# HARE: Explainable Hate Speech Detection with Step-by-Step Reasoning

Warning: This paper contains examples of content that is offensive and may be
upsetting.

Yongjin Yang Joonkee Kim11footnotemark: 1 Yujin Kim11footnotemark: 1 Namgyu Ho
James Thorne† Se-Young Yun†
KAIST AI
{dyyjkd, joonkeekim, yujin399, itsnamgyu, thorne<EMAIL_ADDRESS>equal contribution †corresponding authors

###### Abstract

With the proliferation of social media, accurate detection of hate speech has
become critical to ensure safety online. To combat nuanced forms of hate
speech, it is important to identify and thoroughly explain hate speech to help
users understand its harmful effects. Recent benchmarks have attempted to
tackle this issue by training generative models on free-text annotations of
implications in hateful text. However, we find significant reasoning gaps in
the existing annotations schemes, which may hinder the supervision of
detection models. In this paper, we introduce a hate speech detection
framework, HARE, which harnesses the reasoning capabilities of large language
models (LLMs) to fill these gaps in explanations of hate speech, thus enabling
effective supervision of detection models. Experiments on SBIC and Implicit
Hate benchmarks show that our method, using model-generated data, consistently
outperforms baselines, using existing free-text human annotations. Analysis
demonstrates that our method enhances the explanation quality of trained
models and improves generalization to unseen datasets. Our code is available
at https://github.com/joonkeekim/hare-hate-speech.git.

##  1 Introduction

The increase in the use of online media has intensified the exposure to hate
speech, prompting the need for effective detection systems (Schmidt and
Wiegand, 2017; Fortuna and Nunes, 2018). While early works have been limited
to the classification of explicit hate speech (Caselli et al., 2020; Mathew et
al., 2021), recent works have drawn our attention to implicit forms of hate
speech which are more prevalent, yet subtle. (Jurgens et al., 2019).

To tackle these nuanced forms of hate speech, it is important for systems to
not only identify hate speech but also provide interpretable explanations (Liu
et al., 2019). This can help mitigate distributional biases inherent in simple
classification, allowing people to understand and reason about the potential
harms of hateful text (Sap et al., 2019b). Explanations can also improve the
transparency of content moderation on social media (Gillespie, 2018).

Figure 1: HARE uses large language models (LLMs) to generate hate speech
explanations step-by-step. (a) Recent benchmarks on understanding hate speech
provide free-text annotations on the implications of hate speech, but gaps in
reasoning hinder the supervision of generative detection models. (b) We
propose the use of LLMs to fill in the gaps and enable detection models to
understand and explain hate speech.

Recent works on hate speech understanding (Sap et al., 2019b; ElSherief et
al., 2021; Huang et al., 2022) have considered training autoregressive
language models to generate underlying explanations on hate speech. The models
are trained on human-written free-text rationales such as implied statements
and targeted groups. However, despite the use of novel benchmark datasets,
i.e., SBIC (Sap et al., 2019b) and Implicit Hate (ElSherief et al., 2021), the
trained models struggle to generate detailed and comprehensive explanations.
Moreover, we observe that the provided rationales give marginal improvement to
detection performance under joint training.

A potential cause of the limited supervision provided by existing annotations
on understanding and explaining hate speech may be the existence of critical
gaps in reasoning. For example, as shown in Figure 1, the implied statement of
the post “How dark is my humour? It picks cotton” is annotated as “black folks
are slaves”, in SBIC. To understand this implication, one must understand that
“dark” implies “black folks”, and the phrase “picks cotton” relates to the
historical background of African Americans. While this may be obvious to human
annotators, language models are known to lack societal knowledge and
commonsense reasoning skills to understand these nuances (Talmor et al., 2019;
Li et al., 2022; Choi et al., 2023). This leaves a significant gap between the
training objectives of classification and generating annotated implications,
which may harm supervision (Wiegreffe et al., 2021b; Wang et al., 2023a).

Drawing inspiration from the reasoning capabilities of large language models
(LLMs) improved with chain-of-thought (CoT) reasoning Wei et al. (2022), we
present our novel approach “Explainable HAte Speech Detection with Step-by-
Step REasoning (HARE)”. We leverage LLM-generated free-text rationales using
CoT prompts to fill in the gaps of reasoning in existing hate speech
annotations and enhance supervision of generative detection models. To create
these rationales, we propose two approaches: (1) adopt CoT prompts to create
comprehensive rationales that align with the given texts and (2) incorporate
existing human annotations from benchmarks in the CoT prompts to bridge the
logical gap between the input text and human annotations. When tested on the
challenging SBIC and Implicit Hate datasets, our approach outperforms standard
fine-tuning with given human annotations and provides enhanced explanations
behind the detection results.

##  2 Method

###  2.1 Preliminaries

The task of hate speech detection can be framed as a generative task that
inputs the text $P$ and outputs a prediction class $C$, formulated as
$p(C|P)$, indicating whether the speech is classified as “hate” or “not
hate”.111 We refer to the “offensive” category in SBIC as “hate” class. In the
experiment, we use the wording “offensive” as is.  Furthermore, by
incorporating human-written rationales of the target groups $T$ and implied
statements $I$ of hate speech from the SBIC and Implicit Hate datasets, the
task can also be formulated to sequentially output $C$, $T$, and then $I$, as
$p([C;T;I]|P)$. Hence, the model trained with human-written annotations is
designed to generate $C$ and then provide explanations using annotations $T$
and $I$. However, due to a logical gap between the speech $P$ and the
annotations $T$ and $I$, training a model with these annotated rationales does
not significantly enhance the model’s ability to comprehend hate speech.

Chain-of-thought prompting (Wei et al., 2022) refers to a prompting method
that generates a chain of reasoning to derive answers. Kojima et al. (2022)
introduce a method of including the phrase “Let’s think step by step” at the
end of the prompt to generate reasoning in a zero-shot setting.

###  2.2 Fr-HARE & Co-HARE

In this paper, we propose HARE, a framework that fine-tunes using rationales
generated by LLMs through CoT prompts. By leveraging these rationales, the
model is trained to discern whether a text is classified as hate speech or
not. This approach of utilizing the LLM-generated rationales not only improve
the model performance as in previous works (Ho et al., 2022; Wang et al.,
2023a) but also enables the model to generate more plausible explanations.
Subsequently, we introduce two variants that employ different methods for
extracting rationales from LLMs.

####  Fr-HARE

Our first strategy, Free-HARE (Fr-HARE), extracts rationales using LLM from
the LLM without referencing the annotated rationales. In line with Kojima et
al. (2022), we employ zero-shot prompts to prompt the LLM to extract
rationales, as follows:

Determine whether the following post is offensive, and explain why.
Post: $P$
Answer: Let’s explain step by step.

From the given prompt above, both the rationale $R$ without any logical leap
and the classification result $C$ are generated. Then, we utilize the
generated rationale $R$ by training the model to sequentially generate $C$ and
$R$, represented as $p([C;R]|P)$. Utilizing these rationales enhances the
model’s ability to comprehend the context of hate speech, thus leading to
explanations that are more intelligible to humans.

