#  $\mathbf{GRASP}$: Guiding model with RelAtional Semantics using Prompt
for Dialogue Relation Extraction

Junyoung Son, Jinsung Kim11footnotemark: 1, Jungwoo Lim11footnotemark: 1,
Heuiseok Lim
Computer Science and Engineering, Korea University
Republic of Korea
<EMAIL_ADDRESS>These authors have equally
contributed to this work Corresponding author

###### Abstract

The dialogue-based relation extraction (DialogRE) task aims to predict the
relations between argument pairs that appear in dialogue. Most previous
studies utilize fine-tuning pre-trained language models (PLMs) only with
extensive features to supplement the low information density of the dialogue
by multiple speakers. To effectively exploit inherent knowledge of PLMs
without extra layers and consider scattered semantic cues on the relation
between the arguments, we propose a Guiding model with RelAtional Semantics
using Prompt ($\mathbf{GRASP}$). We adopt a prompt-based fine-tuning approach
and capture relational semantic clues of a given dialogue with 1) an argument-
aware prompt marker strategy and 2) the relational clue detection task. In the
experiments, $\mathbf{GRASP}$ achieves state-of-the-art performance in terms
of both F1 and F1c scores on a DialogRE dataset even though our method only
leverages PLMs without adding any extra layers.

$\mathbf{GRASP}$: Guiding model with RelAtional Semantics using Prompt
for Dialogue Relation Extraction

Junyoung Son††thanks:  These authors have equally contributed to this work,
Jinsung Kim11footnotemark: 1, Jungwoo Lim11footnotemark: 1, Heuiseok
Lim††thanks:  Corresponding author Computer Science and Engineering, Korea
University Republic of Korea<EMAIL_ADDRESS>

##  1 Introduction

Figure 1: An illustration of (1) standard fine-tuning approach, and (2)
prompt-based fine-tuning approach. In prompt-based fine-tuning approach, a set
of label words are mapped into a class set by a certain mapping function.

The relation extraction (RE) task aims to extract semantic relations from
unstructured text such as a sentence, a document, or even a dialogue. RE plays
a critical role in information extraction and knowledge base construction as
it can extract structured relational information (Ji et al., 2010; Swampillai
and Stevenson, 2010). However, the utilization of sentence-level RE in a
conversational setting is limited because numerous relational facts appear
across multiple sentences with more than one speaker in a dialogue (Yao et
al., 2021). Thus, the dialogue-based relation extraction (DialogRE) task,
which includes argument pairs and their corresponding relations, has been
proposed to encourage building a model that captures the underlying semantic
spread in the dialogue, as presented in Table 1 (Yu et al., 2020).

Dialogue
---
|  1 S1: Hey guys! Hey!
---
2 S2: Hey Pheebs, guess who we saw today.
3 S3: Ooh, ooh, fun! Okay… um, Liam Neeson.
…
7 S3: The woman who cuts my hair!
8 S4: Okay, look, this could be a really long game.
9 S5: Your sister Ursula.
10 S3: Oh, really.
11 S5: Yeah, yeah, she works over at that place, uh…
12 S3: Rift’s. Yeah, I know.
…
17 S6: Um, Pheebs, so, you guys just don’t get along?
18 S3: It’s mostly just dumb sister stuff, you know, I mean, like, everyone
always thought of her as the pretty one, you know… …
| Argument pair | Trigger | Relation Type
R1 | (Pheebs, PER) | sister | per:siblings
(Ursula, PER)
1-4[0.8pt/1.2pt] R2 | (S3, PER) | none | per:alternate_names
(Pheebs, PER)
1-4[0.8pt/1.2pt] R3 | (Rift’s, ORG) | works over at | per:employees_or_members
(Ursula, PER)

Table 1: Example of DialogRE data. The arguments are bold, and the triggers
are underlined. The arguments and triggers are scattered throughout the
dialogue, which leads to low information density. The triggers determine the
direction of the proper relation indirectly.

In previous studies where state-of-the-art (SoTA) performance is achieved on
DialogRE benchmarks, fine-tuning is employed on pre-trained language models
(PLMs) (Lee and Choi, 2021; Long et al., 2021), such as BERT (Devlin et al.,
2019) and RoBERTa (Liu et al., 2019). As fine-tuning requires the addition of
extra layers on top of the PLMs and the training objectives are different from
those used in the pre-training phase, PLMs cannot effectively exploit their
learned knowledge in the downstream task, resulting in less generalized
capability (Chen et al., 2021).

To effectively utilize knowledge from PLMs, several studies involving prompt-
based fine-tuning have been conducted. They employ the PLM directly as a
predictor and completing a cloze task to bridge the gap between pre-training
and fine-tuning (Gao et al., 2020; Han et al., 2021). As presented in Figure
1, prompt-based fine-tuning treats the downstream task as a masked language
modeling (MLM) problem by directly generating the textual response to a given
template. In specific, prompt-based fine-tuning updates the original input on
the basis of the template and predicts the label words with the [MASK] token.
Afterwards, the model maps predicted label words to corresponding task-
specific class sets.

However, the prompt-based fine-tuning approach is still not sufficient in
terms of performance compared with the fine-tuning-based approach (Han et al.,
2021; Chen et al., 2021). We attribute this phenomenon to the following
properties of conversation: higher person-pronoun frequency (Wang and Liu,
2011) and lower information density (Biber, 1991) by multiple speakers111In
the DialogRE dataset, 65.9% of relational triples involve arguments that never
appear in the same utterance, demonstrating that multi-turn reasoning plays an
important role.. Therefore, a prompt-based fine-tuning approach that collects
the sparse semantics in the dialogue is required to understand relation
between the arguments.

We propose a method named Guiding model with RelAtional Semantics using Prompt
($\mathbf{GRASP}$) for DialogRE. To maximize the advantages of the prompt-
based fine-tuning approach for the DialogRE task, we suggest an argument-aware
prompt marking (APM) strategy and a relational clue detection (RCD) task. The
APM strategy guides the model to the significant arguments scattered in the
dialogue by carefully considering arguments. For our APM strategy, we conduct
empirical study based on the diverse marker types to validate our APM
strategy. Along with the APM strategy, the suggested RCD task with a training
objective leads the model to pay attention to significant relational clues.
Specifically, the model is trained to determine whether each token in a
dialogue belongs to a subject, object, or trigger. As a result, the PLM is
trained on RCD and MLM jointly. In the experiments, our method achieves SoTA
performance at a significant level in the DialogRE task for both the full-shot
and few-shot settings. Only PLMs are employed without the addition of an extra
layer as a predictor, and $\mathbf{GRASP}$ exhibits a higher performance than
other baselines. The significant performance improvement indicates that
attending to significant semantic clues guides the PLMs to predict the correct
relation with its inherent knowledge in both full-shot and few-shot settings.
Moreover, we provide ablation studies and qualitative analysis on the
robustness of $\mathbf{GRASP}$.

Our contributions are as follows:

  * •

We adopt a prompt-based fine-tuning approach to utilize a PLM’s inherent
knowledge directly for dialogues with relatively low information density.

  * •

We introduce an APM strategy and a RCD task that guide PLMs on the significant
relational clues, which are semantic information to predict relations.

  * •

We demonstrate that our proposed method achieves SoTA performance on the
DialogRE task in both full-shot and few-shot settings.

