.6 | 21.7
SentHAN† Ma et al. (2019) | 45.7 | 45.5 | 45.6 | 22.6 | 20.0 | 21.2
SBERT♢ Kotonya and Toni (2020) | 51.1 | 46.0 | 48.4 | 24.1 | 22.1 | 23.1
GenFE♢ Atanasova et al. (2020) | 44.3 | 44.8 | 44.5 | 28.0 | 26.2 | 27.1
CofCED† Yang et al. (2022) | 53.0 | 51.0 | 52.0 | 29.5 | 29.6 | 29.5
Few-shot Models w/ GPT3.5
Standard Prompt Brown et al. (2020) | 48.5 | 48.5 | 48.5 | 29.1 | 25.1 | 27.0
Vanilla CoT Wei et al. (2022) | 42.4 | 46.6 | 44.4 | 22.6 | 24.2 | 23.7
Search-Augmented CoT† | 47.2 | 51.4 | 49.2 | 27.5 | 23.6 | 25.4
ReAct† Yao et al. (2023) | 51.2 | 48.5 | 49.8 | 33.2 | 29.0 | 31.0
HiSS† (ours) | 53.4 | 54.4∗ | 53.9∗ | 46.8∗ | 31.3∗ | 37.5∗
Table 2: Experimental results of claim verification. Bold denotes the best
performance. ∗ means significantly better than the previous SoTA (CofCED) with
$p<0.01$. † uses external information obtained via search engines. ♢ uses gold
evidence from fact-check reports. Results of fully supervised models are
quoted from Yang et al. (2022).

#### Supervised baselines.

We compare with seven strong supervised models in claim verification: 1) CNN
Wang (2017) uses a convolutional neural model to integrate claim information
and available metadata features (e.g. subject, speaker, and party) to get the
prediction; 2) RNN Rashkin et al. (2017) uses recurrent neural networks to
learn representation from word sequences of the claim. 3) DeClarE Popat et al.
(2018) considers word embedding from both the claim and searched external
information as evidence. 4) SentHAN Ma et al. (2019) proposes a hierarchical
attention network to represent external evidence as well as their semantic
relatedness with the claim. 5) SBERT-FC Kotonya and Toni (2020) uses Sentence-
BERT to encode both claim and evidence for classification. 6) GenFE Atanasova
et al. (2020) predicts fact-check results and generates explanations in the
multi-task setup. 7) CofCED Yang et al. (2022) uses a hierarchical encoder for
text representation and two coarse-to-fine cascaded selectors to extract key
evidence for news claim verification.

#### Few-shot baselines.

We employ the following few-shot baselines for comparison:

1) Standard Prompting Brown et al. (2020) directly asks the LLM to determine
the class label of the claim. 2) Vanilla CoT Prompting Wei et al. (2022) asks
the LLM to output a thought chain before outputting the class label of the
claim. The demonstration examples for both standard prompting and vanilla CoT
are shown in Table 6 and B, respectively (see Appendix B). 3) Search-Augmented
CoT Prompting: To compare with the baselines that can access extra knowledge,
we augment vanilla CoT with search engine by using the claim as query to
retrieve the background information, and then let the LLM output thought chain
and class label based on the information retrieved. 4) ReAct Prompting Yao et
al. (2023) is a variant of CoT that explores the use of LLMs to generate both
reasoning traces and task-specific actions (e.g., search) in an interleaved
manner. For a fair comparison, we employ the same demonstration examples and
search engine across the different systems.

#### Implementation Details

To ensure reproducibility, we generate outputs using greedy decoding by
setting the temperature to $0$. We also freeze the search results for the same
queries involved in the experiments for fair comparison, as search engine
results can potentially change over time across different runs. We utilize the
GPT-3.5 series API text-
davinci-003101010https://platform.openai.com/docs/models/gpt-3-5. as a
backbone LLM. Following Wei et al. (2022), we tune the hyperparameter of the
shot number within $\\{1,2,4,6,8\\}$ on the validation set, and find that the
model achieved the best results with $4$ demonstration examples. Therefore, we
set $K=4$ throughout the experimentation.

###  4.2 Results of Claim Veracity Classification

Table 2 summarizes the performance of verification, and we have the following
findings:

  * •

HiSS with LLM is comparable with or even better than the fully supervised
SoTA. As shown in Table 2, HiSS outperforms previous SoTA (i.e., CofCED) by
1.9% and 8% in F1 on RAWFC and LIAR, respectively. This indicates that few-
shot ICL is promising for news claim verification, owing to the capabilities
of LLM that benefit from its parameterized knowledge with a tremendous amount
of facts. In addition, HiSS is specially designed to better elicit LLM and
guide it through a step-by-step examination of the claim, covering both
explicit and implicit aspects and allowing for more comprehensive and thorough
verification. Moreover, HiSS enables evidence acquisition via web search when
necessary, mitigating the risk of hallucination.

  * •

The performance of few-shot ICL methods varies. Despite utilizing the same
backbone, HiSS surpasses standard prompting, vanilla CoT, and ReAct by 7.95%,
11.65%, and 5.3% in F1 on average, respectively. This observation highlights
the importance of specific methods prompting LLM for news claim verification.
After conducting an in-depth error analysis on 40 randomly selected samples
for vanilla CoT, ReAct, and HiSS111111We omit standard prompting as it
directly outputs the final prediction without providing intermediate or
reasoning steps., as shown in Table 3, we classify the errors observed in the
verification traces into two categories: (1) fact hallucination and (2)
omission of necessary thoughts. We find that vanilla CoT exhibits substantial
issues of both hallucination and thought omission. Although the Search-
Augmented CoT improves its performance, it still falls short of meeting the
standard prompting method. This suggests that using the original claim as a
search query may end up with insufficiently detailed and informative search
results, which explains its subpar performance. In contrast, ReAct, with its
ability to autonomously generate search queries and access external knowledge,
effectively mitigates failures caused by hallucinations. However, it
encounters challenges of thought omission as it may ignore noteworthy points
of a claim due to the lack of claim decomposition and a fine-grained step-by-
step process. Our HiSS prompting method instead effectively addresses both
issues, thanks to its ability to cover both explicit and implicit points of
the claim to get checked and the ability to seek necessary external knowledge
supported by the search engine.

Error Types | CoT | ReAct | HiSS
---|---|---|---
Fact Hallucination | 43% | 28% | 5%
Thoughts Omission | 60% | 53% | 13%
Table 3: Distribution of errors based on 40 examples from RAWFC, where Vanilla
CoT, ReAct, and HiSS give incorrect verification results.

###  4.3 Ablation Study

To analyze the impact of different configurations of HiSS, we conducted an
ablation analysis on RAWFC as shown in Figure 3.

Figure 3: Ablation results on RAWFC dataset.

#### Effect of claim decomposition:

Firstly, we consider HiSS without claim decomposition, where we directly pose
probing questions based on the original claim, bypassing the claim
decomposition step while keeping the step-by-step process. In this setting,
the performance of HiSS decreases by 1.5%. This result demonstrates that claim
decomposition, which separates the claim based on explicit points, is helpful
in improving the final predictions.

#### Effect of subclaim step-by-step verification:

Next, we conduct an ablation study on the step-by-step verification process
for each subclaim. Instead of generating probing questions, we let LLM
directly verify subclaims by searching for relevant background information.
Subsequently, the LLM made predictions based on the subclaims and the
retrieved information. Notably, this modification resulted in a 2.9%
performance drop, underscoring the importance of employing the subclaim step-
by-step verification approach to address the implicit points associated with
each subclaim.

#### Effect of strategy using search:

We compared three approaches to explore the effect of different strategies on
using the search function or not and how it is used: 1) HiSS w/o search, which
relies solely on the internal knowledge from LLM, 2) HiSS always search, which
always queries the search engine to access the external knowledge, and 3)
HiSS, which lets LLM self-decide whether to use the search results in each
step based on its own confidence (see Section 3.3).

As expected, the performance of HiSS w/o search is poor which achieves only an
F1 of 49.8%, indicating that reliance solely on LLM’s internal knowledge is
unreliable and insufficient. An interesting finding is that HiSS prompted to
decide whether to leverage search results or not based on the self-confidence
of LLM achieves an F1 of 54.4%, which is just slightly worse than the HiSS
always search by 1.0%. Our further inspection reveals that out of the 200 test
claims on RAWFC, a total of 934 questions are generated, and LLM flags 690 of
them as being confident to answer. This indicates that in cases where the
model is confident, external knowledge from the search engine can only
marginally improve its performance, as the model is capable of providing
accurate answers. In contrast, for the cases where the model lacks confidence,
leveraging search results can enhance its performance much more greatly.
Assuming that we can basically trust the factuality of search results from the
web, this suggests that the model demonstrates a reasonably good estimation of
its own confidence.

###  4.4 Human Evaluation

We conduct a human evaluation to study the explanation quality of three
different types of explanations: Gold justification given by human
journalists, explanations generated by the strongest supervised explainable
model CofCED, and the reasoning trajectory generated from the HiSS method. We
ask three English-speaking judges to rate these explanations with scores of 1,
2, and 3 (higher is better) according to the following criteria:

  * •

Coverage. The explanation and reasoning does not miss any important points
that contribute to the check.

  * •

Non-redundancy. The explanation and reasoning provided only includes relevant
information that is necessary for understanding the claim and fact-checking
it, without any redundant or repeated information.

  * •

Readability. The explanation and reasoning is straightforward and simple to
read.