####  Co-HARE

We also propose a second strategy, Conditioned-HARE (Co-HARE), which utilizes
human-written rationales to formulate CoT prompts. Fr-HARE generates complete
rationales entirely from scratch, disregarding human annotations included in
the datasets. Although Fr-HARE offers the advantage of being universally
applicable to any hate speech dataset, the LLM-generated rationales might lack
consistency due to wide variation in reasoning sequences. To address this
inconsistency, Co-HARE integrates human-written rationales about target groups
$T$ and implied statements $I$ into the CoT prompt as follows:

Determine whether the following post is offensive, and explain why. Provide a
brief, step-by-step explanation of how the post targets the specified group
and how it leads to the implied statement provided.
Post: $P$
Target: $T$
Implied statement: $I$
Answer: Let’s explain step by step.

Then, we train the model using extracted rationales, as in Fr-HARE. For
samples labeled as “not hate” that do not include human-written rationales, we
apply the prompt used in Fr-HARE. While Co-HARE requires human-written
rationales, it generates rationales that are more tailored to the specific
requirements and features of the dataset, due to its guided nature. Therefore,
the model trained with Co-HARE can provide explanations that align more
closely with the forms of rationales that humans construct.

#### Details of HARE

Once we have extracted the rationales from the LLMs, we follow the approach of
Kojima et al. (2022) to have the LLMs predict the class. Specifically, we
employ a two-stage extraction process. In the first stage, we extract both the
class $C$ and the rationale $R$ from the LLMs using our HARE method,
represented as $p([C;R]|P)$, as previously outlined. In the second stage, we
prompt the LLMs again, this time to predict the class $C$ given the extracted
rationales $R$ and the post $P$, denoted as $p(C|R,P)$. During fine-tuning on
hate speech datasets, if the predicted class $C$ coincides with the true
answer $C$, we concatenate $C$ with the extracted rationale $R$. If the
predicted labels are incorrect, the models are solely trained to predict the
class $C$. Furthermore, following the findings of Ho et al. (2022), we
generate multiple distinct rationales to facilitate the learning process.

##  3 Experiments

###  3.1 Experimental Setup

We utilize SBIC and Implicit Hate datasets for our fine-tuning experiments.
Our models are trained to classify the offensiveness and hatefulness of posts,
using SBIC and Implicit Hate, respectively. It is noteworthy that in our
Implicit Hate experiments, we combine both the explicit and implicit hate
classes into a single “hate” category. We set up baselines with two families
of models: $C$, a model trained exclusively for classification, and
$C$+$T$+$I$, a model trained using human-written rationales. For Fr-HARE and
Co-HARE, by using gpt-3.5-turbo-0613 that is known for its reasoning
capabilities (Ouyang et al., 2022), we extract four and eight different
rationales per each sample in SBIC and Implicit Hate, respectively, following
the hyperparameter setting of Ho et al. (2022). Subsequently, we fine-tune the
model, setting LLM-generated rationales $R$ and class $C$ as target sequence.
For performance evaluation, we measure detection accuracy and compute the F1
score of classification, regarding “hate” as the positive class. We make use
of Flan-T5 (Wei et al., 2021) with different model configurations: small, base
and large. We also conduct experiments using the large models of T5 (Raffel et
al., 2020) and GPT-2 (Radford et al., 2019). A more detailed explanation of
our experimental setup can be found in Appendix B.

Table 1: The performance of fine-tuning on SBIC and Implicit Hate dataset with
various models and size.

Model | Method | SBIC | Implicit Hate
---|---|---|---
Acc | F1 | Acc | F1
GPT-3.5- turbo-0613 | ZS | 80.06 | 81.75 | 73.58 | 65.66
ZS-CoT | 73.48 | 79.07 | 73.98 | 67.19
Flan-T5 small | $C$ | 82.56 | 84.05 | 77.58 | 71.98
$C$+$T$+$I$ | 82.99 | 84.05 | 77.63 | 72.39
Fr-HARE | 84.18 | 85.18 | 79.33 | 73.29
Co-HARE | 84.44 | 85.35 | 78.54 | 73.49
Flan-T5 base | $C$ | 82.35 | 83.71 | 78.03 | 72.17
$C$+$T$+$I$ | 82.54 | 84.41 | 79.77 | 73.15
Fr-HARE | 84.20 | 85.46 | 79.84 | 74.84
Co-HARE | 84.65 | 85.76 | 80.38 | 75.69
Flan-T5 large | $C$ | 81.70 | 82.84 | 78.42 | 72.92
$C$+$T$+$I$ | 83.48 | 83.70 | 80.14 | 73.10
Fr-HARE | 85.21 | 86.16 | 80.49 | 74.62
Co-HARE | 84.93 | 85.57 | 81.49 | 76.71
T5 large | $C$ | 83.03 | 83.53 | 78.79 | 72.50
$C$+$T$+$I$ | 84.23 | 85.21 | 79.61 | 73.80
Fr-HARE | 85.27 | 86.32 | 81.61 | 75.59
Co-HARE | 85.35 | 85.93 | 80.98 | 75.88
GPT-2 large | $C$ | 81.39 | 82.68 | 73.32 | 66.68
$C$+$T$+$I$ | 82.80 | 83.43 | 75.95 | 65.25
Fr-HARE | 83.92 | 85.48 | 78.47 | 71.35
Co-HARE | 84.64 | 85.67 | 80.07 | 71.58

###  3.2 Results and Discussions

#### Do LLM-generated rationales improve detection performance?

Table 1 presents the performance of hate speech detection according to
different methods on the SBIC and Implicit Hate datasets. Our strategies Fr-
HARE and Co-HARE consistently exhibit superior performance over other baseline
methods, regardless of the model size. This suggests that even though the
baseline method is trained using human-written rationales, the more detailed
and logically-sequenced LLM-generated rationales of HARE can further aid the
model in understanding the input text and accurately classifying it as hate
speech. Therefore, the results demonstrate that the quality of rationales has
a strong impact on classification. Furthermore, the performance of our method
consistently improves as the model size increases, in contrast to baselines.
This suggests that diverse reasoning becomes increasingly beneficial as scale
grows. This notable improvement with HARE is achieved by using only 40$ for
each method in our approach, demonstrating that the ability to reason can be
effectively trained with rationales from LLMs.

Additionally, while Fr-HARE and Co-HARE exhibit similar performance, Co-HARE
has a slight edge in most cases. This is because Co-HARE is guided by human-
written annotations, which results in better alignment with the setting of the
datasets, as we mentioned in Section 2.2. It is also noteworthy that all the
fine-tuned models surpass both Zero-Shot (ZS) and Zero-Shot CoT (ZS-CoT,
Kojima et al. (2022)) classification performance of GPT-3.5-turbo, indicating
that merely employing LLM with CoT prompts is not sufficient to tackle this
task.