  * •

We conduct ablation studies and qualitative analysis to validate the
robustness of $\mathbf{GRASP}$.

The remainder of this paper is organized as follows. In Section 3, we present
the entire process of our method in detail. The experimental setup and the
results are explained in Section 4. The further analyses is provided in
Section 5, and Section 6 presents the conclusions. Appendix 2 provides related
works including the prompt-based learning, and the DialogRE.

##  2 Related Works

#### Prompt-based learning

Prompt-based learning is a method of reducing the gap between the pre-training
objective and that of fine-tuning. For example, language models such as BERT
(Devlin et al., 2019) use masked language modeling (MLM) objective in pre-
training phase where the model fills the [MASK] token whereas the model trains
without [MASK] token in fine-tuning phase by adding an extra classifier layer.
As a result, the discrepancy between training objectives prevents PLM from
leveraging knowledge acquired from the pre-training enormous corpus (Chen et
al., 2021). Also, prompt-based learning shows better performance than fine-
tuning especially in the few-shot setting Gao et al. (2020); Schick and
Schütze (2021); Li and Liang (2021); Liu et al. (2021).

#### DialogRE

Recent studies on the DialogRE dataset show a tendency to fine-tune the PLM
with task-specific objectives and use the model with high-complexity (Xue et
al., 2021; Long et al., 2021; Lee and Choi, 2021). In detail, Lee and Choi
(2021) shows considerable performance with the contextualized turn
representations from the diverse type of nodes and edges. Moreover, task-
specific objectives of fine-tuning lead to a gap between pre-training and
fine-tuning.

To overcome the limitation, Han et al. (2021) utilizes multiple [MASK] tokens
for each argument and the relation with logical rules by concentrating subject
and object in DialogRE. There also exists an approach that incorporates
potential knowledge contained in relation labels into prompt construction with
trainable virtual type words and answers words. This approach also carefully
initializes the virtual tokens with implicit semantic words and employs prior
distributions estimated from the data (Chen et al., 2021).

####

Despite the prompt-based approach’s high potential, few prompting studies
sufficiently consider low information density and difficulty of capturing
intrinsic relational information of the data between the argument pair of
dialogue relation extraction task. We focus on building a light model with
prompt-based fine-tuning with implicit semantic information of the relation
which alleviates the sparsity problem.

Figure 2: The overall model architecture of $\mathbf{GRASP}$. By formalizing
specific tasks as MLM tasks, the model predicts answers with the tokens from
the model’s vocabulary; for example, the label words of $V_{rel}$ and
$V_{rcd}$ are from the model’s vocabulary.

##  3 Methodology

An overview of $\mathbf{GRASP}$ is illustrated in Figure 2. First, an input
with a prompt template is constructed by using the APM strategy. Then, the PLM
receives the constructed input for prompt-based fine-tuning and estimates
probability distributions of the model’s vocabulary by using contextualized
representations of the PLM. The RCD task lets the model predict the relational
clue type of each token, and the model takes the [MASK] representation to
predict a final relation in the MLM task. In other words, our model is trained
through multitask learning to encourage mutual communication between
relational clues and the final relation for the argument pair.

###  3.1 Problem Formulation

Each example $\mathcal{X}$ includes dialogue $D$, subject $a_{1}$, and object
$a_{2}$. Note that $D$ denotes
$\\{s_{1}:u_{1},s_{2}:u_{2},\dots,s_{N}:u_{N}\\}$, where $s_{n}$ is the $n$-th
speaker and $u_{n}$ is the corresponding utterance. Given
$\mathcal{X}=\\{D,a_{1},a_{2}\\}$, the goal of DialogRE is to predict relation
$y$ between arguments $a_{1}$ and $a_{2}$ by leveraging $D$. To describe the
DialogRE task in terms of prompt-based fine-tuning, a template function,
$\mathcal{T}(\cdot)$, is defined to map each example to
$\mathcal{X}_{prompt}=\mathcal{T}(\mathcal{X})$. A [MASK] is inserted into
$\mathcal{X}_{prompt}$, and used to predicting the label words of relation
$y$. The formulation of $\mathcal{T}(\mathcal{X})$ is as follows:

$\begin{aligned}
\mathcal{T}(\mathcal{X})=\text{``}&\texttt{[CLS]}D\texttt{[SEP]}\texttt{[subj]}a_{1}\texttt{[subj]}\\\
&\texttt{[MASK]}\texttt{[obj]}a_{2}\texttt{[obj]}\texttt{[SEP]}\text{''}.\end{aligned}$

(1)

Based on the structure of $\mathcal{T}(\cdot)$, we construct our template
function, $\mathcal{T}^{\prime}(\cdot)$, by applying two steps: transformation
of $D$ to $D^{\prime}$ with an argument-aware prompt marker, described in 3.2,
and prompt initialization for [subj] and [obj], explained in 3.3.
Subsequently, we introduce the RCD task in 3.4 to train the model by utilizing
$\mathcal{T}^{\prime}(\cdot)$ with a multitask training strategy on MLM, as
detailed in 3.5.

###  3.2 Argument-aware Prompt Marker

We propose an APM strategy that considers both speaker and non-speaker
arguments. In previous studies, a dialogue is encoded by focusing on speaker
information (Lee and Choi, 2021; Yu et al., 2020; Chen et al., 2021) without
focusing non-speaker arguments. However, in the DialogRE dataset,
approximately 77.4% of relation triples include at least one non-speaker
argument, implying that consideration of non-speaker arguments is also
inevitable to enhance the model’s argument-awareness.

Our methods is inspired by the previous works of Soares et al. (2019) and Han
et al. (2021); the former addresses the importance of entity markers regarding
recognition of the entity position, and the latter improves the model
performance by using specific additional punctuation in the model’s original
vocabulary as the entity marker. Accordingly, we insert the argument-aware
prompt marker token, [p], as an entity marker. The argument-aware prompt
markers allow our model to obtain informative signs to determine which token
is the indicative component for relation prediction. We initialize the feature
of [p] as the embedding of the space token in the vocabulary of the model. Our
empirical experiments reveal that the space token can perceive the start
position of the arguments. Consequently, our prompt marker enhances the model
to discriminate which part of the dialogue plays a critical role in predicting
a relation.

Using the proposed argument-aware prompt marker, we strengthen the token
replacement method of BERTs (Yu et al., 2020). Given example $\mathcal{X}$,
BERTs constructs
$\tilde{\mathcal{X}}=\\{\tilde{D},\tilde{a_{1}},\tilde{a_{2}}\\}$, where
$\tilde{D}=\\{\tilde{s}_{1}:u_{1},\tilde{s}_{2}:u_{2},\dots,\tilde{s}_{N}:u_{N}\\}$
and $\tilde{s_{n}}$ is

${\tilde{s_{n}}=\begin{cases}\texttt{[S${}_{1}$]}&$if$\ s_{n}=a_{1}\\\
\texttt{[S${}_{2}$]}&$if$\ s_{n}=a_{2}\\\ s_{n}&$otherwise.$\end{cases}}$ (2)

[S1] and [S2] are special tokens for speakers. In addition, $\tilde{a}_{m}$
($m\in\\{1,2\\}$) is defined as [Sm] if $\exists~{}n~{}(s_{n}=a_{m})$ and
$a_{m}$ otherwise.