  * •

Overall. The overall quality of the generated explanation and reasoning.

We randomly sample 34 claims from the LIAR test set. Three annotators rate
them independently. We compute Krippendorff’s $\alpha$ inter-annotator
agreement (IAA) Hayes and Krippendorff (2007) and get $0.36$ for coverage,
$0.42$ for non-redundancy, $0.30$ for readability and $0.38$ for overall.

| RAWFC
---|---
| Gold | CofCED | HiSS
Readability | 2.75 | 1.63 | 2.44
Coverage | 2.65 | 1.99 | 2.63
Non-redundancy | 2.72 | 1.28 | 2.25
Overall | 2.69 | 1.74 | 2.54
Table 4: Average human ratings on explanations of verification for the claims
in the RAWFC dataset. Gold, CofCED, and HiSS correspond to the explanations
produced by human journalists, CofCED and HiSS, respectively. A higher score
means a better explanation. The highest score is in bold, and the second is
underlined.

Table 4 shows the averaged scores of human evaluation. We find that the gold
explanations are slightly better than HiSS-based explanations, while the
state-of-the-art automatic explainable claim verification model CofCED is the
worst. In particular, for the coverage criteria, HiSS can elicit explanations
that are on par with the human-written ones. This explains that our HiSS
elicits GPT-3.5 to generate more fine-grained checking points and steps. In
addition, the non-redundancy score is relatively lower, since GPT-3.5 may
generate repeated subclaims. We conjecture that this may be due to the
intrinsic problem of greedy sampling of language models Holtzman et al.
(2019).

##  5 Conclusion and Future Work

In this paper, we study different prompting methods for using LLMs in news
claim verification. We introduce a hierarchical step-by-step (HiSS) method
that prompts LLM to perform the verification in fine-grained steps, aiming to
mitigate the omission of thoughts and fact hallucination. Validated on two
public datasets, HiSS prompting improves the performance of LLMs on the task
over fully-supervised SoTA models and its strong few-shot ICL-based
counterparts. HiSS prompted explanations show superior explainability in their
coverage and readability.

In the future, we will build a conversational fact-checking model based on
LLMs which can be user-friendly and incorporate human fact-checkers in the
loop.

##  6 Limitations

Despite the promising performance of LLMs based on few-shot ICL, fact
verification is a challenging research problem given the fact that performance
scores are still quite low in general. There are a few limitations. Firstly,
in this work, we highlight that all the baselines and our proposed method
solely rely on textual information. We focus on an unimodal approach utilizing
language models and do not consider the potential assistance from other
modalities, such as images and videos, for this task. Although the exploration
of multimodal approaches has gradually drawn some research attention Wang et
al. (2018); Silva et al. (2021); Bu et al. (2023), it falls outside the scope
of our current work.

Meanwhile, the scope of this study is limited to the verification of news
claims, which represents only a subset of the broader issue of misinformation.
Misinformation encompasses a wide range of false or misleading information,
including rumors, fake news articles, and spams Wu et al. (2019). While our
focus was specifically on news claims, future research could explore the
detection and mitigation of misinformation in other formats.

Further, our proposed prompting method heavily relies on the capabilities of
backbone LLMs, which can come with substantial computational costs. Our method
leverages the advancements in multi-step reasoning exhibited by these LLMs,
necessitating high-performance expectations. However, it is worth noting that
most state-of-the-art LLMs are currently not open-source and only available as
services. For instance, GPT-3.5 can only be accessed via API. The reliance on
such LLMs makes deep model control infeasible, and the need for API access
poses challenges in terms of cost.

Finally, while our approach leverages search engines to mitigate the fact
hallucination issue in LLMs, it operates under the assumption that pertinent
information is readily accessible through web search. However, not all
information is indexed or available in search engines. For instance, if
someone claims to have witnessed a rare meteorological phenomenon in a small
town, such event might not be reported on major news websites or databases.
Such firsthand, non-digitized accounts might be retrieved or fact-checked.
This underscores the limitation in relying solely on search engines as a
primary source of external knowledge for fact-checking with LLMs. Another
limitation of our method lies in the claims that are beyond established world
knowledge when necessary relevant knowledge is not complete or even not
available. This necessitates the model’s ability to infer novel knowledge by
formulating and subsequently validating appropriate hypotheses, a task that
remains beyond the capabilities of existing technologies.

## Acknowledgement

We thank the anonymous reviewers for their helpful comments during the review
of this paper.

