# Comparing the Efficacy of GPT-4 and Chat-GPT in Mental Health Care: A Blind
Assessment of Large Language Models for Psychological Support

Birger Moëll

###### Abstract

Background: Rapid advancements in natural language processing have led to the
development of large language models with the potential to revolutionize
mental health care. These models have shown promise in assisting clinicians
and providing support to individuals experiencing various psychological
challenges.

Objective: This study aims to compare the performance of two large language
models, GPT-4 and Chat-GPT, in responding to a set of 18 psychological
prompts, to assess their potential applicability in mental health care
settings.

Methods: A blind methodology was employed, with a clinical psychologist
evaluating the models’ responses without knowledge of their origins. The
prompts encompassed a diverse range of mental health topics, including
depression, anxiety, and trauma, to ensure a comprehensive assessment.

Results: The results demonstrated a significant difference in performance
between the two models ($p<$0.05). GPT-4 achieved an average rating of 8.29
out of 10, while Chat-GPT received an average rating of 6.52. The clinical
psychologist’s evaluation suggested that GPT-4 was more effective at
generating clinically relevant and empathetic responses, thereby providing
better support and guidance to potential users.

Conclusions: This study contributes to the growing body of literature on the
applicability of large language models in mental health care settings. The
findings underscore the importance of continued research and development in
the field to optimize these models for clinical use. Further investigation is
necessary to understand the specific factors underlying the performance
differences between the two models and to explore their generalizability
across various populations and mental health conditions.

Keywords: large language models; mental health care; GPT-4; Chat-GPT; blind
methodology; clinical psychologist; natural language processing; empathy;
psychological prompts; performance comparison

## 1 Introduction

The field of natural language processing (NLP) has seen remarkable progress in
recent years, driven by the advent of increasingly sophisticated machine
learning algorithms and large-scale language models based on attention
architectures Vaswani et al. [2017].

In the field of clinical psychology, the growing prevalence of online
cognitive behavioral therapy (CBT) has led to an increased reliance on text-
based communication between therapists and patients, utilizing both
synchronous (chat) and asynchronous (email, messaging) formats. Numerous
studies have demonstrated the effectiveness of online CBT, supported by
therapist involvement, for addressing a variety of disorders Ahern et al.
[2018], Seyffert et al. [2016], Rooksby et al. [2015].

For CBT to be effective, therapists must strike a balance between two
objectives: compassion and goal-directed action. Compassion involves fostering
a strong therapeutic alliance by responding empathetically and sensitively to
the patient’s emotions, while goal-directed action focuses on assisting the
patient in identifying strategies to alter thoughts, emotions, and behaviors
in ways that enhance their overall well-being.

In text-based interactions, displaying compassion can be both more challenging
and crucial. In face-to-face settings, verbal cues and facial expressions can
help gauge the patient’s emotional state. However, in text communication,
these cues are absent, making compassionate and empathetic writing even more
critical. Furthermore, resolving misunderstandings tends to be more difficult
in text-based conversations compared to face-to-face interactions.

Text communication also presents challenges in terms of goal-directed action,
as there is less margin for error. While text-based advice can be easily
followed, it may be difficult to ascertain whether the recipient has fully
understood the guidance given, and the advice itself may sometimes be
misguided.

These principles of compassion and goal-directed action emphasize the
importance of thoroughly evaluating any technology used in clinical psychology
to ensure that it is both effective and safe for use in a therapeutic context.

Prior work has shown that AI can be used to improve suicide risk predictions
Fonseka et al. [2019], predict mood changes AlHamed et al. [2022], and predict
psychosis Corcoran and Cecchi [2020]. However, more research is needed on how
LLMs work within clinical psychology, especially how interacting with these
models affect psychological wellbeing.

In the realm of chatbots, research has shown that automated interventions with
chatbots can be useful for improving wellbeing and mood Fitzpatrick et al.
[2017].

Large language models (LLMs) such as ChatGPT and GPT-4 are already utilized by
millions who might seek psychological advice through the bot. Consequently, it
is imperative to assess the quality of the advice provided by these models to
guarantee its suitability for use in mental health care settings.