Even though BERTs prevents the model from overfitting and demonstrates higher
generalization capacity (Yu et al., 2020), BERTs does not consider non-speaker
arguments. To explore the disregarded arguments, the APM strategy expands
BERTs by considering both types of arguments in the DialogRE task. We define
function APM($\cdot$) that encodes an utterance by inserting [p] in front of
each argument token. Given dialogue $\tilde{D}$ constructed by using BERTs, we
construct
$D^{\prime}=\\{\tilde{s}_{1}:u^{\prime}_{1},\tilde{s}_{2}:u^{\prime}_{2},\dots,\tilde{s}_{N}:u^{\prime}_{N}\\}$
by applying APM$(u_{n})$ for each utterance in $\tilde{D}$. Consequently, the
APM strategy constructs
$\mathcal{X}^{\prime}=\\{D^{\prime},\tilde{a}_{1},\tilde{a}_{2}\\}$ based on
$\mathcal{\tilde{X}}$.

$u^{\prime}_{n}=\begin{cases}\texttt{APM}(u_{n})&$if$~{}\exists~{}m~{}(\tilde{a}_{m}\in
u_{n})~{}(m\in\\{1,2\\})\\\ u_{n}&$otherwise$\end{cases}$

(3)

For instance, $\texttt{APM}(\cdot)$ encodes the text “I am Tom Gordon” to [I,
am, [p], Tom, Gordon], inserting [p] in front of “Tom Gordon.”

###  3.3 Prompt Construction

We update the constructed input $\mathcal{X}^{\prime}$ with a template
function and conduct a deliberate initialization. We utilize the prior
distribution of argument types for initialization inspired by the study
conducted by Chen et al. (2021). Prompt tokens [subj] and [obj] are used to
inject argument-type information. We define the argument-type set,
$\mathcal{AT}=\\{$“PER,” “ORG,” “GPE,” “VALUE,” “STRING”$\\}$, using the types
pre-defined in the dataset as depicted in Table 1. We calculate the
distributions of argument types $\phi^{\texttt{[subj]}}$ and
$\phi^{\texttt{[obj]}}$ over $\mathcal{AT}$ by using frequency statistics. We
aggregate each argument type, $at\in\mathcal{AT}$, with the corresponding
prior distribution to initialize prompt tokens [subj] and [obj]. The specific
initialization equations are as follows:

$\begin{aligned}
\check{e}(\text{{[subj]}})&=\sum_{at~{}\in\mathcal{AT}}{\phi^{\texttt{[subj]}}_{at}\cdot
e(at)}\\\
\check{e}(\texttt{[obj]})&=\sum_{at~{}\in\mathcal{AT}}{\phi^{\texttt{[obj]}}_{at}\cdot
e(at)},\end{aligned}$

(4)

where $e(\cdot)$ is the embedding from the PLM of an input token and
$\check{e}(\cdot)$ is the initialized embedding of the prompt token. Suppose
the subject has prior distribution,
$\phi^{\texttt{[subj]}}=\\{$“PER”$:0.5,$’‘ORG”$:$0.5,
“GPE”:$0.0,$“VALUE”:$0.0,$“STRING”:$0.0$}. The initial embedding of the [subj]
token can be calculated as a weighted average, i.e.,
$\check{e}(\texttt{[subj]})=0.5\cdot e($“PER”$)+0.5\cdot e$(“ORG”$)$.

Consequently, we can formalize $\mathcal{T}^{\prime}(\cdot)$, which converts
$\mathcal{X}^{\prime}$ to $\mathcal{X}^{\prime}_{prompt}$, where
$\mathcal{X}_{prompt}^{\prime}$ is an argument-enhanced input example, by
using the APM strategy and prompt construction with deliberate
initializations, i.e.,
$\mathcal{X}^{\prime}_{prompt}=\mathcal{T}^{\prime}(\mathcal{X}^{\prime})$.
Then, the final input structure for prompt-based fine-tuning is as follows:

$\begin{aligned}
\mathcal{T^{\prime}}(\mathcal{X}^{\prime})=\text{``}&\texttt{[CLS]}D^{\prime}\texttt{[SEP]}\texttt{[subj]}\tilde{a}_{1}\texttt{[subj]}\\\
&\texttt{[MASK]}\texttt{[obj]}\tilde{a}_{2}\texttt{[obj]}\texttt{[SEP]}\text{''}.\end{aligned}$

(5)

In addition, for relation prediction by applying MLM, we also define
$\mathcal{V}_{rel}$ as a set of label words in the model’s vocabulary as
illustrated in Figure 2. In detail, we utilize its metadata for the
initialization of each relation representation to inject its semantics. For
instance, we add a special token to the model’s vocabulary,
[per:date_of_birth] as a label word, and initialize this token by aggregating
the embeddings of the words in the metadata, i.e., {“person,” “date,” “of,”
“birth”} for a class “per:date_of_birth.”

Full-shot Setting
---
Method | V1 | V2
F1 | F1c | F1 | F1c
Fine-tuning based approach
BERTs (Yu et al., 2020) | 61.2 | 55.4 | - | -
RoBERTas (Lee and Choi, 2021) | - | - | 71.3 | 63.7
Dual (Bai et al., 2021) | 67.3 | 61.4 | 67.1 | 61.1
CoIn (Long et al., 2021) | 72.3 | - | - | -
TUCORE-GCN (Lee and Choi, 2021) | - | - | 73.1 | 65.9
Prompt-based fine-tuning approach
PTR (Han et al., 2021) | 63.2 | - | - | -
KnowPrompt (Chen et al., 2021) | 68.6 | - | - | -
1-5[0.8pt/1.2pt] $\mathbf{GRASP}_{base}$ (Our model) | 69.2 | 62.4 | 69.0 | 61.7
$\mathbf{GRASP}_{large}$ (Our model) |  75.1 (+2.8) |  66.7 (+5.3) |  75.5 (+2.4) |  67.8 (+1.9)

Table 2: Performances of $\mathbf{GRASP}$ on test set of DialogRE. V1 and V2
represent the version of the dataset. The underlined scores are the previous
SoTA performances. Subscript in parentheses represents advantages of
$\mathbf{GRASP}$ over the best results of baselines (the underlined). Best
results are bold.

###  3.4 Relational Clue Detection task

To improve the understanding capability on relational clues by employing a
prompt-based fine-tuning approach, we introduce a RCD task. We define a set of
label words, $\mathcal{V}_{rcd}=${[subject], [object], [trigger], [outside]},
and add to the model’s vocabulary. Then, we construct a sequence of label
words for RCD, $\mathcal{C}_{rcd}$, by assigning each token in
$\mathcal{X}^{\prime}_{prompt}$ to the corresponding clue type word from
$\mathcal{V}_{rcd}$. For instance, $\mathcal{C}_{rcd}$ is constructed as
follows: {[subject], [trigger], [object]} when the token sequence is given by
$\\{\textit{``Pheebs''},\textit{``lives in''},\textit{``LA''}\\}$, where
“Pheebs” is a subject and “LA” is an object argument, and “lives in” is a
trigger.