## References

  * Anil et al. (2023) Rohan Anil, Andrew M. Dai, Orhan Firat, Melvin Johnson, Dmitry Lepikhin, Alexandre Passos, Siamak Shakeri, Emanuel Taropa, Paige Bailey, Zhifeng Chen, Eric Chu, Jonathan H. Clark, Laurent El Shafey, Yanping Huang, Kathy Meier-Hellstern, Gaurav Mishra, Erica Moreira, Mark Omernick, Kevin Robinson, Sebastian Ruder, Yi Tay, Kefan Xiao, Yuanzhong Xu, Yujing Zhang, Gustavo Hernandez Abrego, Junwhan Ahn, Jacob Austin, Paul Barham, Jan Botha, James Bradbury, Siddhartha Brahma, Kevin Brooks, Michele Catasta, Yong Cheng, Colin Cherry, Christopher A. Choquette-Choo, Aakanksha Chowdhery, Clément Crepy, Shachi Dave, Mostafa Dehghani, Sunipa Dev, Jacob Devlin, Mark Díaz, Nan Du, Ethan Dyer, Vlad Feinberg, Fangxiaoyu Feng, Vlad Fienber, Markus Freitag, Xavier Garcia, Sebastian Gehrmann, Lucas Gonzalez, Guy Gur-Ari, Steven Hand, Hadi Hashemi, Le Hou, Joshua Howland, Andrea Hu, Jeffrey Hui, Jeremy Hurwitz, Michael Isard, Abe Ittycheriah, Matthew Jagielski, Wenhao Jia, Kathleen Kenealy, Maxim Krikun, Sneha Kudugunta, Chang Lan, Katherine Lee, Benjamin Lee, Eric Li, Music Li, Wei Li, YaGuang Li, Jian Li, Hyeontaek Lim, Hanzhao Lin, Zhongtao Liu, Frederick Liu, Marcello Maggioni, Aroma Mahendru, Joshua Maynez, Vedant Misra, Maysam Moussalem, Zachary Nado, John Nham, Eric Ni, Andrew Nystrom, Alicia Parrish, Marie Pellat, Martin Polacek, Alex Polozov, Reiner Pope, Siyuan Qiao, Emily Reif, Bryan Richter, Parker Riley, Alex Castro Ros, Aurko Roy, Brennan Saeta, Rajkumar Samuel, Renee Shelby, Ambrose Slone, Daniel Smilkov, David R. So, Daniel Sohn, Simon Tokumine, Dasha Valter, Vijay Vasudevan, Kiran Vodrahalli, Xuezhi Wang, Pidong Wang, Zirui Wang, Tao Wang, John Wieting, Yuhuai Wu, Kelvin Xu, Yunhan Xu, Linting Xue, Pengcheng Yin, Jiahui Yu, Qiao Zhang, Steven Zheng, Ce Zheng, Weikang Zhou, Denny Zhou, Slav Petrov, and Yonghui Wu. 2023.  Palm 2 technical report.
  * Atanasova et al. (2020) Pepa Atanasova, Jakob Grue Simonsen, Christina Lioma, and Isabelle Augenstein. 2020\.  Generating fact checking explanations.  In _ACL_ , pages 7352–7364.
  * Bang et al. (2023) Yejin Bang, Samuel Cahyawijaya, Nayeon Lee, Wenliang Dai, Dan Su, Bryan Wilie, Holy Lovenia, Ziwei Ji, Tiezheng Yu, Willy Chung, et al. 2023.  A multitask, multilingual, multimodal evaluation of chatgpt on reasoning, hallucination, and interactivity.  _arXiv preprint arXiv:2302.04023_.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020.  Language models are few-shot learners.  In _NeurIPs_ , volume 33, pages 1877–1901.
  * Bu et al. (2023) Yuyan Bu, Qiang Sheng, Juan Cao, Peng Qi, Danding Wang, and Jintao Li. 2023.  Online misinformation video detection: a survey.  _arXiv preprint arXiv:2302.03242_.
  * Fan et al. (2020) Angela Fan, Aleksandra Piktus, Fabio Petroni, Guillaume Wenzek, Marzieh Saeidi, Andreas Vlachos, Antoine Bordes, and Sebastian Riedel. 2020.  Generating fact checking briefs.  In _EMNLP_ , pages 7147–7161.
  * Gad-Elrab et al. (2019) Mohamed H Gad-Elrab, Daria Stepanova, Jacopo Urbani, and Gerhard Weikum. 2019.  Exfakt: A framework for explaining facts over knowledge graphs and text.  In _WSDM_ , pages 87–95.
  * Hayes and Krippendorff (2007) Andrew F Hayes and Klaus Krippendorff. 2007.  Answering the call for a standard reliability measure for coding data.  _Communication methods and measures_ , 1(1):77–89.
  * Holtzman et al. (2019) Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. 2019.  The curious case of neural text degeneration.  In _ICLR_.
  * Ji et al. (2023) Ziwei Ji, Nayeon Lee, Rita Frieske, Tiezheng Yu, Dan Su, Yan Xu, Etsuko Ishii, Ye Jin Bang, Andrea Madotto, and Pascale Fung. 2023.  Survey of hallucination in natural language generation.  _ACM Computing Surveys_ , 55(12):1–38.
  * Jiang et al. (2023) Zhengbao Jiang, Frank F Xu, Luyu Gao, Zhiqing Sun, Qian Liu, Jane Dwivedi-Yu, Yiming Yang, Jamie Callan, and Graham Neubig. 2023.  Active retrieval augmented generation.  _arXiv preprint arXiv:2305.06983_.
  * Jin et al. (2022) Yiqiao Jin, Xiting Wang, Ruichao Yang, Yizhou Sun, Wei Wang, Hao Liao, and Xing Xie. 2022.  Towards fine-grained reasoning for fake news detection.  In _AAAI_ , volume 36, pages 5746–5754.
  * Kenton and Toutanova (2019) Jacob Devlin Ming-Wei Chang Kenton and Lee Kristina Toutanova. 2019.  Bert: Pre-training of deep bidirectional transformers for language understanding.  In _NAACL_ , pages 4171–4186.
  * Kotonya and Toni (2020) Neema Kotonya and Francesca Toni. 2020.  Explainable automated fact-checking for public health claims.  In _EMNLP_ , pages 7740–7754.
  * Kruengkrai et al. (2021) Canasai Kruengkrai, Junichi Yamagishi, and Xin Wang. 2021.  A multi-level attention model for evidence-based fact checking.  In _ACL_ , pages 2447–2460.
  * Lazer et al. (2018) David MJ Lazer, Matthew A Baum, Yochai Benkler, Adam J Berinsky, Kelly M Greenhill, Filippo Menczer, Miriam J Metzger, Brendan Nyhan, Gordon Pennycook, David Rothschild, et al. 2018.  The science of fake news.  _Science_ , 359(6380):1094–1096.
  * Lee et al. (2021) Nayeon Lee, Yejin Bang, Andrea Madotto, and Pascale Fung. 2021.  Towards few-shot fact-checking via perplexity.  In _NAACL_ , pages 1971–1981.
  * Lee et al. (2020) Nayeon Lee, Belinda Z Li, Sinong Wang, Wen-tau Yih, Hao Ma, and Madian Khabsa. 2020\.  Language models as fact checkers?  In _FEVER Workshop_ , pages 36–41.
  * Lewis et al. (2020) Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, et al. 2020.  Retrieval-augmented generation for knowledge-intensive nlp tasks.  In _NeurIPS_ , volume 33, pages 9459–9474.
  * Lu and Li (2020) Yi-Ju Lu and Cheng-Te Li. 2020.  Gcan: Graph-aware co-attention networks for explainable fake news detection on social media.  In _ACL_ , pages 505–514.
  * Ma et al. (2019) Jing Ma, Wei Gao, Shafiq Joty, and Kam-Fai Wong. 2019.  Sentence-level evidence embedding for claim verification with hierarchical attention networks.  In _ACL_.
  * Ousidhoum et al. (2022) Nedjma Ousidhoum, Zhangdie Yuan, and Andreas Vlachos. 2022.  Varifocal question generation for fact-checking.  _arXiv preprint arXiv:2210.12400_.
  * Popat et al. (2018) Kashyap Popat, Subhabrata Mukherjee, Andrew Yates, and Gerhard Weikum. 2018.  Declare: Debunking fake news and false claims using evidence-aware deep learning.  In _EMNLP_ , pages 22–32.
  * Press et al. (2022) Ofir Press, Muru Zhang, Sewon Min, Ludwig Schmidt, Noah A Smith, and Mike Lewis. 2022.  Measuring and narrowing the compositionality gap in language models.  _arXiv preprint arXiv:2210.03350_.
  * Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. 2019.  Language models are unsupervised multitask learners.  _OpenAI blog_ , 1(8):9.
  * Rashkin et al. (2017) Hannah Rashkin, Eunsol Choi, Jin Yea Jang, Svitlana Volkova, and Yejin Choi. 2017\.  Truth of varying shades: Analyzing language in fake news and political fact-checking.  In _EMNLP_ , pages 2931–2937.
  * Silva et al. (2021) Amila Silva, Ling Luo, Shanika Karunasekera, and Christopher Leckie. 2021.  Embracing domain differences in fake news: Cross-domain fake news detection using multi-modal data.  In _AAAI_ , volume 35, pages 557–565.
  * Soleimani et al. (2020) Amir Soleimani, Christof Monz, and Marcel Worring. 2020.  Bert for evidence retrieval and claim verification.  In _ECIR_ , pages 359–366.
  * Thorne et al. (2018) James Thorne, Andreas Vlachos, Christos Christodoulopoulos, and Arpit Mittal. 2018\.  Fever: a large-scale dataset for fact extraction and verification.  In _NAACL_ , pages 809–819.
  * Wang et al. (2023) Lei Wang, Wanyu Xu, Yihuai Lan, Zhiqiang Hu, Yunshi Lan, Roy Ka-Wei Lee, and Ee-Peng Lim. 2023.  Plan-and-solve prompting: Improving zero-shot chain-of-thought reasoning by large language models.  _arXiv preprint arXiv:2305.04091_.
  * Wang (2017) William Yang Wang. 2017.  “liar, liar pants on fire”: A new benchmark dataset for fake news detection.  In _ACL (short)_ , pages 422–426.
  * Wang et al. (2018) Yaqing Wang, Fenglong Ma, Zhiwei Jin, Ye Yuan, Guangxu Xun, Kishlay Jha, Lu Su, and Jing Gao. 2018.  Eann: Event adversarial neural networks for multi-modal fake news detection.  In _Proceedings of the 24th acm sigkdd international conference on knowledge discovery & data mining_, pages 849–857.
  * Wei et al. (2022) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou. 2022.  Chain of thought prompting elicits reasoning in large language models.  _arXiv preprint arXiv:2201.11903_.
  * Wu et al. (2019) Liang Wu, Fred Morstatter, Kathleen M Carley, and Huan Liu. 2019.  Misinformation in social media: definition, manipulation, and detection.  In _SIGKDD_ , volume 21, pages 80–90.
  * Yang et al. (2019) Fan Yang, Shiva K Pentyala, Sina Mohseni, Mengnan Du, Hao Yuan, Rhema Linder, Eric D Ragan, Shuiwang Ji, and Xia Hu. 2019.  Xfake: Explainable fake news detector with visualizations.  In _WWW_ , pages 3600–3604.
  * Yang et al. (2018) Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William Cohen, Ruslan Salakhutdinov, and Christopher D Manning. 2018.  Hotpotqa: A dataset for diverse, explainable multi-hop question answering.  In _EMNLP_ , pages 2369–2380.
  * Yang et al. (2022) Zhiwei Yang, Jing Ma, Hechang Chen, Hongzhan Lin, Ziyang Luo, and Yi Chang. 2022\.  A coarse-to-fine cascaded evidence-distillation neural network for explainable fake news detection.  In _COLING_ , pages 2608–2621.
  * Yao et al. (2023) Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimhan, and Yuan Cao. 2023.  ReAct: Synergizing reasoning and acting in language models.  In _ICLR_.
  * Zhang et al. (2023) Yue Zhang, Yafu Li, Leyang Cui, Deng Cai, Lemao Liu, Tingchen Fu, Xinting Huang, Enbo Zhao, Yu Zhang, Yulong Chen, et al. 2023.  Siren’s song in the ai ocean: A survey on hallucination in large language models.  _arXiv preprint arXiv:2309.01219_.
  * Zhou et al. (2022) Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Olivier Bousquet, Quoc Le, and Ed Chi. 2022.  Least-to-most prompting enables complex reasoning in large language models.  _arXiv preprint arXiv:2205.10625_.

##  Appendix A Case Study

Table A shows the example from LIAR where CoT 1) generates sentences
containing fact hallucination, with inaccurate percentages (i.e., 57 and 1
percent), and 2) omits necessary verification of the given percentage. While
its final prediction is correct, it still encounters the issue of omitting
necessary thoughts by not verifying the authenticity of the “1 percentage to
food and agriculture” claim before making a judgment on the entire claim’s
correctness. In contrast, our HiSS method examines each component of the
claim, along with leveraging search engine to gather comprehensive
information. However, it is worth noting that even though HiSS can generate a
more detailed reasoning process, it still faces challenges related to repeated
generations.

Ground-truth: False.
---
Claim: Says 57 percent of federal spending goes to the military and just 1
percent goes to food and agriculture, including food stamps.
Standard Prompting
True.
Vanilla CoT
According to the Congressional Budget Office, the federal government spent
$4.1 trillion. Of that, 57 percent went to the military and 1 percent went to
food and agriculture, including food stamps. Thus the claim is classified as
true.
ReAct
Thought 1: I need to search for information about federal spending to verify
the claim that 57 percent goes to the military
and 1 percent goes to food and agriculture, including food stamps.
Action 1: Search[federal spending breakdown]
Observation 1: This means federal spending was equal to 25%
of the total gross domestic product (GDP), or economic …"
Thought 2: The observation contradicts the claim that 57 percent goes to the
military and  1 percent to food and agriculture, including food stamps. The
claim appears to be false.
Action 2: Finish[False]
HiSS
A fact checker will decompose the claim into 2 subclaims that are easier to
verify:
1\. 57 percent of federal spending goes to the military.
2\. Just 1 percent of federal spending goes to food and agriculture, including
food stamps.
To verify subclaim 1, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: What percentage of federal spending goes to the military?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: Yes.
Answer: About one-sixth of federal spending goes to national defense. CBO
estimates the budgetary effects of
legislation related to national security and assesses the cost-effectiveness
of current and proposed defense programs.
CBO also analyzes federal programs and issues related to veterans.
To verify subclaim 2, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: What percentage of federal spending goes to food and agriculture,
including food stamps? Question: What
percentage of federal spending goes to food and agriculture, including food
stamps?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Federal spending on USDA’s food and nutrition assistance programs
totaled $182.5 billion, 49 percent more
than the … USDA’s food and nutrition assistance programs accounted for about 5
percent of total federal outlays.
Based on the answers to these questions, it is clear that among pants-fire,
false, barely-true, half-true, mostly-true, and true, the claim can be
classified as false. Based on the answers to these questions, it is clear that
among pants-fire, false, barely-true, half-true, mostly-true, and true, the
claim can be classified as false.