#### Are HARE models more generalizable?

Table 2: Cross Evaluation results on HateXplain Mathew et al. (2021) and
DynaHate Vidgen et al. (2020). We utilize Flan-T5-large fine-tuned on SBIC
using each method.

Method | HateXplain | DynaHate
---|---|---
Acc | F1 | Acc | F1
$C$ | 64.40 | 74.18 | 64.35 | 67.41
$C$+$T$+$I$ | 68.84 | 74.52 | 64.72 | 67.41
Fr-HARE | 70.69 | 78.91 | 68.06 | 75.15
Co-HARE | 71.62 | 78.52 | 69.98 | 75.01

To assess the ability of our methods to generalize across different datasets,
we evaluate the models fine-tuned on the SBIC datasets using each method on
two distinct datasets, HateXplain (Mathew et al., 2021) and DynaHate (Vidgen
et al., 2020). Both datasets encompass forms of explicit and implicit hate. On
both datasets, our methods Fr-HARE and Co-HARE both outperform baseline
methods, indicating that our methods enhance the generalizability of the
models by improving their reasoning ability. Moreover, the comparable
performance of Fr-HARE and Co-HARE suggests that both techniques of bridging
logical gaps are effective in tackling hate speech detection.

(a) GPT-4 Single answer grading

(b) GPT-4 Pairwise comparison

Figure 2: The result of GPT-4 evaluation following Zheng et al. (2023). The
bar and line represent the average scores that range from 1 to 10 and 95%
confidence interval, respectively. We utilize Flan-T5-large fine-tuned on SBIC
using each method. Figure 3: Model outputs using baseline methods and our
framework HARE. The samples are from SBIC test set. Note that the answers are
abbreviated by (…), and the full context is reported in Appendix D.

#### Does HARE improve the quality of generated explanations?

To evaluate the quality of the generated explanations, we conduct automated
evaluation using GPT-4 (OpenAI, 2023) which has demonstrated its high
alignment with human evaluation (Liu et al., 2023; Zheng et al., 2023). We
randomly sample 50 “hate” instances from SBIC test set where all methods are
correct and apply single answer grading (Zheng et al., 2023) on explanations
generated by models trained with $C$+$T$+$I$, Fr-HARE, and Co-HARE. Detailed
explanation about GPT-4 evaluation can be found on Appendix C.

The results depicted in Figure 2 (a) show that both Fr-HARE and Co-HARE are
capable of generating higher quality explanations, with Fr-HARE exhibiting
slightly superior performance, suggesting that the flexibility of Fr-HARE
leads to a more qualified explanation.

As Co-HARE is designed to create rationales that align better with human-
written rationales, we also conduct pairwise comparison between Co-HARE and
$C$+$T$+$I$ to determine which method aligns better with human-written labels.
Figure 2 (b) clearly shows that the rationales generated by the model trained
using Co-HARE are aligned more to actual human-written rationales than the
ones generated by the model trained directly with human-written rationales.
This also highlights the value of Co-HARE, as it aligns better with natural
intuition, indicating that both Fr-HARE and Co-HARE can be utilized for
different purposes.

#### How does HARE detect and contextualize hate speech?

Figure 3 presents model outputs using HARE and baseline method. A more
detailed qualitative study of randomly selected samples is available in
Appendix D. In the first sample, the post makes light of harming a young girl
using phrases like “like I like cigar” and “burlap sack”. While our model, Fr-
HARE, does not explicitly connect the “burlap sack” to the idea of a kidnapped
girl, it does recognize the harmful implications towards the girl.
Furthermore, Co-HARE understands the historical context behind the term
“burlap sack”. The presence of the positive term “like” and the absence of
overtly derogatory words might lead some models to classify the statement as
non-offensive. However, our approach appropriately identifies the underlying
hateful context.

In the second sample, the post contains hate speech targeting Jewish victims
of the Holocaust by referencing Anne Frank. Our model accurately recognizes
the historical background of Anne Frank as a Holocaust victim. While the
baseline seems to overlook the historical significance associated with Anne
Frank, our method correctly identifies her and assumes that the reference
constitutes harassment against a Jewish victim, even though there is a slight
misunderstanding about the context of “lack of speaking” in Fr-HARE.

#### Is GPT-3.5 a qualified teacher?

Figure 4: A sample of LLM rationale generated by GPT-3.5-turbo using Fr-HARE
from SBIC train set.

Since our framework is based on distillation of generated rationales from
GPT-3.5 to smaller models, it is crucial to verify whether the teacher is
qualified. Figure 4 displays rationales produced by GPT-3.5-turbo, which is
employed to train the student model. This example illustrates that the LLM not
only discerns the hateful nuances towards both white and black individuals,
but also offers more detailed explanations compared to rationales written by
humans. Notably, it accurately correlates the historical context, associating
the word “slaves” with “pets”. More analysis of rationales from GPT-3.5-turbo
can be found in Appendix D.2.

##  4 Conclusion

In this paper, we present HARE framework to improve the ability of the
language model to understand hate speech and provide clearer explanations for
its decisions. We propose utilizing CoT reasonings extracted from LLMs in two
variants to overcome the logical gaps in human-annotated rationales. When
fine-tuned on the SBIC and Implicit Hate datasets, our methods achieve
superior detection performance and better qualified explanations.

## Limitations

While we assess the quality of explanations generated by HARE using GPT-4, we
do not conduct human evaluations, which are crucial for tasks requiring human-
readable explanations. The primary reason for this omission is that the hate
speech content and its respective explanations could be excessively offensive
for annotators and GPT-4 already aligns with the level of inter-human
agreement. In addition, the "verbosity bias", characterized by a preference
for the longer text of GPT-4 as indicated by (Liu et al., 2023), may also
serve as a limitation in our evaluation process.

## Ethics Statement

####

Predicting whether an online post contains hatespeech is both technically and
socially challenging. While methods for automating hatespeech detection have
utility in an online platform, it is critical that these are tuned and used
appropriately. False-positive errors have potential to censor online speech,
further marginalizing specific user groups, for example: use of n***** in AAVE
English may be flagged. It is critical to understand specific reasoning behind
a classification including deeply social reasons. While language models act as
a mechanism to generate reasonable explanations, it is critical that they are
used appropriately to prevent them from inadvertently educating users on how
to craft more subtle and toxic language. We used automated evaluation metrics
in this paper to prevent exposure of toxic language to human annotators.
However, real-world usage would require validation that deeply rooted social
issues are expressed correctly by these models.

It is also important to note that there might be concerns about the inherent
bias in the GPT-3.5 model. While not flawless, GPT-3.5 has demonstrated its
impartiality regarding gender, race, ethnicity, and religion by achieving the
highest grade on the Harmfulness metric within the FLASK evaluation framework
(Ye et al., 2023). Crucially, we only select rationales that align with the
ground truth label for training, thereby mitigating biases not in sync with
human annotators. Analysis of GPT-3.5-turbo can be found in Section 3 and
Appendix D.2.