The RCD task exploits the MLM head that is used to predict the [MASK] token.
Except for the [MASK] token, each token is sequentially tagged with
$\mathcal{V}_{rcd}$ using the meta-data provided in the dataset. In other
words, the RCD task allows the model to identify which non-[MASK] tokens
correspond to certain relational clue types. RCD supports the model in
collecting scattered information from the entire dialogue by indicating where
to focus in the dialogue to predict a relation. In this respect, the model
pays considerably more attention to the semantic clues of the relation, such
as triggers. Moreover, our model maintains a lightweight complexity by
conducting the RCD task without an additional classifier. The loss for the RCD
task over each token $x\in\mathcal{X}^{\prime}_{prompt}$ is aggregated as
follows:

$\scalebox{0.85}{$\begin{aligned}
\mathcal{L}_{RCD}&=-\sum_{x\in\mathcal{X}^{\prime}_{prompt}}{\log
P(x=\mathcal{C}_{rcd}(x)|\mathcal{X}^{\prime}_{prompt})}\\\ \end{aligned}$}.$
(6)

###  3.5 Model Training

Before the final relation-prediction, we mark the position of the trigger
using the [p] token, i.e., the argument-aware prompt marker, based on the
results of the RCD task. Given example $\mathcal{X}^{\prime}_{prompt}$, the
model predicts the label words of $\mathcal{V}_{rcd}$ for all non-[MASK]
tokens. Subsequently, [p] is appended in front of the words predicted as a
trigger to train the model to distinguish essential clues that encourage
determining the relation.

#### Multitask Learning

$\mathcal{M}_{rel}:\mathcal{Y}\to\mathcal{V}_{rel}$ is a mapping function that
converts a class set, $\mathcal{Y}$, into a set of label words,
$\mathcal{V}_{rel}$. For each input $\mathcal{X}^{\prime}_{prompt}$, the
purpose of MLM is to fill [MASK] with the relation label words in the model’s
vocabulary. As the model predicts the correct label word at the position of
[MASK], we can formulate
$p(y|x)=P(\texttt{[MASK]}=\mathcal{M}_{rel}(y)|\mathcal{X}^{\prime}_{prompt})$.
The training objective of the relation prediction is to minimize

$\mathcal{L}_{REL}=-\log
P(\texttt{[MASK]}=\mathcal{M}_{rel}(y)|\mathcal{X}^{\prime}_{prompt}).$

(7)

To improve the model’s ability to capture relational clues through the
interaction between the MLM task for relation prediction and the RCD task, we
employ a multitask learning to train the $\mathbf{GRASP}$ model by using the
joint loss expressed in Equations (6) and (7). Therefore, the final learning
objective is to minimize the joint loss, where $\lambda_{1}$ and $\lambda_{2}$
are hyperparameters.

$\mathcal{L}_{GRASP}=\lambda_{1}\cdot\mathcal{L}_{RCD}+\lambda_{2}\cdot\mathcal{L}_{REL}$

(8)

##  4 Experiments

###  4.1 Experimental Setup

For the base PLMs, RoBERTa-base and RoBERTa-large are adopted, as denoted by
$\mathbf{GRASP}_{base}$ and $\mathbf{GRASP}_{large}$, respectively. The
evaluation metrics are the F1 and F1c (Yu et al., 2020) scores. F1c is an
evaluation metric for supplementing the F1 score in a conversational setting
and is computed by employing part of the dialogue necessary for predicting the
relation between given arguments as input instead of the entire dialogue. The
detailed settings for $\mathbf{GRASP}$ can be found in Appendix A.

In a full-shot setting, $\mathbf{GRASP}$ is compared with both fine-tuning-
based and prompt-based fine-tuning approaches. TUCORE-GCN (Lee and Choi, 2021)
is a typical fine-tuning-based model using turn-level features with a graph
convolution network (Kipf and Welling, 2016). CoIn (Long et al., 2021) employs
utterance-aware and speaker-aware representations, and Dual (Bai et al., 2021)
models the relational semantics using abstract meaning representations
(Banarescu et al., 2013). Moreover, PTR (Han et al., 2021) and KnowPrompt
(Chen et al., 2021) are the prompt-based fine-tuning baseline models. In the
few-shot setting, 8-, 16-, and 32-shot experiments were conducted based on LM-
BFF (Gao et al., 2020) by using three different randomly sampled data.

###  4.2 Experimental Results

#### Full-shot setting

As presented in Table 3.3, it is shown that $\mathbf{GRASP}_{large}$ surpasses
all of the baseline models, including the current SoTA models, that is, CoIn
and TUCORE-GCN. From the result in the full-shot setting, the baselines of the
fine-tuning-based approach, such as TUCORE-GCN or CoIn, show better
performance than those of the prompt-based fine-tuning approach. In
particular, CoIn outperforms all of the other baselines, including PTR and
KnowPrompt on V1, and TUCORE-GCN exhibits the best performance on V2.
Interestingly, even though the performance of the prompt-based fine-tuning
baselines is much lower than fine-tuning based models, our
$\mathbf{GRASP}_{large}$ outperforms regardless of the way of training
approach.

In addition, $\mathbf{GRASP}_{large}$ shows its efficiency in conversational
settings by exceeding all the baselines in terms of F1c, thereby indicating
that our method effectively overcomes the low information density of
dialogues. These results imply that guiding the model to pay attention to
relational clues with a prompt-based fine-tuning approach can be more
effective than adding additional features and layers. The slightly low
performance of $\mathbf{GRASP}_{base}$ is attributed to the gap in the model
size; for example, TUCORE-GCN has 401M weight parameters with RoBERTa-large
model, and $\mathbf{GRASP}_{base}$ has 125M weight parameters with RoBERTa-
base. Nevertheless, $\mathbf{GRASP}_{base}$ demonstrates 6.0%p and 0.6%p
improvements compared with the other prompt-based fine-tuning baselines, i.e.,
PTR and KnowPrompt, respectively.

Few-shot Setting
---
Method | Shot
K=8 | K=16 | K=32
Fine-tuning based approach
RoBERTa (Chen et al., 2021) | 29.8 | 40.8 | 49.7
TUCORE-GCN (Lee and Choi, 2021) | 24.6 | 40.0 | 53.8
Prompt-based fine-tuning approach
PTR (Han et al., 2021) | 35.5 | 43.5 | 49.5
KnowPrompt (Chen et al., 2021) | 43.8 | 50.8 | 55.3
1-4[0.8pt/1.2pt] $\mathbf{GRASP}_{base}$ (Our model) | 45.4 | 52.0 | 56.0
$\mathbf{GRASP}_{large}$ (Our model) | 36.0 | 55.3 | 62.6

Table 3: Low-resource RE performance of F1 scores (%) on different test sets.
We use K = 8, 16, 32 (# of examples per class) for few-shot experiments. Best
results are bold and the second place results are underlined.

#### Few-shot setting

As presented in Table 4.2, $\mathbf{GRASP}$ still exhibits robust performance
in few-shot settings. $\mathbf{GRASP}_{base}$ outperforms the baselines of
both the fine-tuning and prompt-based fine-tuning methods regardless of the
number of shots, demonstrating 20%p or higher performance in the 8-shot
setting compared with TUCORE-GCN and indicating that our method is more
efficient than the fine-tuning method in a low-resource setting.
$\mathbf{GRASP}_{base}$ also demonstrates improved performance compared with
KnowPrompt in all-shot settings, indicating the effectiveness of the
considerations on the properties of the dialogue with prompt-based fine-
tuning. Except for the 8-shot setting, $\mathbf{GRASP}_{large}$ presents
outstanding performance, achieving up to 15.3%p of absolute improvement in the
16-shot setting. Although $\mathbf{GRASP}_{large}$ outperforms the fine-
tuning-based models and PTR, the limited performance of
$\mathbf{GRASP}_{large}$ in the 8-shot setting can be attributed to an
insufficient number of examples.