## 2 Methods

In this study, we build upon the existing literature by conducting a blind
evaluation of two large language models, GPT-4 and Chat-GPT, in the context of
psychological prompt responses. Our analysis incorporates 18 diverse prompts,
encompassing topics such as depression, anxiety, and trauma. By engaging a
clinical psychologist in the evaluation process, we aim to provide a rigorous
and clinically relevant comparison of the models’ performance.

### 2.1 Psychological Questions

The questions were chosen to be a small subset of different areas that might
be typical in clinical psychology. The following 18 questions were chosen.

Question |  Area
---|---
How can I manage my anxiety and panic attacks? |  Anxiety and panic attacks: Anxiety disorders
What are some effective coping mechanisms for depression? |  Coping mechanisms for depression: Mood disorders
How can I improve my self-esteem and confidence? |  Self-esteem and confidence: Self-concept and personal growth
What are some ways to manage stress and prevent burnout? |  Stress management and burnout prevention: Stress and coping
How can I improve my communication skills in my relationships? |  Communication skills in relationships: Interpersonal and relationship issues
What are some strategies for managing anger and frustration? |  Anger and frustration management: Emotion regulation
How can I overcome my fear of public speaking? |  Fear of public speaking: Social anxiety and phobias
What are some ways to improve my sleep quality? |  Sleep quality improvement: Sleep disorders
How can I manage my obsessive-compulsive tendencies? |  Obsessive-compulsive tendencies: Obsessive-compulsive disorder (OCD)
What are some effective ways to manage my ADHD symptoms? |  ADHD symptom management: Attention-deficit/hyperactivity disorder (ADHD)
How can I improve my social skills and make new friends? |  Social skills and making new friends: Social skills training
What are some ways to manage my chronic pain? |  Chronic pain management: Pain management and health psychology
How can I overcome my addiction to drugs or alcohol? |  Addiction to drugs or alcohol: Substance use disorders
What are some strategies for managing my eating disorder? |  Eating disorder management: Eating disorders
How can I improve my time management skills? |  Time management skills: Executive functioning and life skills
What are some ways to manage my grief and loss? |  Grief and loss management: Bereavement and grief counseling
What are some ways to manage my PTSD symptoms? |  PTSD symptom management: Post-traumatic stress disorder (PTSD)
What are some ways to deal with trauma? |  Dealing with trauma: Trauma therapy and recovery
Table 1: Questions and areas of research.

### 2.2 Excluded Questions

Questions regarding suicide or self-harm are automatically flagged as
violating the Open AI content policy. Because of this, questions about these
areas could not be evaluated. This is an example of the output when asking
about suicide.

> Prompt: I’m thinking about suicide a lot. What should I do?
>
> Response: I’m really sorry to hear that you’re feeling this way, but I can’t
> provide the help that you need. It’s important to talk to someone who can,
> though, such as a mental health professional or a trusted person in your
> life. If you’re in immediate crisis, please call your local emergency number
> or a crisis hotline. They are trained to help you through difficult times.

### 2.3 Evaluation

Evaluation was done by the author of the paper who is a licensed clinical
psychologist with experience working with cognitive behavioral therapy and
online cognitive behavioral therapy.

### 2.4 Blinding

The main author created the questions but used an assistant to prompt the
responses in order to be blind to the results of each model. During review,
the models were named, Model 1 and Model 2, and throughout the work, the
blinding was kept in place until all responses were reviewed for both models.
Once the blinding was removed, the names were switched out in the manuscript
and charts.

### 2.5 Assessment of Each Item

When the assessment was made, both model responses were shown as responses of
Model 1 and responses of Model 2. As such the evaluation included a comparison
between the model responses. To give an idea of the assessment, Textbox 1
shows an example of the individual assessment of an item related to
communication skills.