## Acknowledgement

This work was supported by Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by Korea government
(MSIT) [No. 2021-0-00907, Development of Adaptive and Lightweight Edge-
Collaborative Analysis Technology for Enabling Proactively Immediate Response
and Rapid Learning, 90%] and [No. 2019-0-00075, Artificial Intelligence
Graduate School Program (KAIST), 10%].

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##  Appendix A Related Work

#### Hate Speech Detection

Hate speech (Waseem et al., 2017) is a form of language intended to offend
particular individuals or groups. In this study, we expand this definition by
incorporating the broader concept of offensive language as in (Burnap and
Williams, 2016; Ribeiro et al., 2018). Early research (Mozafari et al., 2020;
Caselli et al., 2020; Sarkar et al., 2021) on hate speech focused mostly on
improving the classification score with pre-trained transformer encoder, such
as BERT (Devlin et al., 2018).

Recent works on hate speech detection have delved into providing underlying
explanations for predictions on hate speech (Sap et al., 2019a, b; Mathew et
al., 2021; ElSherief et al., 2021; Lin, 2022). One line on research focuses on
keyword-based explanations (Sap et al., 2019a; Davidson et al., 2019; Mathew
et al., 2021; Kim et al., 2022), but this approach often fails to capture
implicit hatefulness that is not explicitly present in the text. Other studies
involve training generative models with human-written free-text rationales
(Sap et al., 2019b; ElSherief et al., 2021; Huang et al., 2022) present in
multiple benchmarks (Sap et al., 2019b; ElSherief et al., 2021). Nevertheless,
due to the existence of logical gaps in these human-annotated rationales
(Aggarwal et al., 2021; Sun et al., 2022), relying solely on these rationales
results in sub-optimal detection and explanation quality. An alternative
approach involves using explanations that utilize external knowledge sources
(Sridhar and Yang, 2022; Lin, 2022) or leveraging task decomposition and
knowledge infusion, with framing hate speech detection as a few-shot task, to
improve performance and generalize better (AlKhamissi et al., 2022). However,
these methods primarily aim to enhance classification performance, and their
explanations cannot go beyond the limitations of incomplete, human-written
free-text rationales. Our proposed HARE demonstrates its effectiveness by
incorporating LLM-generated rationales, which include logical completeness and
abundant explanatory power extracted with our CoT prompting.

#### Self-Rationalization

Self-rationalization, a technique where models provide explanations for their
predictions, has been extensively studied to make models more understandable
and transparent (Marasović et al., 2021; Wiegreffe et al., 2021a, b). Recent
studies leverage rationale-augmented exemplars to few-shot prompt LLMs (Wei et
al., 2022; Wang et al., 2022; Lampinen et al., 2022), while others fine-tune
smaller models using the rationales generated by LLMs (Ho et al., 2022; Wang
et al., 2023a; Chan et al., 2023; Wang et al., 2023b). Several works apply
their self-rationalization methods to detect hate speech and generate
underlying explanations (Marasović et al., 2021; Ludan et al., 2023; Chen et
al., 2023). However, they directly use human-written rationales in their
framework, which may lack logical completeness. Instead, our method adopts the
approach of utilizing the rationales generated by LLMs.

##  Appendix B Implementation Details

#### Training Details

All the experiments utilize an Adafactor (Shazeer and Stern, 2018) optimizer
with batch size of 32 and learning rate $\in$ {5e-3, 5e-4, 5e-5} without
learning rate scheduling. We train all models for 10 epochs and utilize the
checkpoint with the highest validation F1 score to evaluate on the test set.
Top-k sampling with $k$=20 is adopted as a decoding method. For dataset split,
we use official dataset split for SBIC and random split with 6:2:2 ratio for
train, validation, and test sets for Implicit Hate. All our experiments are
conducted using a single NVIDIA A6000 GPU.

##  Appendix C Details on GPT-4 Evaluation

To conduct GPT-4 evaluation, we utilize the prompts proposed by Zheng et al.
(2023). For single answer grading, the prompt format for GPT-4 is written as
follows:

[Instruction]
Please act as an impartial judge and evaluate the quality of the response
provided by an AI assistant to the user question displayed below. Your
evaluation should consider factors such as the helpfulness, relevance,
accuracy, depth, creativity, and level of detail of the response. Begin your
evaluation by providing a short explanation. Be as objective as possible.
After providing your explanation, please rate the response on a scale of 1 to
10 by strictly following this format: "[[rating]]"
[Question]
Determine whether the following post is offensive or not. You must choose one
of the options.
Post: $P$
Options:
(A) Offensive
(B) Not offensive
[The Start of Assistant’s Answer]
$C,R$
[The End of Assistant’s Answer]

To compare the alignment of generated rationales with human-written
rationales, we conduct a pairwise comparison of explanations against ground-
truth rationales. These ground-truth rationales encompass the annotated target
groups and implied statements from the instruction, and we compare the
results. Additionally, to minimize bias from the order of candidate answers,
we adopt the approach of Zheng et al. (2023), considering both original and
swapped orders of predicted explanations. If $C+T+I$ and Co-HARE are chosen
alternately, it is deemed a tie. Should one method be selected following a
tie, that method is considered as the chosen one. The prompt format for the
pairwise comparison is provided below:

[Instruction]
Please act as an impartial judge and evaluate the quality of the responses
provided by two AI assistants to the user question displayed below. You should
choose the assistant that follows the user’s instructions and answers the
user’s question more accurate. When choosing the assistant, please consider
the true answers below:
Target: $T$
Implied Statement: $I$
Your evaluation should consider which response is more similar to the true
answers. Begin your evaluation by comparing the two responses and provide a
short explanation. Avoid any positional biases and ensure that the order in
which the responses were presented does not influence your decision. Do not
allow the length of the responses to influence your evaluation. Do not favor
certain names of the assistants. Be as objective as possible. After providing
your explanation, output your final verdict by strictly following this format:
"[[A]]" if assistant A is more accurate, "[[B]]" if assistant B is more
accurate, and "[[C]]" for a tie.
[Question]
Determine whether the following post is offensive or not. You must choose one
of the options.
Post: $P$
Options:
(A) Offensive
(B) Not offensive
[The Start of Assistant A’s Answer]
Answer from one method
[The End of Assistant A’s Answer]
[The Start of Assistant B’s Answer]
Answer from another method
[The End of Assistant B’s Answer]

##  Appendix D Qualitative Study

###  D.1 Qualitative Study of HARE

Figures 5, 6, 7, and 8 showcase results generated by the fine-tuned
Flan-T5-large model using HARE and $C$+$T$+$I$, based on test samples from
SBIC. Although a brief explanation is provided in Section 3.2, we delve deeper
with an extended analysis of the 20 examples from our qualitative study. These
20 samples were randomly chosen in proportion to their correct and incorrect
predictions across the different methods.