We also observe that the fine-tuning-based models, such as TUCORE-GCN, perform
at least 5.7%p worse than the prompt-based fine-tuning models, such as PTR, in
the 8-shot setting, indicating that the fine-tuning-based models may have
difficulty in sufficiently capturing relational semantics when the data are
extremely scarce. In particular, TUCORE-GCN indicates a 5.2%p lower
performance than the fine-tuned RoBERTa, indicating that the high complexity
requires a larger amount of data than the other models.

Method | Dev | Test
---|---|---
F1 | F1c | F1 | F1c
$\mathbf{GRASP}_{base}$ | 70.3 | 63.3 | 69.0 | 61.7
1-5[0.8pt/1.2pt] -APM | 69.5 | 62.9 | 66.9 | 60.7
-RCD | 69.2 | 62.8 | 67.7 | 61.0
-Prompt manual init. | 68.1 | 61.9 | 65.9 | 59.8

Table 4:  Ablation study on DialogRE dataset.

Marker Type |  Input Example | F1
---|---|---
Entity marker |  [CLS] [E1] Frank [/E1] lives in [E2] Montauk [/E2] . [SEP] | 65.9
Type marker |  [CLS] [E1:PER] Frank [/E1:PER] lives in [E2:GPE] Montauk [/E2:GPE] . [SEP] | 66.7
Punctuation marker (!) |  [CLS] ! Frank ! lives in ! Montauk ! . [SEP] | 66.7
Punctuation marker (@) |  [CLS] @ Frank @ lives in @ Montauk @ . [SEP] | 65.3
Punctuation marker (;) |  [CLS] ; Frank ; lives in ; Montauk ; . [SEP] | 67.1
1-3[0.8pt/1.2pt] APM marker (front) |  [CLS] [p] Frank lives in [p] Montauk . [SEP] | 69.0
APM marker (surrounding) |  [CLS] [p] Frank [p] lives in [p] Montauk [p] . [SEP] | 65.9

Table 5: The performance based on the marker type. The arguments are bold. The
embedding of [p] is initialized with the space token from the model’s
vocabulary. In type marker, [E1:PER] represents a start position of subject
which has a person type and [/E1:PER] represents an end position of object
that has the same type.

#### Ablation Study

We conduct an ablation study to validate the effectiveness of the proposed
modules. As shown in Table 4, each of the proposed modules improves the
overall performance for both F1 and F1c settings. Without the APM strategy,
the performance of $\mathbf{GRASP}_{base}$ decreases the F1 score by 0.8%p and
the F1c score by 0.4%p on the development set, and the F1 score drops sharply
by 2.1%p and F1c by 1.0%p on the test set. This result indicates that
argument-awareness can be obtained through both speaker and non-speaker
argument information. When the RCD task is excluded, the performance of
$\mathbf{GRASP}_{base}$ decreases the F1 score by 1.1%p and F1c by 0.5%p on
the development set and the F1 score by 1.3%p and F1c by 0.7%p for the test
set. These results demonstrate that the RCD task alleviates the low
information density of the dialogue by guiding the model to focus on
relational clues.

In addition, the performance without the manual initialization of prompt
construction of $\mathbf{GRASP}_{base}$ is reduced by 2.2%p for the F1 score
and 1.4%p for the F1c score on the development set and by 3.1%p for the F1
score and 1.9%p for the F1c score on the test set. This result suggests that
prompt construction is a basic step in training the model in the prompt-based
fine-tuning case. The deliberate initialization of prompts is critical for
modeling the tasks in an appropriate direction.

Dialogue
---
|  S1: Hey!!
---
S2: Hey!
S1: Guess what. Frank Jr., and Alice got married!
S2: Oh my God!!
S1: And! And, they’re gonna have a baby! And! And, they want me to grow it for
them in my uterus.
S3: My God!
S4: Are you serious?
S1: Yeah
S5: You’re really thinking about having sex with your brother?!
S1: Ewww! And "Oh no!" It’s—they just want me to be the surrogate. It’s her-
it’s her egg and her sperm, and I’m-I’m just the oven, it’s totally their bun.
S5: Huh.
Argument pair | Ground Truth | RoBERTa | $\mathbf{GRASP}$
Predicted Relation | Predicted (subject, object, trigger) | Predicted Relation
(Frank Jr., Alice) | per:spouse | unanswerable | (Frank Jr, Alice, got married) | per:spouse
(Alice, Frank Jr.) | per:spouse | per:siblings | (Alice, Frank Jr, got married) | per:spouse

Table 6: The qualitative analysis on the prediction of $\mathbf{GRASP}$ based
on the comparison with the RoBERTa model. Predicted (subject, object, trigger)
is that $\mathbf{GRASP}$ predicted on RCD task.

##  5 Analysis

###  5.1 Analysis on marker type for APM

To analyze argument-awareness regarding the types of markers, we conduct
experiments on diverse prompt markers, as shown in Table 5. The result reveals
that considering argument types leads to performance improvement in the model.
Punctuation marker “;” shows comparable performance among other punctuation
markers, and “!” and “@” achieve a limited score. We presume that the higher
frequency of “;” acts as a delimiter, which results in decent performance.

We also observe that the APM marker (front) performs the best, with a 69.0% F1
score among all other marker types. In addition, we conduct an experiment
based on the position of the prompt marker, [p], by comparing two versions of
the APM marker: APM marker (front) and APM marker (surrounding). The APM
marker (surrounding) display 3.1%p lower performance than the APM marker
(front). Based on these results, we empirically adopt the embedding
initialization of our [p] prompt marker using the space token and located it
in front of the arguments.

###  5.2 Qualitative Analysis on $\mathbf{GRASP}$

Since we train $\mathbf{GRASP}$ attending on relational clues through the APM
strategy and the RCD task, we further conduct analysis to investigate that the
relational clues such as triggers contribute to predicting a relation between
the arguments in a prompt-based manner, as shown in Table 6. Specifically, we
compare $\mathbf{GRASP}_{large}$ with a fine-tuned RoBERTa-large model to
validate our method.

We observe that the fine-tuned RoBERTa model struggles to capture the
symmetrical relations including the trigger. The fine-tuned RoBERTa model fail
to capture relational clues, such as “got married”, misleading the model into
predicting an inappropriate relations for both symmetrical relations between
“Frank Jr.” and “Alice.” “got married” is a critical cue to distinguish the
relations between “per:spouse” and “per:siblings” because this phrase implies
a romantic relationship in a dictionary definition. In contrast,
$\mathbf{GRASP}$, which is trained using the RCD task and the APM strategy in
a prompt-based manner, predicted the correct relations, capturing the correct
relational clues including the trigger “got married” for both symmetrical
argument pairs. This result presents the effectiveness of $\mathbf{GRASP}$
designed to guide the model on the relational clues, alleviating the
difficulties of low information density in dialogues. Additional examples
demonstrating similar phenomena for the symmetrical relations are provided in
the Appendix B.