Table 5: Case study. Labels and predictions from different methods are in
pink, generated sentences containing fact hallucination are in purple, and
repeated sentences are in yellow.

##  Appendix B Demonstration Examples

We show the demonstration examples used in the LIAR dataset. Table 6 and B
present the prompts we used for standard prompting and vanilla CoT,
respectively. Table 8 (a) and 8 (b) present the prompts used for HiSS.

Q: Among pants-fire, false, barely-true, half-true, mostly-true, and true, the
claim "Emerson Moser, who was Crayola’s top crayon molder for almost 40 years,
was colorblind." is classified as
---
A: mostly-true.
Q: Among pants-fire, false, barely-true, half-true, mostly-true, and true, the
claim "Bernie Sanders said 85 million Americans have no health insurance." is
classified as
A: half-true.
Q: Among pants-fire, false, barely-true, half-true, mostly-true, and true, the
claim "JAG charges Nancy Pelosi with treason and seditious conspiracy." is
classified as
A: pants-fire.
Q: Among pants-fire, false, barely-true, half-true, mostly-true, and true, the
claim "Cheri Beasley “backs tax hikes — even on families making under
$75,000." is classified as
A:barely-true.
Table 6: Demonstration examples of standard prompting.

Q: Choose a label from pants-fire, false, barely-true, half-true, mostly-true,
and true for the following claim.
---
Claim: "Emerson Moser, who was Crayola’s top crayon molder for almost 40
years, was colorblind."
A: Moser did have a type of colorblindness. However he said he only had blue-
green colorblindness, So only some colors were confusing to him. Thus the
claim is classified as mostly-true.
Q: Choose a label from pants-fire, false, barely-true, half-true, mostly-true,
and true for the following claim.
Claim: "Bernie Sanders said 85 million Americans have no health insurance."
A: Sanders accurately cited the Commonwealth Fund report in an October op-ed.
His statement on CNN was partially accurate but left out important details and
context. Sanders omitted that his figure included people who either have no
health insurance or are underinsured. Thus the claim is classified as half-
true.
Q: Choose a label from pants-fire, false, barely-true, half-true, mostly-true,
and true for the following claim.
Claim: "JAG charges Nancy Pelosi with treason and seditious conspiracy."
A: A spokesperson for the U.S. Navy Judge Advocate General’s Corps told
PolitiFact it isn’t true that Pelosi has been charged. The original claim
appeared on a website that posts fantastical, false stories and has a
disclaimer that the Thus the claim is classified as pants-fire.
Q: Choose a label from pants-fire, false, barely-true, half-true, mostly-true,
and true for the following claim.
Claim: "Cheri Beasley “backs tax hikes — even on families making under
$75,000."
A: Beasley does support the Inflation Reduction Act but the claim is wrong to
suggest it includes a change in tax rates for that income bracket. Expert
analyses show that any effect on those earners would be on the scale of $350
and, by one estimate, delayed for several years. Thus the claim is classified
as barely-true.

Table 7: Demonstration examples of vanilla CoT prompting. The CoT process is
highlighted in green.

Q: Claim: “Emerson Moser, who was Crayola’s top crayon molder for almost 40
years, was colorblind.”
---
A: A fact checker will decompose the claim into 4 subclaims that are easier to
verify:
1\. Emerson Moser was a crayon molder at Crayola.
2\. Moser worked at Crayola for almost 40 years.
3\. Moser was Crayola’s top crayon molder.
4\. Moser was colorblind.
To verify subclaim 1, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Is there any official record or documentation indicating that
Emerson Moser worked as a crayon molder
at Crayola?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: Yes.
Answer: Yes.
To verify subclaim 2, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Are there any official records or documentation confirming Emerson
Moser’s length of employment at
Crayola?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Emerson Moser, who is retiring next week after 35 years, isn’t
colorblind in the sense that he can’t see
color at all. It’s just that some …
To verify subclaim 3, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Are there credible sources or publications that mention Emerson
Moser as Crayola’s top crayon molder?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: Yes.
Answer: Yes.
To verify subclaim 4, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Are there any credible sources or records indicating that Emerson
Moser was colorblind?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: Yes.
Answer: Yes.
Question: Was Emerson Moser’s colorblindness only confusing for certain
colors?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Moser has had tritanomaly, a type of colorblindness that makes it
difficult to distinguish between blue and
green and between yellow and red.
Based on the answers to these questions, it is clear that among pants-fire,
false, barely-true, half-true, mostly-true, and true, the claim can be
classified as mostly-true.
Q: Claim: “Bernie Sanders said 85 million Americans have no health insurance.”
A: A fact checker will not split the claim since the original claim is easier
to verify.
To verify the claim, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: How many Americans did Bernie Sanders claim had no health insurance?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: “We have 85 million Americans who have no health insurance,” Sanders
said Dec. 11 on CNN’s State of
the Union.
Question: How did Bernie Sanders define “no health insurance”?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Sanders spokesperson Mike Casca said the senator was referring to the
number of uninsured and
under-insured Americans and cited a report about those numbers for adults.
Question: How many Americans were uninsured or under-insured according
to the Commonwealth Fund survey?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: The Commonwealth Fund survey found that 43% of working-age adults 19
to 64, or about 85 million
Americans, were uninsured or inadequately insured.
Question: Is the statement “we have 85 million Americans who have no health
insurance” partially accurate according
to the information in the passage?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Bernie Sanders omitted that his figure included people who either have
no health insurance or are
under-insured.
Based on the answers to these questions, it is clear that among pants-fire,
false, barely-true, half-true, mostly-true, and true, the claim is classified
as half-true.
Table 8 (a): Demonstration examples of HiSS prompting. The CoT process is
highlighted in green.

Q: Claim: “JAG charges Nancy Pelosi with treason and seditious conspiracy.”
---
A: A fact checker will decompose the claim into 2 subclaims that are easier to
verify:
1\. JAG has made a claim or accusation against Nancy Pelosi.
2\. The specific charges or allegations made against Nancy Pelosi are treason
and seditious conspiracy.
To verify subclaim 1, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Is it true that JAG has made a claim or accusation against Nancy
Pelosi?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: There is no evidence to support this claim and a spokesperson for the
U.S. Navy Judge Advocate General’s
Corps has stated that it is not true.
To verify subclaim 2, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Is it true that the specific charges or allegations made against
Nancy Pelosi are treason and seditious
conspiracy?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: There is no evidence to support this claim.
Question: Where is the source of the claim?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: Real Raw News, a disclaimer stating that it contains “humor, parody
and satire” and has a history of
publishing fictitious stories.
Based on the answers to these questions, it is clear that among pants-fire,
false, barely-true, half-true, mostly-true, and true, the claim is classified
as pants-fire.
Q: Claim: “Cheri Beasley “backs tax hikes — even on families making under
$75,000.”
A: A fact checker will decompose the claim into 2 subclaims that are easier to
verify:
1\. Cheri Beasley supports tax increases.
2\. Cheri Beasley supports tax increases for families with an income under
$75,000.
To verify subclaim 1, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Does Cheri Beasley supports tax increases?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: Yes.
Answer: Beasley supports student loan bailouts for the wealthy.
To verify subclaim 2, a fact-checker will go through a step-by-step process to
ask and answer a series of questions relevant to its factuality. Here are the
specific steps he/she raise each question and look for an answer:
Question: Does the ad accurately link Beasley’s position on student loan debt
forgiveness with her stance on tax hikes
for families making under $75,000 per year?
Tell me if you are confident to answer the question or not. Answer with “yes”
or “no”: No.
Answer: The ad makes a misleading connection between the two issues and does
not accurately represent Beasley’s
position on tax hikes for families making under $75,000 per year.
Answer: No.
Based on the answers to these questions, it is clear that among pants-fire,
false, barely-true, half-true, mostly-true, and true, the claim is classified
as barely-true.
Table 8 (b): Demonstration examples of HiSS prompting. The CoT process is
highlighted in green.


# The volume conjecture for polyhedra implies the Stoker conjecture

Giulio Belletti

###### Abstract

We show that the Volume Conjecture for polyhedra implies a weak version of the
Stoker Conjecture; in turn we prove that this weak version of the Stoker
conjecture implies the Stoker conjecture. The main tool used is an extension
of a result of Montcouquiol and Weiss, saying that dihedral angles are local
coordinates for compact polyhedra with angles $\leq\pi$.

##  1 Introduction

To what extent dihedral angles determine a hyperbolic polyhedron is a long
standing question, first posed in [Sto68, Page 152].