When comparing human-written annotations with HARE, it becomes evident that
the annotated rationales in SBIC often take the form of implied statements,
following a simple Hearst-like pattern (Sap et al., 2019b). Learning from such
rationales, which are closely tied to the conclusion, creates a logical gap
for the model and makes interpretation challenging for humans. For instance,
understanding hate speech without background knowledge references, such as
’burlap sack’, can make it difficult to see the connection between the
statement "girls are not worthy of equal life" and the provided sentence.
Figures 5 and 6 showcase successful cases where models have attempted to
bridge this reasoning gap through HARE, offering more detailed rationales that
encompass the context. Furthermore, these models exhibit capabilities not seen
in previous research, such as detecting terms with historical significance
(e.g., ’burlap sack’ or ’Anne Frank’) or common words that may carry hateful
connotations (e.g., ’reds’), thus enhancing the intermediate reasoning
process.

However, when examining the failure cases in Figures 7 and 8, the results show
that HARE sometimes fails due to increased sensitivity to potentially harmful
terms, thereby classifying them as offensive. While this increased sensitivity
can be viewed as a drawback, there are instances, such as with the Alzheimer
example, where an expression might be interpreted as hateful depending on the
individual. This suggests that HARE aims to classify a post as hateful if it
could be considered offensive to certain groups. Moreover, considering the
David Bread Katz example, it is also challenging for HARE to decide if the
post is offensive if it post with background that it hasn’t encountered,
possibly due to a lack of background knowledge regarding the implied shooting
incident, illustrating the limitation of LLM distillation.

###  D.2 Qualitative Study on GPT-3.5 rationales

When comparing annotations with rationales generated by GPT-3.5, we observe
that human-written rationales from SBIC use implied statements that follow
simple Hearst-like patterns (Sap et al., 2019b). In contrast, models like LLM
(e.g., GPT-3.5) tend to provide detailed, step-by-step explanations, often
complemented by relevant social background information, which is immensely
beneficial. For example, while earlier rationales might omit mentioning Bill
Cosby’s conviction of sexual assault, GPT-3.5 explicitly informs us of this
fact, greatly enhancing comprehension. A particularly striking example is the
"pet" case. While a human annotator perceived it as hate speech targeted at
black individuals, GPT-3.5 points out that it could also be used derogatorily
against white individuals, thereby emphasizing the potential biases in hate
speech detection.

When rationales are categorized and structured, as seen in SBIC, instead of
being tailored to individual posts, they may not be sufficient for learning
implications. This could explain why, as suggested by Table 2, there’s a
decrease in generalization for $C$+$T$+$I$. Our approach offers aligned
rationales for each post at minimal cost, enabling the learning of diverse
reasons for potential hate, which in turn leads to enhanced generalization.

Figure 5: Success cases of HARE

Figure 6: Success cases of HARE

Figure 7: Failure cases of HARE

Figure 8: Failure cases of HARE

(a) Success cases of GPT-3.5-turbo

(b) Failure cases of GPT-3.5-turbo

Figure 9: Success cases and failure cases of GPT-3.5-turbo when prompted with
our CoT prompt.


June 2022

The Physics of Time: Current Knowledge and Unanswered Challenges111Based on
talk delivered at the 13th Bial Symposium, “The mystery of time”, Behind and
Beyond the Brain, 6-9 April 2022, Porto, Portugal.

Orfeu Bertolami1,2

1Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

2Centro de Física das Universidades do Minho e do Porto, Faculdade de
Ciências, Universidade do Porto

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

###### Abstract

In contemporary physics space and time are intertwined entities so that
kinematical and dynamical quantities are expressed in the four-dimensional
space-time. This formulation seems to contradict our every-day experience and
perception according to which space and time are distinct entities. In this
brief report we shall discuss these apparently antagonist views and analyse
the underlying physical property of time, namely that it evolves from the past
to the present, from the present to the future.

…

Time passed, turning everything to ice.
Under the ice, the future stirred.
If you fell into it, you died. It was a time
of waiting, of suspended action.

I lived in the present, which was
that part of the future you could see.
The past floated above my head,
like the sun and moon, visible but never reachable.

It was a time
governed by contradictions, as in
I felt nothing and
I was afraid.

…

in Averno (2006)

Louise Glück

##  1 Introduction

According to the Big-Bang theory, the origin of the Universe is, in fact, the
process of creation of matter, space and time. Despite its modernity and
sophistication, the science that describes this extraordinary event, shares
the idea of a primordial beginning with the very first manifestations of
rational thinking of distinct civilisations, the so-called myths of creation
(see, for instance [1, 2] for extensive discussions). The earliest forms of
philosophical thinking, have invariably regarded space and time as essential
features of reality, even though quite distinct ones.

Indeed, at first glance, space and time seem to be quite different. Space can
be freely experienced as one can move in any direction without restriction.
Time however, has a well defined direction. Past and future are clearly
distinct as our actions can affect only the latter. We have memory, but not
precognition. Matter, organic or otherwise, eventually decay ceaselessly
rather than getting itself organised. As far as we can detect through our most
powerful microscopes, the particle accelerators, the fabric of space-time is
made of at least three distinct spatial dimensions and only one time
dimension222Notice that if the number of time dimensions were greater than
one, one would encounter all types of complications as, on quite general
grounds, the Partial Differential Equations that describe the physical
phenomena would be ultra-hyperbolic, leading to unpredictability, or to
backward causality (see e.g. Ref. [3] for a discussion)..

Actually, in physics, time allows for describing the evolution and
transformation of systems and processes in a very convenient way, being for
this reason an “invisible” or “hidden” dimension. Despite of that, time is
similar to space, as both have always been thought as the arena of all
manifestations of Nature. Everything lies in space, as well as in time, and
the intrinsic and fundamental relationships between the most basic elements of
all physical entities could be decomposed into points, straight lines and
geometrical figures in two or three dimensions and whose properties were
remarkably systematised by Euclid (ca. 330 - 275 B.C.) geometry. These
relationships would in turn reveal the intrinsic properties of space itself.
These geometrical properties are shared by time with the unification of space
and time proposed by Hermann Minkowski [4] in 1908, inspired by Einstein’s
Theory of Special Relativity (for a discussion see, for instance, [5, 6]).

On the other hand, by any account, time is a mysterious ingredient of the
Universe and has been the subject of continuous philosophical debate over the
centuries (we refer the interest reader to the brief discussion in Ref. [5]).
Time is intuitively related to change, and it is subjectively perceived as a
permeating entity that flows. This view can be traced back as far as to
Aristotle (384 BC - 322 BC), who regarded “time as the measure of change”.
Throughout history, one encounters a wide range of reflections and
considerations about the nature of time. For many civilisations, cycles in
Nature, were an evidence of the circular and repetitive pattern of time. It
was only in the 17th century that Francis Bacon (1561 - 1626) posed the
concept of linear time, and through the influence of Newton (1643 - 1727),
Barrow (1630 - 1677), Leibniz (1646 - 1716), Locke (1632 - 1704) and Kant
(1724 - 1804) amongst others, about the 19th century, the idea of a linear
time evolution was the dominant one in science and philosophy (for discussions
see, for instance, [2, 5, 7, 8, 9, 10, 11, 12]). However, despite all evidence
and arguments, for some physicists, Einstein being amongst the finest, the
irreversible flow of time is no more than a “persistent illusion”.