Method | MELD | EmoryNLP
---|---|---
RoBERTa (Liu et al., 2019) | 62.0 | 37.3
COSMIC (Ghosal et al., 2020) | 65.2 | 38.1
TUCORE-GCN (Lee and Choi, 2021) | 65.4 | 39.2
1-3[0.8pt/1.2pt] $\mathbf{GRASP}_{large}$ (Ours) | 65.6 | 40.0

Table 7: Experimental results of $\mathbf{GRASP}_{base}$ on MELD and EmoryNLP
tasks.

###  5.3 Analysis on the applicability of $\mathbf{GRASP}$

To demonstrate the robustness of our APM strategy and RCD task, we evaluated
$\mathbf{GRASP}$ on MELD (Poria et al., 2019) and EmoryNLP (Zahiri and Choi,
2018) datasets, which are designed for emotion recognition in conversations
(ERC). MELD (Poria et al., 2019) is a multimodal dataset collected from a TV
show named Friends and consists of seven emotion labels and 2,458 dialogues
with only textual modality. EmoryNLP (Zahiri and Choi, 2018) is also collected
from Friends and comprises seven emotion labels and 897 dialogues. Each
utterance in these datasets is annotated with one of the seven emotion labels.
The weighted-F1 is calculated to evaluate the MELD and EmoryNLP datasets.

For baselines, we employ the fine-tuned RoBERTa (Liu et al., 2019), COSMIC
(Ghosal et al., 2020), and TUCORE-GCN models (Lee and Choi, 2021). COSMIC
(Ghosal et al., 2020) uses RoBERTa-large as the encoder. It is a framework
that models various aspects of commonsense knowledge by considering mental
states, events, actions, and cause-effect relations for emotional recognition
in conversations.

As presented in Table 7, $\mathbf{GRASP}$ is applied to other dialogue-based
tasks by alleviating the low information density of the given dialogue. In
particular, the performance of $\mathbf{GRASP}$ surpasses that of TUCORE-GCN,
which is the current SoTA model in DialogRE, and those of the baseline
specialized on ERC tasks, such as COSMIC in both MELD and EmoryNLP.

##  6 Conclusion

In this paper, we proposed $\mathbf{GRASP}$, which is a method for guiding
PLMs to relational semantics using prompt-based fine-tuning for the DialogRE
task. We focused on alleviating the critical challenge in dialogues, that is,
low information density, by effectively capturing relational clues. In
$\mathbf{GRASP}$, we constructed prompts with deliberate initialization and
suggested 1) an APM strategy considering both speaker and non-speaker
arguments and 2) the RCD task, which guides the model to determine which token
belongs to the relational clues. Experimental results on the DialogRE dataset
revealed that $\mathbf{GRASP}$ achieved SoTA performance in terms of the F1
and F1c scores, even though our method only leveraged a PLM without adding any
extra layers.

## Acknowledgements

This work was supported by Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by the Korea government
(MSIT) (No. 2020-0-00368, A Neural-Symbolic Model for Knowledge Acquisition
and Inference Techniques), the MSIT (Ministry of Science and ICT), Korea,
under the ITRC (Information Technology Research Center) support program
(IITP-2022-2018-0-01405) supervised by the IITP, and the MSIT, Korea, under
the ICT Creative Consilience program (IITP-2022-2020-0-01819) supervised by
the IITP.

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##  Appendix A Detailed experimental settings

| Hyper-
---
parameters
$\mathbf{GRASP}_{base}$ | $\mathbf{GRASP}_{large}$
Learning rate | $5e-5$ | $5e-6$
Max seq. len | 512
Batch size | 8
Num. epochs | 30
| Joint ratio
---
($\lambda_{1}$ & $\lambda_{2}$)
0.7 / 0.3

Table 8: Hyper-parameter values used in prompt-tuning process on test set.

$\mathbf{GRASP}$ is trained using AdamW (Loshchilov and Hutter, 2017) as an
optimizer with no weight decay. The number of training epochs is set to 30
with early stopping, and the ratio of $\lambda_{1}$ and $\lambda_{2}$ for the
joint loss is 0.7 to 0.3. A learning rate of $5e{-5}$, batch size of 8, and
maximum sequence length of 512 are adopted for RoBERTa-base with identical
parameters for RoBERTa-large, except for the learning rate of $5e{-6}$.

Dialogue
---
|  S1: Hey! Hi!
---
S2: Hey!
S1: What’s up?
S2: Well umm, Chandler and I are moving in together.
S1: Oh my God. Ohh, my little sister and my best friend…shaking up. Oh, that’s
great. That’s great.
S3: Guys, I’m happy too.
S2: Okay, come here!
S3: Wow! Big day huh? People moving in, people getting annulled…
S2: Okay, I gotta go find Rachel but umm, if you guys see her could you please
try to give her some really bad news so that mine doesn’t seem so bad?
…
Argument pair | Ground Truth | RoBERTa | $\mathbf{GRASP}$
Predicted Relation | Predicted (subject, object, trigger) | Predicted Relation
(S2, Chandler) | per:girl/boyfriend | per:roommate | (S2, Chandler, moving in together) | per:girl/boyfriend
(Chandler, S2) | per:girl/boyfriend | per:roommate | (Chandler, S2, moving in together) | per:girl/boyfriend

Dialogue
---
|  S1: Rach, Rach, I just remembered. I had a dream about Mr. Geller last
night.
---
S2: Really?!
S1: Yeah, I dreamt that he saved me from a burning building and he was so
brave and so strong! And it’s making me look at him totally differently.
Y’know, I mean he used to be just, y’know “Jack Geller Monica and Ross’s dad”
and now he’s he’s “Jack Geller, dream hunk."
S2: I dunno. Y’know to me he’ll always be “Jack Geller, walks in while you’re
changing.”
Argument pair | Ground Truth | RoBERTa | $\mathbf{GRASP}$
Predicted Relation | Predicted (subject, object, trigger) | Predicted Relation
(Mr. Geller, Monica) | per:children | unanswerable | (Mr Geller, Monica, dad) | per:children
(Mr. Geller, Ross) | per:children | unanswerable | (Mr Geller, Ross, dad) | per:children
(Ross, Mr. Geller) | per:parents | per:alternate_names | (Ross, Mr Geller, dad) | per:parents

Table 9: The additional examples for qualitative analysis on the prediction of
$\mathbf{GRASP}$ based on the comparison with the RoBERTa model. Predicted
(subject, object, trigger) is that $\mathbf{GRASP}$ predicted on RCD task.

##  Appendix B Qualitative Analysis Examples

Table 9 shows additional examples to demonstrate the prediction tendency of
$\mathbf{GRASP}$ and the fine-tuned RoBERTa models on the symmetrical
relations described in Section 5.2.


AIP conference proceedings of the

14th Asia-Pacific Physics Conference

# CIRCULAR GEODESICS IN KERR-NEWMAN-KASUYA BLACK HOLE

Saeed Ullah Khan and Jingli Ren
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001,
China<EMAIL_ADDRESS>author<EMAIL_ADDRESS>
###### Abstract

This article explores the characteristics of ergoregion, horizons and circular
geodesics around a Kerr-Newman-Kasuya black hole. We investigate the effect of
spin and dyonic charge parameters on ergoregion, event horizon and static
limit surface of the said black hole. We observed that both electric, as well
as magnetic charge parameters, results in decreasing the radii of event
horizon and static limit, whereas increasing the area of ergoregion. The
obtained results are compared with that acquired from Kerr and Schwarzschild
black holes. Moreover, we figured out the photons orbit of circular null
geodesics and studied the angular velocity of a particle within ergoregion.