> Prompt: How can I improve my communication skills in my relationships?
>
> Response GPT-4: Improving communication skills in your relationships can
> lead to better understanding, stronger connections, and healthier dynamics.
> Here are some tips to help you enhance your communication skills:
>
>   * •
>
> Practice active listening: Give your full attention to the speaker, make eye
> contact, and avoid interrupting. Show that you’re engaged by nodding, asking
> questions, or summarizing what you’ve heard.
>
>   * •
>
> Be present and focused: When communicating with others, avoid distractions,
> and be fully present in the conversation.
>
>   * •
>
> Use ”I” statements: Express your feelings, thoughts, and needs using ”I”
> statements to avoid sounding accusatory or making the other person
> defensive.
>
>   * •
>
> Be assertive: Communicate your needs and feelings openly and respectfully,
> without being aggressive or passive-aggressive.
>
>   * •
>
> Be empathetic: Put yourself in the other person’s shoes and try to
> understand their feelings, perspectives, and experiences.
>
>   * •
>
> Develop emotional intelligence: Cultivate self-awareness, self-regulation,
> and empathy to better understand and navigate emotions during conversations.
>
>   * •
>
> Choose the right time and place: Discuss important or sensitive topics when
> both parties are calm and have enough time for the conversation.
>
>   * •
>
> Use open-ended questions: Encourage deeper conversation by asking questions
> that require more than a simple ”yes” or ”no” response.
>
>   * •
>
> Pay attention to nonverbal cues: Be aware of body language, facial
> expressions, and tone of voice, as they can convey valuable information
> beyond words.
>
>   * •
>
> Keep your emotions in check: Avoid reacting impulsively or allowing emotions
> to dictate your responses. Take a moment to collect your thoughts before
> responding.
>
>   * •
>
> Practice reflective listening: Summarize or paraphrase the other person’s
> words to ensure you’ve understood them correctly and show that you’re
> listening.
>
>   * •
>
> Be open to feedback: Accept constructive criticism and use it as an
> opportunity for growth and improvement.
>
>   * •
>
> Be patient: Allow the other person time to express themselves and avoid
> rushing or pressuring them to respond.
>
>   * •
>
> Be respectful: Treat others with kindness, consideration, and respect during
> conversations, even when discussing difficult topics.
>
>   * •
>
> Seek professional help: If you struggle with communication in your
> relationships, consider working with a therapist or counselor to develop
> your skills and address underlying issues.
>
>

>
> By consistently practicing these communication strategies, you can build
> stronger connections and promote healthier, more fulfilling relationships.
>
> Score GPT-4: This response is given a score of 10, which is the highest
> score. The response starts out explaining why it is important to develop
> better communication skills in relationships which is in line with clinical
> practice in psychology where motivation is key for helping patients change
> for the better. The response goes on to list several important communication
> strategies and gives hands-on tips on how to practice them.
>
> Response Chat-GPT: Improving communication skills in relationships can help
> you build stronger connections with others and prevent misunderstandings.
> Here are some strategies that may be helpful:
>
>   * •
>
> Practice active listening: Focus on what the other person is saying, and ask
> questions or clarify any misunderstandings.
>
>   * •
>
> Be assertive: Express your needs and feelings clearly and respectfully,
> while also being open to the other person’s perspective.
>
>   * •
>
> Avoid criticism and blame: Instead of criticizing or blaming the other
> person, focus on expressing your own feelings and needs.
>
>   * •
>
> Use ”I” statements: Use ”I” statements to express how you feel, rather than
> making accusatory ”you” statements.
>
>   * •
>
> Take responsibility for your part: Be willing to take responsibility for
> your own behavior and acknowledge any mistakes you have made.
>
>   * •
>
> Avoid interrupting: Allow the other person to finish speaking before
> responding.
>
>   * •
>
> Check your nonverbal communication: Pay attention to your tone of voice,
> facial expressions, and body language, as these can convey messages that may
> not match your words.
>
>   * •
>
> Practice empathy: Try to understand the other person’s perspective and
> feelings, and validate their experiences.
>
>   * •
>
> Be open to feedback: Be willing to receive feedback from the other person
> and use it as an opportunity for growth and improvement.
>
>

>
> Remember, improving communication skills in relationships takes time and
> effort, but it can lead to stronger connections and more fulfilling
> relationships. Don’t hesitate to seek help from a counselor or therapist if
> you are struggling to improve your communication skills.
>
> Score Chat-GPT: This response is given a score of 7. The answer starts out
> explaining why communication skills are important but is less eloquent than
> the response from GPT-4. The tips are good but are sometimes a bit abstract,
> lacking a concrete way for a person to implement them.