In what concerns its ontological nature, it is still a matter of debate
whether time is a real fundamental quantity or, instead, a composite suitable
parameter, like temperature to describe the physical laws (see, for instance,
Ref. [14] for a discussion in the context of string theory) and to frame, in
unequivocal terms, causation. Causation being, of course, a crucial feature
not only in physics, but according to some philosophers, like David Hume (1711
- 1776), a fundamental element for the human understanding of reality. In
fact, a relevant connection to physics arises here as the decomposition of
human perception down to the physiology of nervous tissues, down to its
chemistry and then down to the causal character of the physical laws, renders
Hume’s proposition quite plausible.

Back to physics, one could state that reflections about the nature of time
were polarised by Newton’s concept of absolute time, who assumed that time
flowed at the same rate for all observers in the Universe [13]. For Newton,
time and space were an infinitely large vessel, containing events, and
existing independently of the latter. This picture was completely changed in
1905 by Albert Einstein (1879 - 1955), through the formulation of the Special
Theory of Relativity, in the context of which it was shown, in particular,
that time flows at different rates for different observers. Three years later,
Hermann Minkowski (1879 - 1909) [4] suggested the unification of the time and
space, giving rise to the notion of a fundamental four-dimensional entity,
space-time mentioned above. The Relativity revolution was concluded in 1915
when Einstein put forward the Theory of General Relativity where it was shown
that the flat spacetime of Special Relativity is curved by energy-matter.
Since then the discovery of new forms of energy-matter in physics has given
rise to new space-time geometries (see, for instance, [3] and references
therein).

The merging of space and time into the space-time concept is particularly
relevant and useful for the physical description of any phenomena in Nature
independently of the frame of reference used to parametrize change-evolution-
transformation. It is interesting that this picture that is so attractive for
physicists was strongly rejected, for instance, by the philosopher Henri
Bergson (1859 - 1941). According to Bergson, intelligence and intellect can
only have a clear idea of discontinuity and immobility, being thus unable to
fully grasp life and to think about evolution. For Bergson, intellect tends to
represent becoming as a series of states. Geometry and logic, as products of
intelligence, must be verified against common sense. He believed that
intelligence and the origin of material bodies are correlative and have
evolved by reciprocal adaptation. For him, intellect is the power to
distinguish different things. However, in reality, there are no separate solid
things, only a continuous stream of becomings. Becoming leads to life when it
is an ascendant movement, and to matter when it is a descendant one.

For Bergson, space and time are profoundly dissimilar. The intellect is
associated with space, whilst instinct and intuition are connected with time.
Space, a feature of matter, arises from a dissection of the flux which is
really illusory, although useful in practice. Time, on the contrary, is the
essential element of life. Mathematical time, according to Bergson, is
actually a form of space, but the time, as the essence of life, is what he
refers to as duration. Duration merges past and present into an organic whole,
where there is mutual entanglement and succession with distinction. To an
extensive account of the differences between Bergson’s ideas and the ones
pertaining physics and the debate with Einstein in 1922 in Paris, the reader
is referred to the superb account by Jimena Canales [15] and to her talk at
this Symposium.

In what follows we shall further delve in the two main issues related to time.
The first one refers to its flow, which concerns the so-called arrows of time,
associated to the fundamental and ubiquitous Second Law of Thermodynamics. The
second one on whether the fabric of space-time admits reversible motion in
time through closed time-like curves (CTCs) or the so-called “time machine
solutions” of the field equations of General Relativity.

This brief account about the main physical features of time has the following
outline: In Section 2, we discuss the issue of directionality of the time
flow. In Section 3, we address the view of the time in Relativity, the CTCs
and the ensued putative violations of causality. Finally in Section 4, we
briefly discuss some particular issues such as the idea of time crystals, of a
Block Universe, where all moments in past time still exist and not only the
present, and the specious present asymmetry. Of course, this list is quite
short and many other issues could be addressed, such as for instance, the
putative connection among the various arrows of time and the problem of time
in quantum gravity. They certainly deserve a discussion, however, given their
technicality, we refer the interested reader to Ref. [6]. Finally in Section 5
we present our closing remarks.

##  2 Arrows of time

A fundamental feature of the nature of time concerns its flow. Modern physics
describes phenomena through dynamical laws, from which the time evolution is
determined after specifying a suitable set of initial conditions. The
fundamental dynamical equations of classical and quantum physics are symmetric
under a time reversal, i.e., mathematically, one could instead specify the
final conditions and evolve the physical system back in time. The resulting
would be indistinguishable from the one obtained by the usual procedure.
However, in macroscopical phenomena, which are described by thermodynamics, as
well as some instances in general relativity and quantum mechanics (at the
measurement process) , the evolution of the systems is essentially asymmetric
in time. This distinguishes past from future. In the context of the Second Law
of Thermodynamics, which states that in an isolated system, entropy, a measure
of microscopic disorder, is a strictly increasing function of state providing
itself a direction for the flow of time. It is an interesting conjecture the
one which the Second Law of Thermodynamics and the thermodynamical arrow of
time follow from the initial conditions of the Universe, meaning that the flow
of time is set by the expansion of the Universe and it would revert in case of
contraction.

In fact, an intense discussion on the physics of macroscopic phenomena took
place in the second half of XIX century, when Maxwell (1831 - 1879) and
Boltzmann (1844 - 1906) became aware of the tension between the linear time
evolution and the reversibility of the fundamental equations of motion and
later on with the eternal recurrence of motion in the phase space (the
composed space of positions and velocities of the particles), shown by
Poincaré (1854 - 1912) in 1890.

Before that, in the 1870s, Boltzmann understood that getting an arrow of time
out the mechanics of atoms was impossible without using averaging arguments.
He eventually obtained in 1872 a time-asymmetric evolution equation, now
referred to as Boltzmann’s equation, whose solution was a single-particle time
dependent distribution function of a set of molecules in a diluted gas. From
this distribution function he got a time decreasing function of time, the so-
called ${\cal H}$-function, which he identified with minus the entropy,
obtaining then a solution for the irreversibility problem at molecular level!

The persistent objections of influential opponents such as Ernest Mach (1838 -
1916) and Friedrich Ostwald (1853 - 1932), led Boltzmann into depression and
eventually to his suicide in 5th September 1906 at the age of 62 at Duino, the
seaside resort on the Adriatico coast nearby Trieste.

After Boltzmann’s work, the irreversibility problem was revisited in 1907 by
Paul Ehrenfest (1880 - 1933) and his wife, Tatyana Afanasyeva (1876 - 1964),
who further developed Boltzmann’s idea of averaging over a certain region of
the phase space and showed that the averaged ${\cal H}$-function would remain
strictly decreasing in the thermodynamical limit.