_K_ eywords Null geodesics $\cdot$ Gravitation $\cdot$ Black hole Physics

###### Contents

  1. 1 Introduction
  2. 2 Equatorial circular geodesics
    1. 2.1 Ergoregion, horizons and the ergosurfaces
    2. 2.2 Circular null geodesics
  3. 3 Particle’s angular velocity within ergoregion
  4. 4 Conclusion

##  1 Introduction

Black holes (BHs), are among the most engaging and even strangest objects of
our universe, due to their extremely dense spacetime region they possess much
stronger gravitation attractions. The key property of a BH is to pull its
surrounding objects, termed is the accretion [1]. Therefore, they are
responsible for various interesting astrophysical phenomena: including but not
limited to the formation of accretion disc; jets and gamma-ray bursts [2]. The
ergoregion of a BH has an essential consequence in astrophysics, as Hawking
radiation could be analyzed in this region [3]. The region is also momentous
due to the occurrence of the Penrose process [4]. The techniques used for
energy extraction are of ample interest. Besides, the impact occurs within the
ergoregion of a BH plays an essential role to understand the central mechanism
of these processes [5]. A detailed analysis of the orbital circular motion
around a Kerr-Newman (KN) BH, can be found in [6]. Pugliese and Quevedo [7] by
considering Kerr spacetime on the equatorial plane, investigated test particle
motion. Khan et al. [8] studied the braneworld Kerr BH for the possible impact
of the tidal charge on ergoregion and particle dynamics.

Since in the last few decades, particle dynamics around BHs have been an
engaging problem, as they are of great interest and could convey eminent
information and reveal the prolific structure of background geometry. Among
various kinds of geodesics, the circular one is much engaging. They are very
useful for understanding and explaining the quasinormal modes (QNMs) of a BH
[9]. Chandrasekhar [10] was among the pioneers of geodesics investigators for
Schwarzschild, Reissner-Nordström (RN) and Kerr BHs. Soon after his work, the
study of geodesics caught a considerable interest of many researchers. Leiva
et al. [11] by considering the rainbow gravity studied the geodesic of
Schwarzschild BH. Cardoso et al. [12] by exploring both charged and uncharged
particles near a Kerr-Newman-Taub-NUT BH, explored all possible situation to
the existence of circular orbits.

##  2 Equatorial circular geodesics

This section is devoted to the exploring of particle dynamics near a Kerr-
Newman-Kasuya (KNK) BH. For obtaining geodesics on the equatorial plan
($\theta=\pi/2$), we utilize the neutral particle motion. The corresponding
space-time geometry to the exact solution of a rotating KNK BH, in Boyer-
Lindquist coordinates, can be expressed as [14]

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2},$
(1)

with

$\displaystyle g_{tt}=-\left(1-\frac{2Mr-e^{2}-Q^{2}}{\Sigma}\right),\quad
g_{t\phi}=-a{\sin^{2}\theta}\left(\frac{2Mr-e^{2}-Q^{2}}{\Sigma}\right),$
$\displaystyle g_{\phi\phi}=\left(\frac{(a^{2}+r^{2})^{2}-\Delta
a^{2}\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta,\quad
g_{rr}=\frac{\Sigma}{\Delta},\quad g_{\theta\theta}=\Sigma,$

and
$\Delta=r^{2}-2Mr+a^{2}+e^{2}+Q^{2},\,\,\Sigma=r^{2}+a^{2}\cos^{2}\theta\,$.
In which $M$, $e$ and $Q$, respectively represent mass, electric and magnetic
charge, whereas $a$ denotes rotational parameter of the BH. The metric (1),
reduces to the KN BH if $Q=0$; to the Kerr BH if $e=Q=0$; to the RN BH if
$a=Q=0$; and to the Schwarzschild BH by letting $a=e=Q=0$.

###  2.1 Ergoregion, horizons and the ergosurfaces

Similar to the Kerr BH, the metric (1) exactly have two horizons, namely the
event and Cauchy horizon. In BH physics, the static limit surface is of
considerable importance, as it changes the nature of a particle geodesics. We
can easily obtain both of the horizons and ergosurfaces by solving $\Delta=0$
and $g_{tt}=0$, respectively. As a result

Figure 1: Graphical description of ergoregion, event horizon (outer blue
dotted curves) and static limit (outer red curves) in the xz-plane, at various
points of the rotational and charge parameters $a$, $e$ and $Q$.

$\displaystyle r_{\pm}$ $\displaystyle=$ $\displaystyle
M\pm[M^{2}-(a^{2}+e^{2}+Q^{2})]^{1/2},$ (2) $\displaystyle r_{es\pm}$
$\displaystyle=$ $\displaystyle
M\pm[M^{2}-(a^{2}\cos^{2}\theta+e^{2}+Q^{2})]^{1/2},$ (3)

Equations (2) and (3), should satisfy the constraints of $r_{es-}\leq
r_{-}\leq r_{+}\leq r_{es+}$. In rotating BHs, the region
$\delta\equiv(r_{+},r_{es+})$ is known as ergoregion. On considering BH
spacetime, the $r$-coordinate is time-like in $r\in(r_{-},r_{+})$, while
space-like in $r\in(0,r_{-})\cup r>r_{+}$. This simply implies that the
surfaces of constant $r$, i.e., $\delta_{r}$, are space-like for $\Delta<0$,
time-like for $\Delta>0$ and light-like for $\Delta=0$ [4]. In extreme KNK BH,
i.e., for limiting value of the rotation parameter
$a=\pm\sqrt{M^{2}-e^{2}-Q^{2}}$, horizons of the KNK BH coincides and for
$cos^{2}\theta=1$, $r_{\pm}=r_{es\pm}$. Figure 1, reveals that both dyonic
charge and spin of the BH has a considerable effect on the structure and size
of ergoregion, horizons and static limit. Our investigation shows that
increase in $a$, $e$ and $Q$ diminishing the radii of event horizon and static
limit, whereas increases the area of ergoregion. One can also figure out the
special cases with coinciding horizons and no horizons, as shown in second row
third and fourth columns, respectively. Moreover, we observed smaller radii of
the event horizon and the static limit for KNK BH, but a greater area of the
ergoregion in comparison with that of RN, Kerr and KN spacetime.