######  Conjecture (The Stoker conjecture).

Let $P_{1}$ and $P_{2}$ be two hyperbolic polyhedra with $1$-skeleton $\Gamma$
and the same dihedral angle at each corresponding edge. Then $P_{1}$ and
$P_{2}$ are isometric.

Over the next half century there have been several results in this direction:
as soon as some conditions are required of the polyhedra (for example, acute-
angled in [And70], simple in [RH93] or hyperideal in [BB02]) then the
conjecture is known to be true. Moreover [Mon13] and [Wei13] independently
proved that the conjecture is true locally, i.e. the dihedral angles are local
coordinates for the space of polyhedra with fixed $1$-skeleton.

In this paper, we propose a weak version of the Stoker conjecture:

######  Conjecture (Weak Stoker conjecture).

Let $P_{1}$ and $P_{2}$ be two polyhedra with $1$-skeleton $\Gamma$ and the
same dihedral angle at each corresponding edge. Then $P_{1}$ and $P_{2}$ have
the same volume.

Clearly the Stoker conjecture implies the weak Stoker Conjecture; in Corollary
4.4 we prove that in fact the converse also holds.

The interest of the weak Stoker Conjecture is two-fold: firstly, it could be a
more tractable problem than the Stoker conjecture; secondly, it is actually a
consequence of the Volume Conjecture for polyhedra, providing a connection
between a long standing open problem in hyperbolic geometry and quantum
invariants.

Since the introduction of quantum invariants in [Wit89], many conjectures have
been proposed to link their asymptotic behavior to several topological and
geometric properties. Among them the subject of the most research has been the
plethora of volume conjectures, relating the growth of quantum invariants of
objects to the volume of hyperbolic structures on them. The first such
conjecture was proposed by Kashaev ([Kas97], see also [MM01]) relating the
growth of the Jones polynomial of a knot evaluated at a certain root of unity
to the volume of the complement of the knot. More recently, Chen and Yang
[CY18] proposed a volume conjecture for the Turaev-Viro invariant of compact
manifolds. Both of these conjectures have extensions relating higher order
terms of the asymptotic expansion of a quantum invariant to different
quantities such as the Reidemeister torsion. These conjectures have been
proven for certain families of knots or manifolds (see for example [Oht18] and
[BDKY22] for results about the Volume Conjecture of Chen and Yang, and
[Oht16],[OY18], [Oht17] for results on the Kashaev volume conjecture).

A related series of conjectures connects the quantum invariants of a planar
graph $\Gamma$ in $S^{3}$ to the volume of hyperbolic polyhedra that have
$\Gamma$ as a $1$-skeleton. The first such conjecture was given in [CGvdV15]
for trivalent planar graphs and simple hyperideal polyhedra; it was extended
in [KM18] and later in [Bel20] to any polyhedral (i.e. planar and
$3$-connected) graph and any hyperbolic polyhedron.

######  Conjecture (The volume conjecture for polyhedra).

Let $P$ be a hyperbolic polyhedron with dihedral angles
$\alpha_{1},\dots,\alpha_{m}$ at the edges $e_{1},\dots,e_{m}$, and
$1$-skeleton $\Gamma$. Let $col_{r}$ be a sequence of $r$-admissible colorings
of the edges $e_{1},\dots,e_{m}$ of $\Gamma$ such that

$2\pi\lim_{r\rightarrow+\infty}\frac{col_{r}(e_{i})}{r}=\pi-\alpha_{i}.$

Then

$\lim_{r\rightarrow+\infty}\frac{\pi}{r}\log\left\lvert
Y_{r}(S^{3},\Gamma,col_{r},e^{2\pi i/r})\right\rvert=\mathrm{Vol}(P).$

where $Y_{r}\left(S^{3},\Gamma,col,e^{2\pi i/r}\right)$ is the Yokota
invariant of the graph $\Gamma$ with edges colored with $col_{r}$ evaluated at
$e^{2\pi i/r}$ (see Section 3 and [Bel20, Section 2] for more details and
definitions).

In Section 2 we give an overview of hyperbolic polyhedra and we prove the
technical results about the space of hyperbolic polyhedra with fixed
$1$-skeleton that are needed in the proof of the main result. In Section 3 we
give a brief review about the Yokota invariant and the Volume conjecture for
polyhedra. Finally in Section 4 we prove the equivalence of the Weak Stoker
Conjecture and the Stoker Conjecture. Quantum invariants are only involved in
Section 3; a reader only interested in the geometric results of this paper can
safely skip it if so desired.

##  2 Hyperbolic polyhedra

######  Definition 2.1.

A _hyperbolic polyhedron_ is a compact intersection of half spaces of
$\mathbb{H}^{3}$.

######  Remark 2.2.

Throughout the literature, other types of hyperbolic polyhedra have also been
considered, for example polyhedra with ideal or hyperideal vertices. The
results of this paper can be easily extended to _proper generalized hyperbolic
polyhedra_ (see [Bel21]) with no ideal vertices, by simply truncating them to
obtain a hyperbolic polyhedron of Definition 2.1. However the techniques here
presented cannot be directly applied to the case of polyhedra with some ideal
vertices, since it is unknown in this case whether angles are local
coordinates; the Stoker conjecture for polyhedra with only ideal vertices is
known to be true by [Riv96].

######  Definition 2.3.

If $\Gamma$ is a graph we say that a polyhedron $P$ has $1$-skeleton $\Gamma$
if there exists a simplicial map $\phi:\Gamma\rightarrow P$ sending vertices
to vertices injectively and edges to edges.

######  Remark 2.4.

A graph $\Gamma$ is the $1$-skeleton of a polyhedron if and only if it is
planar and $3$-connected (that is to say, it has more than three vertices and
cannot be disconnected by removing two edges) [Ste22]. Such a graph admits a
unique embedding in $S^{2}$ up to planar isotopy and reflections [Fle73,
Corollary 3.4]; therefore, even though $\Gamma$ is just an abstract graph, it
is convenient to treat it as an embedded graph in the sphere. As such, we will
often speak of its faces, meaning the faces of the cellularization of $S^{2}$
induced by an embedding.

######  Conjecture 1 (The Stoker conjecture).

Let $P_{1}$ and $P_{2}$ be two polyhedra with the same $1$-skeleton $\Gamma$.
If for every edge $e$ of $\Gamma$, the dihedral angles of $P_{1}$ and $P_{2}$
at their corresponding edge are equal, then $P_{1}$ and $P_{2}$ are isometric.

######  Conjecture 2 (A weak version of the Stoker conjecture).

Let $P_{1}$ and $P_{2}$ be two polyhedra with the same $1$-skeleton $\Gamma$.
If for every edge $e$ of $\Gamma$, the dihedral angles of $P_{1}$ and $P_{2}$
at their corresponding edge are equal, then $P_{1}$ and $P_{2}$ have the same
volume.

######  Theorem 2.5.

Conjecture 1 is equivalent to Conjecture 2.

######  Theorem 2.6.

If Conjecture 3 is true, then the Stoker conjecture is true.

To prove Theorem 2.6 we will need to take a polyhedron $P$ with $1$-skeleton
$\Gamma$, choose a diagonal $e$ of a non-triangular face, and deform $P$ to
make $e$ into an edge. To do this, it is convenient to treat $P$ as a
polyhedron with $1$-skeleton equal to $\Gamma\cup e$; thus we introduce a
concept of _weak_ $1$-skeleta.

######  Definition 2.7.

Let $e\subseteq\Gamma$ be an edge. A polyhedron $P$ has _weak $1$-skeleton_
$(\Gamma,e)$ if there exists a simplicial map $\phi:\Gamma\rightarrow P$
sending vertices to vertices injectively, edges different from $e$ to edges of
$P$ and the edge $e$ to a face of $P$.

Put in other terms, in a polyhedron with weak $1$-skeleton $(\Gamma,e)$ we
allow the two faces intersecting in $e$ to lie in the same plane. If this
happens, the $1$-skeleton of $P$ is actually the graph obtained from $\Gamma$
by removing the edge $e$; otherwise, the $1$ skeleton of $P$ is simply
$\Gamma$. Notice that if the $1$-skeleton of $P$ is $\Gamma$, then $P$ has
weak skeleton $(\Gamma,e)$ for any edge $e$.

######  Definition 2.8.

The space of polyhedra with $1$-skeleton $\Gamma$ is denoted
$\mathcal{P}_{\Gamma}$. The space of polyhedra with weak $1$-skeleton
$(\Gamma,e)$ is denoted $\mathcal{P}_{\Gamma}^{e}$.

######  Remark 2.9.

A polyhedron with $F$ faces is an intersection of $F$ half spaces. The set
$\mathcal{H}$ of half spaces in $\mathbb{R}^{3}$ is naturally a manifold; thus
both $\mathcal{P}_{\Gamma}$ and $\mathcal{P}_{\Gamma}^{e}$ are naturally
subsets of the manifold $\mathcal{H}^{F}$. Thus they are topological spaces;
we will prove next that they are actually smooth manifolds.

Let $\Gamma^{\prime}$ be any $3$-connected planar graph and $\Gamma$ be the
graph obtained from $\Gamma^{\prime}$ by adding an edge $e$ to a non-
triangular face of $\Gamma^{\prime}$.