In 1928, Wolfgang Pauli (1869 - 1958) when considering the problem of
transitions in the context of quantum mechanical perturbation theory, showed
that satisfying the Second Principle of Thermodynamics would require a master
equation, which confirmed the conviction of Maxwell and Boltzmann that
irreversibility should emerge from the procedure of inferring macroscopic
behaviour out of microscopic physics.

More recently, Prigogine (1917 - 2003) and collaborators considered the
radical proposal that irreversible behaviour should be already incorporated at
the microscopic level (see e.g. Ref. [19] for a general discussion). However,
this suggestion is not universally accepted.

Of course, besides understanding how the irreversible behaviour of macroscopic
systems did arise from microphysics, physicists identified several phenomena
whose behaviour exhibit an immutable flow from past to present, from present
to future. The British astrophysicist and cosmologist Arthur Eddington (1882 -
1944) [16], coined the term “arrow of time” to refer to this evolutionary
behaviour (see also Refs. [17, 18] for an updated discussion). Let us briefly
describe and enumerate these phenomena:

1) The direction of the growth of the entropy in irreversible and dissipative
macroscopic phenomena coincides with the flow of time;

2) The propagation of electromagnetic radiation, converging from the infinite
to a source is not observed, even though this being a possible solution of the
Maxwell’s equations of the electromagnetic field;

3) The irreversible collapse of the wave function of a quantum system after
the measurement process;

4) The exponential degradation in time of systems and the exponential growth
of self-organised systems (for a sufficiently abundant supply of resources).
The development of complex systems has led some authors to refer to this
behaviour as “creative evolution”, “arrow of life” or “physics of becoming”
[8, 12, 19, 20];

5) The discovery of the CP-symmetry violation in the $K^{0}-\bar{K}^{0}$
system due to weak interactions, allows for inferring from the CPT-theorem, a
fundamental cornerstone of quantum field theory, that the T-symmetry is also
violated333C, denotes charge conjugation, that is, particle-antiparticle
operation; P, denotes parity or mirror image operation; T, means time
inversion operation. The CPT-theorem states that the result of these three
operations together leave all local quantum field theories unchanged. This
means that there exists, at a fundamental level, an intrinsic irreversibility.

It is relevant to point out that the breaking of the CP-symmetry and the
violation of the baryon number444The number that counts the content of quarks
and anti-quarks of a given hadronic particle in an expanding Universe are
conditions from which the observed baryon asymmetry of the universe (BAU) can
be set up (see, for instance, Ref. [21]). An alternative route to explain the
BAU is through the violation of the CPT-symmetry itself. This is possible in
the context of string theory (see Ref. [22] and references therein);

6) Psychological time is obviously irreversible and historical. The past can
be recognised, while the future is unknown. Presumably this can be related
with the issue of causation. A fascinating question is how this fundamental
feature of Nature has shaped, through Evolution, the structure of our brain
which is known to have a common cortical metrics for time, space and quantity
[23];

7) Systems bound gravitationally exhibit the so-called gravito-thermal
catastrophic behaviour [24], that is, their entropy grows as they contract,
which in turn implies that their specific heat is negative. This feature is
also shared by black holes, whose entropy is proportional to their horizon’s
area. This feature raises the extremely relevant question of determining
whether the gravitational field should have an intrinsic entropy [25]. In
fact, it has been recently proposed that an entropy should be assigned even to
the vacuum [26].

On the largest scales, the expansion of the Universe, which is itself
adiabatic, is a unique phenomenon, which was conjectured to be the master
arrow of time from which all others should be subordinated via the putative
entropy of the gravitational field [25].

##  3 Time in Relativity

The conceptual standing of time in Relativity implies, at first, that it has
the same status as space, even though their specific properties are somewhat
different. The main consequence of Relativity is that it endows space-time
with plastic features. In Special Relativity, it follows from its fundamental
tenets, that time in a moving frame of reference stretches with respect to the
time in a frame of reference at rest. This means that clocks in motion slow
down with respect to clocks at rest. Under the same conditions, space
contracts. These are purely kinematical effects and follow from the
fundamental requirement that the laws of physics are invariant, or are said to
be covariant, in whatever frame of reference that move with respect to each
other at constant velocity. These frames of reference are called inertial
frames.

In the General Theory of Relativity, the covariance of the laws of physics are
generalised to frames of reference that are accelerated with respect to each
other, that is, the General Theory concerns non-inertial frames of reference.
Einstein first showed that these frames are equivalent to a gravitational
field and this discovery allowed him to turn the General Theory into a theory
of gravity. Furthermore, after reasoning that a light ray in an accelerated
frame of reference would be seen by an observer at rest as following a curved
trajectory, he concluded that the same would happen in the presence of a
gravitational field. He went on to conclude that matter-energy curves space-
time. It follows that in General Relativity the time flow is affected by
gravitational fields.

Special and General Relativity are extremely successful theories from the
experimental point of view. General Relativity, for instance, is very well
established at Solar System level [27, 28, 29] and its predictions are
multifold, ranging from the existence of black holes and gravitational
radiation to cosmological descriptions, such as the Hot Big-Bang model, in the
context of which space-time and matter were created about 13.7 thousand
millions of years ago (for extensive discussions, see, for instance, Refs.
[30, 31, 32]).

###  3.1 Closed time-like curves and the ensued menace of causality violation

As seen above, General Relativity admits a quite rich lore of conceptual and
mathematical solutions. Among them one finds non-trivial geometries where time
folds into itself and gives origin closed time-like curves (CTCs). These
solutions potentially violate causality as they allow for an observer who
travels along this type of curve to return to an event that coincides with the
departure event giving origin to time travel paradoxes [35]. Obviously, the
arrow of time leads to a forward local flow of time, but globally an observer
can, when travelling along a CTC, return to an event in the past.

The existence of CTCs imply paradoxes like the observer that kills an ancestor
making his/her very existence impossible as well as the possibility of the
flow of information from the future or from the present to the past. These
violations of causality have also quite clear implications, for instance, to
the issue of free-will.

A great number of solutions of Einstein’s field equations contain CTCS, but
there are two quite conspicuous types of solutions [37]. Solutions with a
tipping over of the light cones due to a rotation about a cylindrically
symmetric axis [38, 39, 40, 41, 42]; and solutions that violate the energy
conditions of General Relativity. The latter admit matter-energy distributions
where local observers effectively measure negative energy densities. Although
classical forms of matter do not allow for this behaviour, it is argued that
quantum fields may admit this possibility. In fact, this condition can be
relaxed in alternative theories of gravity (see, for instance, Refs. [33,
34]).

Furthermore one could admit the reverse procedure and consider geometries
which could lead, after some simple manipulations, to CTCs [48, 49, 50, 51].
These include the so-called traversable wormholes [43, 36], the warp drive
[44, 45, 46], and the Krasnikov tube [47].

Let us briefly discuss the case traversable wormholes [43]. A wormhole is
essentially constituted by two mouths at different regions of spacetime
connected by a tunnel or handle. Several types of manipulations can give
origin to CTCs. The simplest one involves a single wormhole mouth [48] which
is moving with respect to the other mouth so to create a time shift (cf.
Special Relativity) without affecting the internal geometry of the wormhole.
More elaborate variations of this procedure can be envisaged [36, 51].