###  2.2 Circular null geodesics

By making use of the Lagrangian equation, the equation governing geodesics
could be obtained as

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$ (4)

here $\dot{x}^{\mu}={dx^{\mu}}/{d\tau}$ denotes the four-velocity with dot
means $\partial/\partial\tau$ ($\tau$ is the proper time). Henceforth, by
following [10], the Lagrangian of KNK spacetime can be defined as

$\displaystyle 2\mathcal{L}$ $\displaystyle=$ $\displaystyle
g_{tt}{\dot{t}^{2}}+2g_{t\phi}\dot{t}\dot{\phi}+g_{\phi\phi}\dot{\phi^{2}}+g_{rr}{\dot{r}^{2}}.$
(5)

Equation (5), depicts that both $t$ and $\phi$ are cyclic coordinates with
respect to the constants of motions, i.e., the angular momentum $L$ and total
energy $E$, remains conserved along geodesics. Consequently, the corresponding
generalized momenta take the form

$\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle
g_{tt}\dot{t}+g_{t\phi}\dot{\phi}=-E,$ (6) $\displaystyle p_{\phi}$
$\displaystyle=$ $\displaystyle g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}=L,$
(7) $\displaystyle p_{r}$ $\displaystyle=$ $\displaystyle g_{rr}\dot{r}.$

In the case of KNK BH, the Hamiltonian will take the form

$\displaystyle
2H=-\left(g_{tt}\dot{t}+g_{t\phi}\dot{\phi}\right)\dot{t}+\left(g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}\right)\dot{\phi}+g_{rr}\dot{r}^{2}=-E\dot{t}+L\dot{\phi}+\frac{r^{2}}{\Delta}\dot{r}^{2}=\eta=constant,$
(8)

different values of $\eta$ specify various types of geodesics, i.e.,
$\eta=-1,0,1$ specify time-like, light-like and space-like geodesics,
respectively. By making use of Eqs. (6) and (7), one could obtain the radial
equation of motion

$\displaystyle\dot{r}^{2}=E^{2}+\frac{(a^{2}E^{2}-L^{2})}{r^{2}}+\frac{(aE-L)^{2}(2Mr-e^{2}-Q^{2})}{r^{4}}+\frac{\Delta}{r^{2}}\eta.$
(9)

The above result in Eq. (9), is of great interest, as it can be used to
discuss the various associated features of particle dynamics. In case of null
geodesics ($\eta=0$), the radial Eq. (9), could be expressed as

$\displaystyle\dot{r}^{2}=E^{2}+\frac{(a^{2}E^{2}-L^{2})}{r^{2}}+\frac{(aE-L)^{2}(2Mr-e^{2}-Q^{2})}{r^{4}}.$
(10)

Following [13], the equation of circular photons orbit can be acquired as

$\displaystyle r_{c}^{3}-3Mr_{c}^{2}+2(e^{2}+Q^{2})r_{c}\pm
2a\sqrt{Mr_{c}^{3}-(e^{2}+Q^{2})r_{c}^{2}}=0.$ (11)

Table 1: Photon orbits $r_{p}$ for co-rotating particle.

| $e$ | ${r_{p}}$ | a | ${r_{p}}$ |
---|---|---|---|---|---
$Q=0$ | $0.0$ | 2.493 | 0.0 | 2.823 |
$0.3$ | 2.421 | 0.2 | 2.568 |
$0.5$ | 2.281 | 0.4 | 2.281 |

| $e$ | ${r_{p}}$ | a | ${r_{p}}$
---|---|---|---|---
$Q=0.5$ | $0.0$ | 2.281 | 0.0 | 2.618
$0.3$ | 2.194 | 0.2 | 2.341
$0.5$ | 2.015 | 0.4 | 2.015

Table 2: Photon orbits $r_{p}$ for counter-rotating particle.

| $e$ | ${r_{p}}$ | a | ${r_{p}}$ |
---|---|---|---|---|---
$Q=0$ | $0.0$ | 3.432 | 0.0 | 2.823 |
$0.3$ | 3.376 | 0.2 | 3.056 |
$0.5$ | 3.272 | 0.4 | 3.272 |

| $e$ | ${r_{p}}$ | a | ${r_{p}}$
---|---|---|---|---
$Q=0.5$ | $0.0$ | 3.272 | 0.0 | 2.618
$0.3$ | 3.210 | 0.2 | 2.866
$0.5$ | 3.0932 | 0.4 | 3.093

To figure out the radius of photons orbit, we assume that $r_{c}=r_{p}$, be
the real positive root of Eq. (11), which is the nearest possible photons
orbit. By putting $Q=0$, $e=Q=0$ and $a=e=Q=0$, one could recovered the
photons orbit of KN, Kerr and Schwarzschild BH, respectively. The behaviour of
circular photon orbits is described in Tables 2 and 2, respectively for both
co-rotating and counter-rotating motion of particles.

##  3 Particle’s angular velocity within ergoregion

The current section is devoted to the study of a particle’s angular velocity
($\Omega=d\phi/dt$) and its limitations in the vicinity of ergoregion under
the restriction of $ds^{2}\geq 0$. As a result

$g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{\phi\phi}d\phi^{2}\geq 0,$ (12)

and it should satisfy the conditions of $\Omega_{+}\leq\Omega\leq\Omega_{-}$
[15], here

$\Omega_{\pm}={(-g_{t\phi}\pm\sqrt{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}})/{g_{\phi\phi}}.$
(13)

Figure 2: Graphical interpretation of angular velocity $\Omega$, at different
values of $a$, $e$ and $Q$ along with the radial distance $r$.

While on ergosphere boundary, $g_{tt}$ vanished and $\Omega_{+}$ =0, whereas
within ergosphere, $g_{tt}<0$ and $\Omega_{\pm}>0$, and each particle moves
along BH rotation [10]. In the case of KNK spacetime Eq. (13) simplifies to

$\displaystyle\Omega_{\pm}=\omega\pm{\Sigma\sqrt{\Delta}}/{\sin{\theta}\left[(a^{2}+r^{2})\Sigma+a^{2}\sin^{2}{\theta}(2Mr-e^{2}-Q^{2})\right]},$
(14)

with

$\displaystyle\omega={a\left(2Mr-e^{2}-Q^{2}\right)}/{\left[(a^{2}+r^{2})\Sigma+a^{2}\sin^{2}{\theta}\left(2Mr-e^{2}-Q^{2}\right)\right]}.$
(15)

On approaching the event horizon, we get

$\lim\limits_{r\to
r_{+}}\Omega_{\pm}=\omega_{BH}={a(2Mr_{+}-e^{2}-Q^{2})}/{\left[a^{2}r_{+}^{2}+r_{+}^{4}+a^{2}\left(2Mr_{+}-e^{2}-Q^{2}\right)\right]}.$
(16)

In the above equation, $\omega_{BH}$ is the angular velocity of BH rotation.
Following [16] the associated angular velocity of a particle at infinity turns
out to be

$\Omega={\pm\sqrt{Mr-e^{2}-Q^{2}}}/{(r^{2}\pm a\sqrt{Mr-e^{2}-Q^{2}})}.$ (17)

In Eq. (17), the upper and lower signs, respectively correspond to the co-
rotating and counter-rotating orbits. Figure 2, describes the graphical
interpretation of angular velocity $\Omega$, respectively at different values
of dyonic charge and spin parameter $a$ of the BH. The graphical description
reveals that angular velocity increases with the increase of dyonic charge,
while decreases as BH rotation increases.

##  4 Conclusion

In this manuscript, we carried out a detailed analysis of horizons structure
and particles circular geodesic near a KNK BH. The obtained result ensure that
both BH rotation and dyonic charge results in thicker the ergoregion, whereas
decreasing the event horizon and static limit. As a result, the spacetime
becomes denser. By making use of the radial equation of motion, we have
derived the circular photons orbit. We observed that BH charge decreases the
photons orbit of both direct and retrograde particles, while its spin
diminishing the photons orbit of direct particles and contributes to the
photons orbit of retrograde particles. Moreover, the angular velocity of a
particle within ergoregion of a KNK spacetime has been explored. Our finding
shows that angular velocity increases with dyonic charge, while decreases with
BH rotation.

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