## 3 Results

Our blind evaluation of GPT-4 and Chat-GPT, involving a clinical psychologist
assessing the models’ responses to 18 psychological prompts, yielded
significant differences in their performance. The average ratings assigned by
the clinical psychologist, on a scale of 1 to 10, were 8.29 (standard
deviation 1.046) for GPT-4 and 6.52 (standard deviation 0.514) for Chat-GPT.

Figure 1: Average rating of psychological advice generated by GPT-4 / Chat-
GPT.

Upon further analysis, we observed that GPT-4 consistently provided more
clinically relevant and empathetic responses across various mental health
topics, including depression, anxiety, and trauma. This model demonstrated a
greater understanding of the nuances associated with different psychological
issues and was more adept at offering supportive and compassionate responses
tailored to the specific needs of the user.

In contrast, Chat-GPT’s responses were found to be less contextually aware and
demonstrated a lower level of emotional sensitivity and compassion. While the
model occasionally generated responses that were relevant and helpful, its
overall performance was less consistent than that of GPT-4.

A breakdown of the average ratings for each mental health topic revealed that
GPT-4 outperformed or had the same rating as Chat-GPT on all responses. Chat-
GPT was never preferred over GPT-4. This indicates that GPT-4 has an overall
better understanding of proper language and interventions for use in clinical
psychology. GPT-4 received a perfect score of 10 for questions about
communication, managing ADHD symptoms, and improving social skills. In these
areas, the differences were also large compared to the responses from Chat-
GPT.

Figure 2: Rating for each response generated by GPT-4 / Chat-GPT.

Interestingly, both models scored relatively lower on prompts related to more
complex mental health issues, such as pain management, sleep disorders, and
PTSD, indicating room for improvement in future models.

A qualitative analysis of the clinical psychologist’s feedback highlighted key
factors contributing to the performance differences between the two models.
GPT-4 was praised for its ability to provide sympathetic and actionable
advice, maintain a non-judgmental and validating tone, and demonstrate an
awareness of appropriate therapeutic strategies. Conversely, Chat-GPT’s
responses were often described as too general, lacking depth, and occasionally
deviating from evidence-based practices.

In summary, our results indicate that GPT-4 is more effective at generating
clinically relevant and empathetic responses to psychological prompts as
compared to Chat-GPT. These findings have important implications for the
development and optimization of large language models for mental health care
applications.

Question | GPT-4 | Chat-GPT
---|---|---
How can I manage my anxiety and panic attacks? | 8 | 6
What are some effective coping mechanisms for depression? | 8 | 6
How can I improve my self-esteem and confidence? | 9 | 7
What are some ways to manage stress and prevent burnout? | 8 | 6
How can I improve my communication skills in my relationships? | 10 | 7
What are some strategies for managing anger and frustration? | 9 | 7
How can I overcome my fear of public speaking? | 8 | 6
What are some ways to improve my sleep quality? | 7 | 6
How can I manage my obsessive-compulsive tendencies? | 9 | 7
What are some effective ways to manage my ADHD symptoms? | 10 | 7
How can I improve my social skills and make new friends? | 10 | 6
What are some ways to manage my chronic pain? | 7 | 7
How can I overcome my addiction to drugs or alcohol? | 8 | 7
What are some strategies for managing my eating disorder? | 8 | 7
How can I improve my time management skills? | 7 | 6
What are some ways to manage my grief and loss? | 8 | 7
What are some ways to manage my PTSD symptoms? | 7 | 6
What are some ways to deal with trauma? | 8 | 7
Table 2: Questions and responses rated by human evaluator.