###  3.2 Defusing the causality paradoxes

The existence of CTCs can potentially undo irreversible events at the expense
of creating somewhat embarrassing violations of causality. If one keeps the
metric nature of General Relativity, hence the only way to avoid these
violations is through a veto to CTCs. These curves can be rejected on logical
or physical grounds. The latter are obviously preferable. The following
solutions have been proposed:

i) In Novikov’s Principle of Self-Consistency [51], CTCs are admitted if self-
consistent, that is, the usual causal structure might be broken, but the
future can only influence without changing events in the past. This principle
implies that solutions of the laws of physics that are locally admissible are
those that are also globally self-consistent.

ii) Hawking’s Chronology Protection Conjecture is based on the experimental
evidence that “we have not been invaded by hordes of tourists from the future”
[54]. The physical support for the conjecture lies on the observation that a
relevant physical quantity, the renormalised quantum expectation of the
stress-energy tensor diverges close to the CTC’s. It is argued that the
conjecture can be shown in the context of the quantum gravity theory, the
theory that will presumably reconcile General Relativity to Quantum Mechanics.
The typical scale of energy of this conjectured theory is about
$10^{19}~{}GeV$, about fifteen orders of magnitude greater than the energy of
the LHC collider at CERN in Geneva, Switzerland, concentrated at a extremely
small region of space, about $10^{-35}~{}m$.

iii) It has been pointed out the CTCs arise in the limit of theories where
sensible physical properties of matter break down [42, 55]. The generality of
this situation for most of the solutions would render CTCs unphysical.

Thus, it is clear that the existence of CTCs is still the subject of
discussion and the final answer cannot be known before a deeper understanding
of quantum gravity. In fact we shall see that the matter of time travelling
becomes particularly pressing when considering the idea of a Block Universe.

##  4 Some Relevant Open Questions

Let us briefly discuss here some ideas about the nature of space-time that are
still a matter of lively debate:

Time crystals

Time crystals are quantum mechanical systems in their lowest energy state that
exhibit a periodic motion [57, 58], analogously to usual crystals that are a
periodic repetition of atoms in space. Their existence are associated to a
putative spontaneous breaking of discrete time translational symmetry, which
is thought to be impossible in Quantum Mechanics. However, it is believed that
evidence of time crystals have been found experimentally in many systems (see
Ref. [59] for a review). A recent proposal suggests that time crystals are an
emergent feature of phase-space noncommutative quantum mechanics [60].

Block Universe

It is assumed that as time evolves, gone events no longer exist, meaning that
as the future is not there yet, only the “present” is actual. An interesting
alternative description of space-time is the so-called Block Universe, in
which a preferred “present” is non-existent and past is equally present. Thus,
all points in time are equally valid frames of reference and a specific
instant that lies in the past or in the future is frame dependent. In
principle, there is no logical contradiction in a space-time block like that,
however even though observers can experience the subjective flow of time, an
universal present is ruled out by Special Relativity where no simultaneity
exists. Common sense and irreversible phenomena seem to contradict the
possibility of a Block Universe. Indeed, we refer to Ref. [61] a consistent
discussion on the objections to the Block Universe viewpoint.

It is evident that in a Block Universe, the causality paradoxes are
particularly pressing and demand for a categoric answer about the existence of
CTCs.

Specious Present

The so-called “specious present”, has been the subject of discussion in
experimental psychology and philosophy for quite some time. Our perception of
the present is such that is difficult, if not impossible, to precise the span
of time it comprises. It is a well established fact that in our experience,
objects are given to us as being of the present, but the part of time referred
to by the data is a very different one in what it concerns past and future.
The specious present has also been encountered in a completely independently
way when dealing with the engineering problem of transmission of visual and
audio information in a synchronised way.

In fact, experiments show that our brain copes well when a sensory delay
between visual and audio information is smaller than $125~{}ms$ for present-
past transmission, but is much shorter, about $45~{}ms$, for present-future
transmission (see, for instance, Ref. [62] for a discussion). How to interpret
this time asymmetry? Does it reflect the fundamental causality discussed by
the philosopher David Hume? Is it a consequence of Evolution, which causally
wired our brain to cope with sensory data?

##  5 Conclusions

In this brief report we attempted to discuss in a self consistent way the
nature of time and the most conspicuous implications related with various
conceptions of this hidden and mysterious ingredient of the physical
description. After a short discussion of some specific philosophical ideas
about time, we spelled out how the description of macroscopic phenomena
through the Second Law of Thermodynamics evolved so to allow us to
understanding how irreversibility arises from microphysics, despite the
symmetry of the fundamental evolution equations, classical and quantum, under
time reversal. Indeed, in the context of Statistical Mechanics555The branch of
physics that describes microphysics and its matching to the macroscopic
description achieved through the Laws of Thermodynamics., it is assumed that
the absolute deterministic behaviour is lost as the macroscopic description
necessarily averages out the micro-properties of the systems.

We have then discussed how in the Relativity revolution the physical
description suggested a merged space-time depiction of the events so to endow
space-time with plastic properties. We also elucidated how it followed that
time flows at different rates for observers in different frames of reference.
As seen, this is radically different from the description according Newtonian
physics, where time was assumed to flow at a constant rate for all observers.
In the context of General Relativity, space-time is curved by matter-energy
and the flow of time is dependent of the strength of the gravitational field.
The space-time curvature leads to completely new phenomena such as black
holes, gravitational waves and a great variety of cosmological spaces,
including the one with a Big Bang, which about 13.7 thousand millions years
ago, gave origin to matter and space-time itself.

General Relativity also admits solutions of its field equations in which time
folds into itself, giving origin to CTCs. This surprising possibility can
lead, as discussed, to causality paradoxes. We have then presented some
putative solutions to these embarrassing paradoxes. Even though, these
solutions seem quite sensible as they clearly do the job of protecting
causality, presumably, a definite answer about the existence of CTCs requires
an encompassing theory of quantum gravity.

Finally, we have closed our report with some brief discussions about topics
that are intimately related with the nature of time and how it is perceived,
namely, time crystals, the Block Universe proposal and the question about how
our brain processes sensorial information, the specious present question, and
its well established temporal asymmetry.

As a closing remark we could say that any attempt to encompass all aspects of
the problem of time is bound to fail. This author was well aware of this
essential limitation and pursued no more than to present, through some broad
strokes, a brief account on how physicists approach the fundamental problem of
time. The concept of time is at the very heart of the epistemological
questions associated with the physical description of phenomena, both at
macroscopic and at microscopic scales. However, this is necessarily a partial
view of the matter. The ubiquity of time in any form of articulate thinking
puts it conceptually at the very heart of any discussion about consciousness
and its emergence at biological, physical and philosophical levels. Time,
being the History of everything is the horizon of all forms of thinking. The
sought for an explanation about the most essential nature of time can only
lead us to the source of new questions.

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