######  Proposition 2.10.

The space $\mathcal{P}_{\Gamma}$ is a manifold; the space
$\mathcal{P}_{\Gamma}^{e}$ is a manifold with boundary whose interior is
$\mathcal{P}_{\Gamma}$ and whose boundary is $\mathcal{P}_{\Gamma^{\prime}}$.

###### Proof.

The proof for $\mathcal{P}_{\Gamma}$ is in [Mon13, Proposition 17]; however
since the proof for $\mathcal{P}_{\Gamma}^{e}$ is adapted from it we briefly
sketch it. The space $\mathcal{P}_{\Gamma}$ is an open set of the space of
Euclidean polyhedra with $1$-skeleton $\Gamma$ (namely, the Euclidean
polyhedra with every vertex contained in the open unit ball); therefore if we
prove that the latter is a manifold we are done.

Denote with $F=\\{f_{1},\dots,f_{n}\\}$ the set of faces of $\Gamma$, and with
$\mathcal{H}$ the set of half spaces in $\mathbb{R}^{3}$. Then
$\mathcal{P}_{\Gamma}$ is a subset of $\mathcal{H}^{F}$; the inclusion map
sends $P\in\mathcal{P}_{\Gamma}$ to the $F$-tuple
$\left(\Pi_{f_{1}},\dots,\Pi_{f_{n}}\right)$ of half-spaces supporting the
faces of $P$.

The image of this inclusion is characterized as the tuples
$\left(\Pi_{f_{1}},\dots,\Pi_{f_{n}}\right)\in\mathcal{H}^{F}$ such that:

  1. 1.

for every $v\in\Gamma$ lying on the faces $f_{i_{1}},\dots,f_{i_{k}}$, the
corresponding half planes $\Pi_{f_{i_{1}}},\dots,\Pi_{f_{i_{k}}}$ intersect in
a single point $v_{i_{1},\dots,i_{k}}\in\mathbb{R}^{3}$;

  2. 2.

if $j\notin\\{i_{1},\dots,i_{k}\\}$, then $v_{i_{1},\dots,i_{k}}$ is in the
interior of $\Pi_{j}$.

The first condition is open if $v$ is a trivalent vertex and closed otherwise.
The second condition is clearly open; in particular it implies that the
intersection defining a polyhedron must be non-redundant (i.e. if we remove
any half space, the polyhedron changes) and that the intersection of half-
planes around different vertices are different points of $\mathbb{H}^{3}$.

These conditions can also be described by equations. The first condition is
equivalent to the set of equations

$\Psi_{v,j}=\mathrm{det}\left(\partial\Pi_{i_{j}}^{*},\partial\Pi_{i_{j+1}}^{*},\partial\Pi_{i_{j+2}}^{*},\partial\Pi_{i_{j+3}}^{*}\right)=0$

for $j\in\\{1,\dots,k-3\\}$. Put in other words, this equation asks that the
dual vectors to the boundaries of $\Pi_{i_{j}},\dots,\Pi_{i_{j+3}}$ must be
coplanar, which is equivalent to asking that
$\partial\Pi_{i_{j}},\dots,\partial\Pi_{i_{j+3}}$ intersect in a single point.
In [Mon13, Proposition 17] it is proven that these conditions are linearly
independent.

To describe the second condition as an equation we define the _signed
distance_ of $v\in\mathbb{R}^{3}$ from a half space $H$, denoted
$\delta(v,H)$, as the real number having absolute value the Euclidean distance
from $v$ to $\partial H$, positive sign if $v\in H$ and negative sign
otherwise. Then condition 2 is

$\delta(v_{i_{1},\dots,i_{k}},\Pi_{j})>0$

for all $j\notin{i_{1},\dots,i_{k}}$. Since these conditions are open, then
the space $\mathcal{P}_{\Gamma}$ is a manifold.

Consider now the polyhedra with weak $1$-skeleton $(\Gamma,e)$, and suppose
(after reindexing if necessary) that $f_{1}$ and $f_{2}$ are the two faces of
$\Gamma$ adjacent to $e$. Then $\mathcal{P}_{\Gamma}^{e}$ is a subset of
$\mathcal{H}^{F}$ as before, and the tuples
$\left(\Pi_{f_{1}},\dots,\Pi_{f_{n}}\right)\in\mathcal{P}_{\Gamma}^{e}$ are
characterized by the following conditions:

  1. 1.

for every $v\in\Gamma$ lying on the faces $f_{i_{1}},\dots,f_{i_{k}}$, the
corresponding half planes $\Pi_{f_{i_{1}}},\dots,\Pi_{f_{i_{k}}}$ intersect in
a single point $v_{i_{1},\dots,i_{k}}\in\mathbb{R}^{3}$;

  2. 2.

if $j\notin\\{1,2,i_{1},\dots,i_{k}\\}$, then $v_{i_{1},\dots,i_{k}}$ is in
the interior of $\Pi_{j}$;

  3. 3.

if $j=1\notin\\{i_{1},\dots,i_{k}\\}$ and $2\notin\\{i_{1},\dots,i_{k}\\}$ or
viceversa, then $v_{i_{1},\dots,i_{k}}$ is in the interior of $\Pi_{j}$;

  4. 4.

if $j=1\notin\\{i_{1},\dots,i_{k}\\}$ and $2\in\\{i_{1},\dots,i_{k}\\}$ or
viceversa, then $v_{i_{1},\dots,i_{k}}$ belongs to $\Pi_{j}$.

The first and fourth conditions are closed while the second and third are
open. We can describe these conditions as equations in a similar manner as
before; the first conditions carry over without change. The second condition
becomes

$\delta(v_{i_{1},\dots,i_{k}},\Pi_{j})>0$

for all $j\notin{1,2,i_{1},\dots,i_{k}}$; the third condition is analogous.
The fourth condition is instead

$\delta(v_{i_{1},\dots,i_{k}},\Pi_{j})\geq 0$

if $j=1\notin\\{i_{1},\dots,i_{k}\\}$ and $2\in\\{i_{1},\dots,i_{k}\\}$ or
viceversa.

To prove that $\mathcal{P}_{\Gamma}^{e}$ is a manifold, we need to show that
the equations of condition 1 and the equalities in condition 4 are
independent.

Now notice that if $\delta(v_{i_{1},\dots,i_{k}},\Pi_{1})=0$, then
$v_{i_{1},\dots,i_{k}}$ belongs to $\partial\Pi_{1}$. This can only happen if
$2\in\\{i_{1},\dots,i_{k}\\}$ by the aforementioned conditions. Since
$\partial\Pi_{1}$ and $\partial\Pi_{2}$ share three distinct points (the two
vertices that are endpoints of $e$ and $v_{i_{1},\dots,i_{k}}$) they must
coincide; therefore this condition is equivalent to $\Pi_{1}=\Pi_{2}$.

Suppose then that there is a linear dependence for the equations
$\Psi_{v,i}=0$ and $\Pi_{1}=\Pi_{2}$; this would imply that there is a linear
dependence for the analogous system of equations in
$\mathcal{P}_{\Gamma^{\prime}}$ which is false.

Therefore $\mathcal{P}_{\Gamma}^{e}$ is a smooth manifold whose boundary is
$\mathcal{P}_{\Gamma^{\prime}}$ and whose interior is $\mathcal{P}_{\Gamma}$.

∎

######  Corollary 2.11.

The space $\mathcal{A}_{\Gamma}$ is a manifold; the space
$\mathcal{A}_{\Gamma}^{e}$ is a manifold with boundary whose interior is
$\mathcal{A}_{\Gamma}$ and whose boundary is $\mathcal{A}_{\Gamma^{\prime}}$.

###### Proof.

We need to show that the group $\textrm{Iso}(\mathbb{H}^{3})$ acts on
$\mathcal{P}_{\Gamma}$ freely and properly. To see that the action is free
notice that since the polyhedra in $\mathcal{P}_{\Gamma}$ are non degenerate,
an isometry preserving $P\in\mathcal{P}_{\Gamma}$ must fix at least $4$ points
in generic position (i.e. not all contained in a plane) and thus must be the
identity. To prove that the action is proper we can repeat verbatim the proof
of [BB02, Lemma 8]. Notice that the proof in [BB02] only requires that the
polyhedra considered have $4$ faces in generic position. Therefore the same
argument carries over unchanged for $\mathcal{P}_{\Gamma}^{e}$. ∎

The following lemma essentially states that for any polyhedron $P$ and any
diagonal $e$, we can deform $P$ to “add” $e$ as an edge, increasing the angle
at $e$ with a non-zero derivative.

######  Lemma 2.12.

For any $P\in\partial\mathcal{P}_{\Gamma}^{e}$ there exists an $\epsilon>0$
and a family $P_{t}$ for $t\in[0,\epsilon)$ such that

  * •

$P_{0}=P$;

  * •

$P_{t}$ is in the interior of $\mathcal{P}_{\Gamma}^{e}$ for any $t>0$;

  * •

the dihedral angle of $P_{t}$ at $e$ is equal to a certain smooth function
$f(t)$ with non-vanishing derivative at $0$.

###### Proof.

Since $P$ is in $\partial\mathcal{P}_{\Gamma}^{e}$ then the two faces $F_{1}$
and $F_{2}$ intersecting in $e$ actually coincide. As before name
$\Gamma^{\prime}$ the graph obtained from $\Gamma$ by removing $e$. Consider
$P\subseteq\mathbb{H}^{3}\subseteq\mathbb{R}^{3}$ and view it as a Euclidean
polyhedron. The dual polyhedron $P^{*}$ is a Euclidean polyhedron with
$1$-skeleton $\Gamma^{\prime*}$ which is contained in a ball of radius $r$
centered in $0$. Notice that $P^{*}$ has a vertex $V$ which is dual to
$F_{1}=F_{2}$. The graph $\Gamma^{\prime*}$ is obtained from $\Gamma^{*}$ by
shrinking the edge $e^{*}$ down to a point which we denote with $v$.