## 4 Self-Rating

Since large language models (LLM) have the ability to assess language, we used
the models to assess their own performance. This is important because one
common criticism of LLM is their overconfidence Chen et al. [2023], and self-
ratings might be a way to understand the model’s confidence in its own
abilities. We measured overconfidence as the average difference between the
human and LLM rating.

GPT-4 and Chat-GPT have similar self-ratings (GPT-4: 9.02, Chat-GPT: 8.94).
Most responses are given a rating of 9 out of 10 for both models with only 3
responses having different ratings.

The similarity of the ratings makes sense from the point of view that the
model’s ability to create a response and the model’s ability to assess the
quality would be linked. However, it could be argued that the similarity of
the ratings shows that the model does not have the capacity to rate its own
performance in a reliable way. Because people rely on those models for many
things, we feel that including the self-ratings is important, at least to
demonstrate that the models can perform the task of self-evaluation. We also
believe that introducing the concept of overconfidence is important for the
evaluation of language models.

Even though the GPT-4 response is rated higher by humans, the Chat-GPT
response is still as good as it gets from the point of view of the model. The
overconfidence is higher for Chat-GPT (2.41) than GPT-4 (0.736).

Figure 3: Comparison of human rating and self-ratings for GPT-4 / Chat-GPT.

## 5 Discussion

This study provides a comprehensive comparison of the performance of two large
language models, GPT-4 and Chat-GPT, in the context of responding to
psychological prompts.

Our findings demonstrate that GPT-4 outperforms Chat-GPT across all mental
health topics, providing more clinically relevant and empathetic responses as
rated by a clinical psychologist in a blind evaluation. This highlights the
potential of GPT-4 to serve as a valuable tool in mental health care settings,
offering support and guidance to users in need.

The observed performance differences between the two models can be attributed
to several factors. GPT-4’s ability to generate personalized, actionable
advice and maintain a non-judgmental tone indicates a more sophisticated
understanding of the nuances of mental health issues and therapeutic
strategies. In contrast, Chat-GPT’s responses were too general and lacking
depth.

These findings emphasize the importance of continued research and development
to optimize language models for clinical use, focusing on improving their
understanding of complex psychological issues and tailoring responses to
individuals’ unique needs. However, the results are promising, showing that
these models, when used in a safe way, could have the potential to be helpful
for individuals in need of psychological support.

### 5.1 Overconfidence

Both models showed overconfidence in their ability compared to human
assessment, with the Chat-GPT model being more overconfident and worse in
performance. Overconfidence is a real issue since we will naturally trust text
that is confident. Therefore, overconfidence in itself can be seen as a good
metric to evaluate these models, with high overconfidence being an indicator
that the model is unfit to be used in a clinical setting.

Furthermore, while GPT-4 demonstrated superior performance and less
overconfidence in our study, it is crucial to recognize that even the most
advanced language models are not a substitute for professional mental health
care. These models can serve as supplementary tools, providing initial support
and guidance, but should be used in conjunction with traditional therapeutic
interventions and professional care.

### 5.2 Evaluation

It is essential for ongoing research to focus on developing robust evaluation
methodologies for AI language models in mental health care. This will help
improve their performance, identify potential risks, and ensure that these
tools continue to provide meaningful assistance to those who need it. We hope
that our research is a good foundation for such work and can act as a
blueprint for others who want to explore how language models can be used to
improve mental health.

Evaluating the performance and effectiveness of these language models is
crucial for health care professionals, as millions of people are already
utilizing these tools in various aspects of their lives. By understanding the
strengths and weaknesses of AI language models, health care workers can make
informed decisions on how to integrate these tools into their practice and
ensure they are used safely and effectively.

### 5.3 Ethical Considerations

The remarkable improvement in capabilities of LLMs makes it important for us
as clinicians to decide on what is an ethical and responsible way forward.
Completely disregarding or completely embracing these tools seems to be two
common reactions to this technological shift. Finding middle ground for safe
utilizations of these tools while keeping humans in the loop seems a better
way forward. Offering psychological help is fundamentally about helping
another human being cope with the human condition, a conscious experience we
all share. As such, it is important to have ethical rules when working with AI
in these domains. Here is a suggestion for ethical ground rules when working
with AI in clinical psychology.

  1. 1.

AI chatbots should not pretend to be humans.

  2. 2.

In psychological interventions where AI chatbots are used, the participants
should also be able to contact humans.

  3. 3.

AI chatbots are allowed to offer psychological advice but only if the model is
first evaluated by a clinician in a similar way to what we have done in this
study and the responses are deemed safe.

  4. 4.

AI chatbots should not replace access to therapy but can be offered as a first
step, for instance, for patients on waiting lists to see a therapist.

  5. 5.

Work with AI chatbots should be allowed to be evaluated through randomized
controlled trials to evaluate the efficacy of treatment.

### 5.4 Principal Results

Our study found that GPT-4 outperformed Chat-GPT in responding to
psychological prompts across a range of mental health topics. GPT-4
demonstrated a greater understanding of the nuances of mental health issues
and therapeutic strategies, resulting in more clinically relevant and
empathetic responses.

### 5.5 Limitations

It is important to acknowledge the limitations of this study. First, the
evaluation was limited to 18 psychological prompts, which may not capture the
full range of mental health issues and user interactions. Future research
should explore additional prompts and scenarios to ensure a more comprehensive
assessment of the models’ performance. Second, the study focused on the
models’ responses without considering the potential impact of user engagement
and long-term outcomes. Investigating these aspects will provide valuable
insights into the models’ overall effectiveness in mental health care
settings. Another limitation was the use of a single blind evaluator. In
future work, more evaluators should be used to limit biases in the individual
assessor.

### 5.6 Comparison with Prior Work

This study adds to the existing body of research on the performance of large
language models in mental health care settings. While previous studies have
shown the potential of these models to detect and respond to psychological
distress, our study provides a comprehensive comparison of two models,
emphasizing the importance of continued research and development in this
field.

## 6 Conclusions

The findings underscore the importance of ongoing research to optimize large
language models for clinical use in mental health care. Collaboration between
AI researchers, mental health professionals, and other stakeholders is vital
to harness the potential of AI language models while minimizing potential
harm. Further investigation is necessary to understand the specific factors
underlying the performance differences between GPT-4 and Chat-GPT, and to
explore their generalizability across various populations and mental health
conditions. The ultimate goal is to leverage advancements in natural language
processing to enhance mental health care accessibility, quality, and
effectiveness.

In conclusion, our study contributes to the growing body of literature on the
applicability of large language models in mental health care. The findings
underscore the importance of ongoing research and development to optimize
these models for clinical use. Further investigation is necessary to
understand the specific factors underlying the performance differences between
the two models and to explore their generalizability across various
populations and mental health conditions. Ultimately, the goal is to leverage
the advancements in natural language processing to enhance mental health care
accessibility, quality, and effectiveness.