Consider $\Phi$ the homothety centered in $0$ of factor $\frac{1}{2r}$; then
$\Phi(P^{*})$ is a polyhedron contained in a ball centered in $0$ of radius
$\frac{1}{2}$ and thus we can consider it as a hyperbolic polyhedron. By
[MW13, Theorem 1.2 and Section 4.4] there exists a smooth family $Q_{t}$ (for
$t$ in a right neighborhood of $0$) with the following properties:

  * •

$Q_{0}=\Phi(P^{*})$;

  * •

$Q_{t}$ has $1$-skeleton $\Gamma^{*}$;

  * •

the length of the edge of $Q_{t}$ corresponding to $e^{*}$ is equal to $t$.

Then the family $\Phi^{-1}(Q_{t}^{*})$ is smooth and clearly satisfies the
first two requirements in the statement of this lemma. To verify the third,
consider $F^{t}_{1}$ and $F^{t}_{2}$, and their dual points $v^{t}_{1}$ and
$v^{t}_{2}$. Denote with $V_{1}^{t}$ and $V_{2}^{t}$ the points on the one-
sheeted hyperboloid $H_{S}^{3}\subseteq\mathbb{R}^{1,3}$ that project down to
$v_{1}^{t}$ and $v_{2}^{t}$ respectively; up to applying a suitable family of
isometries $g_{t}$ and reparametrizing $Q_{t}$, we can assume that
$v^{t}_{1}=\left(\frac{1}{\sqrt{\alpha^{2}-1}},\frac{\alpha}{\sqrt{\alpha^{2}-1}},0,0\right)\in\mathbb{R}^{1,3}$
and
$v^{t}_{2}=\left(\frac{1}{\sqrt{\alpha^{2}+t^{2}-1}},\frac{\alpha}{\sqrt{\alpha^{2}+t^{2}-1}},\frac{t}{\sqrt{\alpha^{2}+t^{2}-1}},0\right)$.
The angle between $F^{t}_{1}$ and $F_{2}^{t}$ is given by the formula

$\displaystyle\theta_{e}^{t}=\textrm{ArcCos}\left(-\langle
v_{1}^{t},v_{2}^{t}\rangle\right)=\textrm{ArcCos}\left(\frac{1-\alpha^{2}}{\sqrt{\left(\left(\alpha^{2}-1\right)\left(\alpha^{2}+t^{2}-1\right)\right)}}\right)$

The derivative of $\theta_{e}^{t}$ with respect to $t$ evaluated in $t=0$ can
be computed to be

$-\sqrt{\frac{1}{\alpha^{2}-1}}$

which is different from $0$ (notice that $\alpha=1$ would imply that
$v_{1}^{t}$ is an ideal point which is not the case). ∎

######  Theorem 2.13.

The map $\Psi:\mathcal{A}_{\Gamma}^{e}\rightarrow\mathbb{R}^{n}$ assigning to
each polyhedron up to isometry its tuple of dihedral angles is a local
diffeomorphism.

###### Proof.

Consider a polyhedron $P\in\mathcal{A}_{\Gamma}^{e}$. If $P$ is in the
interior of $\mathcal{A}_{\Gamma}^{e}$, then $\Psi$ is a local diffeomorphism
thanks to [Mon13, Theorem 19]. Consider then
$P\in\partial\mathcal{A}_{\Gamma}^{e}\cong\mathcal{A}_{\Gamma^{\prime}}$ where
$\Gamma^{\prime}$ is obtained from $\Gamma$ by removing the edge $e$. Then by
Corollary 2.11 we can choose $v_{2},\dots,v_{n}\in
T_{P}\mathcal{A}_{\Gamma^{\prime}}$ and a vector $v_{1}\in
T_{P}\mathcal{A}^{e}_{\Gamma}$ which is tangent to a family $P_{t}$ whose
existence is guaranteed by Lemma 2.12. In this basis,

$d_{P}\Psi=\left(\begin{matrix}a&\begin{matrix}0&0&\dots&0\end{matrix}\\\
\begin{matrix}*\\\ *\\\ \vdots\\\\[4.30554pt]
*\end{matrix}&\text{\Large$d_{P}\Psi^{\prime}$}\end{matrix}\right)$

where $\Psi^{\prime}$ is the dihedral angle map of
$\mathcal{A}_{\Gamma^{\prime}}$ and $a<0$. Because $d_{P}\Psi^{\prime}$ is
invertible (by [Mon13, Proposition 17]) and $a\neq 0$ (by Lemma 2.12) this
matrix is invertible, hence $\Psi$ is a local diffeomorphism at $P$. ∎

##  3 The volume conjecture for hyperbolic polyhedra

In this section we briefly talk about the Yokota invariant and the Volume
Conjecture for polyhedra. More details about these topics can be found in
[Bel20].

Take a graph $\Gamma$ contained in a manifold $M$, an odd integer $r>2$, a
primitive $r$-th root of unity $q$ and a coloring $col$ of the edges of
$\Gamma$ with natural even numbers less than $r-2$; then the Yokota invariant
$Y_{r}(M,\Gamma,col,q)$ is defined in [Yok96] as a direct generalization of
the Kauffman skein bracket of a trivalent graph.

The simplest case to which these invariants apply is that of a planar graph in
$S^{3}$; in this case, if the graph is $3$-connected (that is to say, it
cannot be disconnected by removing $2$ edges and has more than $3$ vertices)
then it is the $1$-skeleton of a hyperbolic polyhedron. The Volume Conjecture
for polyhedra relates the growth of the invariants
$Y_{r}\left(S^{3},\Gamma,col,e^{2\pi i/r}\right)$ (as $r\rightarrow\infty$) to
the volume of a hyperbolic polyhedron with $1$-skeleton $\Gamma$ and dihedral
angles determined by $col$ in a certain way.

A conjecture relating the growth of the quantum $6j$-symbol to the volume of
hyperbolic tetrahedra first appeared in [Cos07]. Later, various
generalizations to the case of trivalent graphs and simple polyhedra were
proposed in [CGvdV15] and [KM18]; the following version of the conjecture
first appeared in [Bel20] as a direct generalization of these conjectures to
non-simple polyhedra.

######  Conjecture 3 (The volume conjecture for polyhedra).

Let $P$ be a proper polyhedron with dihedral angles
$\alpha_{1},\dots,\alpha_{m}$ at the edges $e_{1},\dots,e_{m}$, and
$1$-skeleton $\Gamma$. Let $col_{r}$ be a sequence of $r$-admissible colorings
of the edges $e_{1},\dots,e_{m}$ of $\Gamma$ such that

$2\pi\lim_{r\rightarrow+\infty}\frac{col_{r}(e_{i})}{r}=\pi-\alpha_{i}.$

Then

$\lim_{r\rightarrow+\infty}\frac{\pi}{r}\log\left\lvert
Y_{r}(S^{3},\Gamma,col_{r},e^{2\pi i/r})\right\rvert=\mathrm{Vol}(P).$

######  Remark 3.1.

Conjecture 3 is stated for proper generalized hyperbolic polyhedra, that is to
say, polyhedra with possibly hyperideal vertices and whose truncations are all
right-angled; for the terminology see [Bel21]. In the present paper we are
only concerned with hyperbolic polyhedra as in Definition 2.1, i.e. compact
polyhedra. Therefore, it is safe to mentally substitute “proper” for
“hyperbolic” in the statement of Conjecture 3 (and any subsequent mention of
the term).

Conjecture 3 holds (see [CM]) for any polyhedron obtained by gluing tetrahedra
with at least one hyperideal vertex along some of their truncation faces (in
particular, some of these polyhedra are compact but all of them are simple).
It was also proved for another, larger family of non-simple polyhedra in
[Bel20] (albeit for a single sequence of colors each). Further numerical
evidence for it appears in the appendix of [Bel20].

######  Proposition 3.2.

If Conjecture 3 is true, then the weak Stoker Conjecture is true.

###### Proof.

If $P_{1}$ and $P_{2}$ have the same $1$-skeleton $\Gamma$ and the same
dihedral angles $\theta_{1},\dots,\theta_{n}$, then we can choose a sequence
of colorings $col_{r}$ satisfying condition 3. Then the Volume Conjecture
implies that

$\lim_{r\rightarrow+\infty}\frac{\pi}{r}\log\left\lvert
Y_{r}(\Gamma,col_{r})\right\rvert=\mathrm{Vol}(P_{1})=\mathrm{Vol}(P_{2}).$

∎

##  4 The weak Stoker conjecture implies the Stoker conjecture

We now proceed to show that the weak version of the Stoker conjecture implies
the strong version. The main tool involved, other than Theorem 2.13, is the
Schläfli identity.