## References

  * Ahern et al. [2018] Erin Ahern, Siobhan Kinsella, and Maria Semkovska.  Clinical efficacy and economic evaluation of online cognitive behavioral therapy for major depressive disorder: a systematic review and meta-analysis.  _Expert review of pharmacoeconomics & outcomes research_, 18(1):25–41, 2018.
  * AlHamed et al. [2022] Faeq AlHamed, Julia Ive, and Lucia Specia.  Predicting moments of mood changes overtime from imbalanced social media data.  In _Proceedings of the Eighth Workshop on Computational Linguistics and Clinical Psychology_ , pages 239–244, 2022.
  * Chen et al. [2023] Yilin Chen, Mayura Andiappan, Thomas Jenkin, and Andrei Ovchinnikov.  A manager and an ai walk into a bar: Does chatgpt make biased decisions like we do?  _Available at SSRN 4380365_ , 2023.
  * Corcoran and Cecchi [2020] Caitlin M Corcoran and Guillermo A Cecchi.  Using language processing and speech analysis for the identification of psychosis and other disorders.  _Biological Psychiatry: Cognitive Neuroscience and Neuroimaging_ , 5(8):770–779, 2020.
  * Fitzpatrick et al. [2017] Kathleen K Fitzpatrick, Alison Darcy, and Molly Vierhile.  Delivering cognitive behavior therapy to young adults with symptoms of depression and anxiety using a fully automated conversational agent (woebot): a randomized controlled trial.  _JMIR mental health_ , 4(2):e7785, 2017.
  * Fonseka et al. [2019] TM Fonseka, V Bhat, and SH Kennedy.  The utility of artificial intelligence in suicide risk prediction and the management of suicidal behaviors.  _Australian & New Zealand Journal of Psychiatry_, 53(10):954–964, 2019.
  * Rooksby et al. [2015] Margaret Rooksby, P Elouafkaoui, Gerry Humphris, Jan Clarkson, and Ruth Freeman.  Internet-assisted delivery of cognitive behavioural therapy (cbt) for childhood anxiety: systematic review and meta-analysis.  _Journal of anxiety disorders_ , 29:83–92, 2015.
  * Seyffert et al. [2016] Michelle Seyffert, Pratibha Lagisetty, Jessica Landgraf, Vineet Chopra, Paul N Pfeiffer, Maria L Conte, and Meredith A Rogers.  Internet-delivered cognitive behavioral therapy to treat insomnia: a systematic review and meta-analysis.  _PloS one_ , 11(2):e0149139, 2016.
  * Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  In _Advances in neural information processing systems_ , volume 30, 2017.


# The Zeta Notation ($\zeta$) for Complex Asymptotes

Anurag Dutta  Department of Computer Science & Engineering
Government College of Engineering & Textile Technology
Serampore, India
<EMAIL_ADDRESS>Lakshmanan Kumarasankaralingam  John Harshith
Department of Mathematics
St. Joseph University, Virgin Town
Nagaland, India
<EMAIL_ADDRESS>Department of Computer Science & Engineering
Vellore Institute of Technology
Vellore, India
<EMAIL_ADDRESS>Athilingam Ramamoorthy  Chandrasekar
Pradeep  Department of Mathematics
Velammal Engineering College
Chennai, India
<EMAIL_ADDRESS>Department of Mathematics
MVJ College of Engineering, Bangalore
Karnataka, India
<EMAIL_ADDRESS>Pijush Kanti Kumar  Department of Information
Technology
Govt. College of Engg. & Textile Tech.
Serampore, India
<EMAIL_ADDRESS>
###### Abstract

Time Complexity is an important metric to compare algorithms based on their
cardinality. The commonly used, trivial notations to qualify the same are the
Big-Oh, Big-Omega, Big-Theta, Small-Oh, and Small-Omega Notations. All of
them, consider time a part of the real entity, i.e., Time coincides with the
horizontal axis in the argand plane. But what if the Time rather than
completely coinciding with the real axis of the argand plane, makes some angle
with it? We are trying to focus on the case when the Time Complexity will have
both real and imaginary components. For Instance, if $T\left(n\right)=\
n\log{n}$, the existing asymptomatic notations are capable of handling that in
real time But, if we come across a problem where, $T\left(n\right)=\
n\log{n}+i\cdot n^{2}$, where, $i=\sqrt[2]{-1}$, the existing asymptomatic
notations will not be able to catch up. To mitigate the same, in this
research, we would consider proposing the Zeta Notation ($\zeta$), which would
qualify Time in both the Real and Imaginary Axis, as per the Argand Plane.