######  Theorem 4.1.

[[Mil94, Chapter: The Schläfli differential equality]] If $P_{t}$ is a smooth
family of hyperbolic polyhedra with the same $1$-skeleton, dihedral angles
$\theta_{1}^{t},\dots,\theta_{n}^{t}$ and edge lengths
$l_{1}^{t},\dots,l_{n}^{t}$, then

$\frac{\partial\textrm{Vol}(P_{t})}{\partial
t}_{\big{\rvert}_{t=t_{0}}}=-\frac{1}{2}\sum_{i}l_{i}^{t_{0}}\frac{\partial\theta^{t}_{i}}{\partial
t}_{\big{\rvert}_{t=t_{0}}}$

######  Proposition 4.2.

If Conjecture 2 is true, then two polyhedra with the same $1$-skeleton and the
same dihedral angles have the same edge lengths.

###### Proof.

Consider $P_{1}$ and $P_{2}$ with the same $1$-skeleton $\Gamma$ and the same
dihedral angles $\\{\theta_{e}\\}_{e\textrm{ edges of }\Gamma}$; fix an edge
$e\subseteq\Gamma$. By Theorem 2.6, for $t$ close to $1$ there are two
families of polyhedra with $1$-skeleton $\Gamma$, denoted with $P_{1}^{t}$ and
$P_{2}^{t}$, such that:

  * •

$P_{1}^{1}=P_{1}$ and $P_{2}^{1}=P_{2}$;

  * •

the dihedral angles of both $P_{1}^{t}$ and $P_{2}^{t}$ at $e$ are equal to
$t\theta_{e}$.

  * •

the dihedral angles of both $P_{1}^{t}$ and $P_{2}^{t}$ at an edge
$e^{\prime}\neq e$ are equal to $\theta_{e^{\prime}}$.

Because of the assumption of the validity of Conjecture 2,
$\textrm{Vol}(P_{1}^{t})=\textrm{Vol}(P_{2}^{t})$ for all $t$ in a
neighborhood of $1$; therefore by the Schläfli identity,
$l_{e}^{1}=\frac{\partial\textrm{Vol}(P_{1}^{t})}{\partial\theta_{e}}|_{t=1}=\frac{\partial\textrm{Vol}(P_{2}^{t})}{\partial\theta_{e}}|_{t=1}=l_{e}^{2}$.
∎

######  Proposition 4.3.

If Conjecture 2 is true, then two polyhedra with the same $1$-skeleton and the
same dihedral angles have congruent faces.

###### Proof.

Consider two polyhedra $P_{1}$ and $P_{2}$ with the same $1$-skeleton $\Gamma$
and the same dihedral angles $\\{\theta_{e}\\}_{e\textrm{ edges of }\Gamma}$.
Fix faces $F_{1}$ and $F_{2}$ (of $P_{1}$ and $P_{2}$ respectively)
corresponding to the same face $F$ of $\Gamma$. If $F$ is triangular, then
$F_{1}$ and $F_{2}$ have the same edge lengths by Proposition 4.2, and
therefore they are congruent. Suppose that $F$ is not triangular: we show that
$F_{1}$ and $F_{2}$ are congruent by showing that their diagonals have the
same length. Choose two non-adjacent vertices $v_{1}$ and $v_{2}$ of $F$, and
add the edge $\tilde{e}=\overline{v_{1}v_{2}}$ to $\Gamma$; we call the new
graph $\Gamma^{\prime}$. Since $P_{1}$ and $P_{2}$ are both elements of
$\mathcal{P}_{\Gamma}=\partial\mathcal{P}_{\Gamma^{\prime}}^{\tilde{e}}$, we
can think of them as polyhedra with weak $1$-skeleton
$(\Gamma^{\prime},\tilde{e})$; in particular their dihedral angle at
$\tilde{e}$ is equal to $\pi$.

By Theorem 2.6, for $t$ in some interval $(1-\epsilon,1]$ there are two
families of polyhedra with $1$-skeleton $\Gamma^{\prime}$, denoted with
$P_{1}^{t}$ and $P_{2}^{t}$, such that:

  * •

$P_{1}^{1}=P_{1}$ and $P_{2}^{1}=P_{2}$;

  * •

the dihedral angles of both $P_{1}^{t}$ and $P_{2}^{t}$ at $\tilde{e}$ are
equal to $t\pi$.

  * •

the dihedral angles of both $P_{1}^{t}$ and $P_{2}^{t}$ at an edge
$e\neq\tilde{e}$ are equal to $\theta_{e^{\prime}}$.

Therefore, by the same argument of Proposition 4.2, the length of $\tilde{e}$
is the same in $P_{1}$ and in $P_{2}$. This shows that every diagonal in
$P_{1}$ has the same length of the corresponding diagonal in $P_{2}$, which
implies that each face of $P_{1}$ is congruent to the corresponding face of
$P_{2}$. ∎

######  Corollary 4.4.

If Conjecture 2 is true, then the Stoker conjecture is true.

###### Proof.

Let $P_{1}$ and $P_{2}$ have the same $1$-skeleton and dihedral angles.
Assuming that Conjecture 2 is true, then by Proposition 4.3, $P_{1}$ and
$P_{2}$ have congruent faces. Then, by [RH93, Theorem 4.1], two polyhedra with
the same $1$-skeleton and congruent faces are isometric. ∎

## References

  * [And70] E. M. Andreev.  On convex polyhedra in Lobachevskii spaces.  Mat. Sb., 123(3):445–478, 1970.
  * [BB02] X. Bao and F. Bonahon.  Hyperideal polyhedra in hyperbolic 3-space.  Bull. Soc. Math. de France, 130(3):457–491, 2002.
  * [BDKY22] G. Belletti, R. Detcherry, E. Kalfagianni, and T. Yang.  Growth of quantum $6j$-symbols and applications to the volume conjecture.  Journal of differential geometry, 120(2):199–229, 2022.
  * [Bel20] G. Belletti.  A maximum volume conjecture for hyperbolic polyhedra.  arXiv:2002.01904, 2020.
  * [Bel21] G. Belletti.  The maximum volume of hyperbolic polyhedra.  Transactions of the American Mathematical Society, 374(2):1125–1153, 2021.
  * [CGvdV15] F. Costantino, F. Guéritaud, and R. van der Veen.  On the volume conjecture for polyhedra.  Geom. Dedicata, 179(1):385–409, 2015.
  * [CM] Q. Chen and J. Murakami.  Asymptotics of Quantum 6$j$-Symbols.  arXiv preprint math.GT/1706.04887.
  * [Cos07] F. Costantino.  $6j$-symbols, hyperbolic structures and the volume conjecture.  Geom. Topol., 11:1831–1854, 2007.
  * [CY18] Q. Chen and T. Yang.  Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants.  Quantum Topol., 9(3):419–460, 2018.
  * [Fle73] H. Fleischner.  The uniquely embeddable planar graphs.  Discrete Math., 4(4):347–358, 1973.
  * [Kas97] R. Kashaev.  The hyperbolic volume of knots from the quantum dilogarithm.  Letters in Mathematical Physics, 39(3):269–275, 1997.
  * [KM18] A. Kolpakov and J. Murakami.  Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra.  Experimental Mathematics, 27(2):193–207, 2018.
  * [Mil94] J.W. Milnor.  Collected papers. 1. Geometry.  Publish or Perish, 1994.
  * [MM01] H. Murakami and J. Murakami.  The colored jones polynomials and the simplicial volume of a knot.  Acta Mathematica, 186(1):85–104, 2001.
  * [Mon13] G. Montcouquiol.  Deformations of hyperbolic convex polyhedra and cone-3-manifolds.  Geom. Dedicata, 166(1):163–183, 2013.
  * [MW13] G. Montcouquiol and H. Weiß.  Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone–3–manifolds.  Geometry & Topology, 17(1):369–412, 2013.
  * [Oht16] T. Ohtsuki.  On the asymptotic expansion of the Kashaev invariant of the $5_{2}$ knot.  Quantum Topol, 7(4):669–735, 2016.
  * [Oht17] T. Ohtsuki.  On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings.  International Journal of Mathematics, 28(13):1750096, 2017.
  * [Oht18] T. Ohtsuki.  On the asymptotic expansion of the quantum $SU(2)$ invariant at $q=\exp(4\pi\sqrt{-1}/N)$ for closed hyperbolic $3$-manifolds obtained by integral surgery along the figure-eight knot.  Algebr. Geom. Topol., 18(7):4187–4274, 2018.
  * [OY18] T. Ohtsuki and Y. Yokota.  On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 165, pages 287–339. Cambridge University Press, 2018.
  * [RH93] I. Rivin and C. Hodgson.  A characterization of compact convex polyhedra in hyperbolic 3-space.  Invent. Math., 111(1):77–111, 1993.
  * [Riv96] I. Rivin.  A characterization of ideal polyhedra in hyperbolic 3-space.  Ann. of Math., pages 51–70, 1996.
  * [Ste22] E. Steinitz.  Polyeder und raumeinteilungen.  Encyk. der Math. Wiss., 12:38–43, 1922.
  * [Sto68] J. Stoker.  Geometrical problems concerning polyhedra in the large.  Communications on pure and applied mathematics, 21(2):119–168, 1968\.
  * [Wei13] H. Weiss.  The deformation theory of hyperbolic cone–3–manifolds with cone-angles less than 2$\pi$.  Geom. Topol., 17(1):329–367, 2013.
  * [Wit89] E. Witten.  Quantum field theory and the Jones polynomial.  Communications in Mathematical Physics, 121(3):351–399, 1989.
  * [Yok96] Y. Yokota.  Topological invariants of graphs in 3-space.  Topology, 35(1):77–87, 1996.

Giulio Belletti, Université Paris-Saclay

E-mail address<EMAIL_ADDRESS>