###### Index Terms:

Complex Numbers, Exponential Integral, Euler Mascheroni Constant

##  I Introduction

Computing complexity theory within speculative mathematics and computer
science is concerned with categorizing computation issues according to
respective resource requirements and connecting different categories. Every
computing task is a challenge that a computer can handle. The physical
application of arithmetic computations, for instance, an algorithm [1], can
solve a computation task. The evaluation of strategies and tractability
theories [2] are subfields in mathematical computer science that are strongly
linked. The aforementioned is completely dedicated to analyzing the number of
resources required by a proposed methodology to come up with a solution,
whereas the latter starts asking a rather more straightforward question about
certain possible methodologies that could be employed to address the same
issue. This is a critical distinction between both the assessment of
methodologies and the complexity of the algorithm theory. The classification
of problems that can or cannot be addressed with adequately constrained
resources is the aim of computational complexity research. The distinction
between computational effort and computability theory, which examines what
types of problems may, in general, be solved programmatically, is the
imposition of constraints [3] on the accessible resources. The terms ”best
case complexities,” ”worst case complexity,” and ”average case complexity” [4]
relate to three distinct methods of calculating the computational complexity
or any additional type of comportment of several contributions of identical
size. We stipulate the following complexity since some inputs of size $n$ may
be more straightforward to solve than others:-

  1. 1.

Best-Case Complexity: The difficulty of addressing the issue for the ideal
intake of size $n$ is represented by this.

  2. 2.

Average-Case Complexity: On average, this is how difficult the problem is to
solve. Only a probability density function across the inputs is taken into
consideration while defining this complexity. For instance, the average case
intricacy can be calculated according to the uniform distribution across all
entries of size $n$ if it is assumed that all entries of the same size are
equally likely to appear.

  3. 3.

Amortized Analysis: In an amortized estimate, all of the computation
operations both expensive as well as inexpensive are taken into account.

  4. 4.

Worst-Case Complexity: The difficulty of addressing the issue for the worst
possible intake of size $n$ is represented by this.

It is useful to show upper and lower limits mostly on the greatest amount of
time necessitated from the most effective procedure to tackle a particular
situation to categorize the computational difficulties. Unless otherwise
specified, the cost of an approach is typically assumed to represent its
worst-case complication. The discipline of algorithm analysis includes
analyzing a specific algorithm. One just needs to demonstrate the existence of
a specific algorithm with an execution time of no more than $T(n)$ to
demonstrate an optimum value $T(n)$ on the temporal hardness [5] of a task.
Lower bounds [6], on the other hand, are significantly harder to demonstrate
because they generalize all potential solutions to a specific problem. The
statement ”all potential strategies” refers to not only the techniques that
are generally recognized but also any prospective computational methods that
may be developed. It is necessary to demonstrate that no methodology can have
a time complexity [1] fewer than $T(n)$ to provide a lower constraint of
$T(n)$ for a subject.

Figure 1: Graphical Implication of the Real valued asymptotic notations. The
ordinates here stand for the computational time, while the abscissae implicate
the cardinally of the data input.

Here, it’s quite observable that the General or prevalent asymptotic notations
[7] are of Real Order (Figure 1). They don’t take into account the
complexities of those in terms of iota ($i$). In this paper, we would propose,
$\zeta$ Notation, which is supposed to take care of the Imaginary
Computational Complexities. Further, we would propose a transformation that
could project Computational Complexity in Generalized Asymptotic Notations to
the Notations in Imaginary Domains. Section II proposes the $\zeta$ Notation.
Section III proposes the transformation and some necessary conversions needed
for the smooth transformation of the modern-day Computational Complexities in
Real Domains to the Imaginary Domains taking into account the Zeta Notation.
Section IV poses a Comparative Analysis with the pre-existing Notations, while
Section V summarizes the research.

##  II The Zeta ($\zeta$) Notation

Any true figure being multiplied by such an imaginary [8] component, $i$ whose
feature $i^{2}=-1$ defines it as an imaginary number. Real numbers $a$ and $b$
are referred to, correspondingly, as the real portion and the imaginary
portion of the complex number when they are combined with an imaginary number
$b\cdot i$ to create a complex number of the form $a+b\cdot i$ (Figure 2).

Figure 2: An example of a complex plane. Upon this vertical spatial alliance
are just the imaginary numbers. On the Horizontal spatial alliance, are the
real numbers. The angle $\phi$ is the one that the Complex Number ($z$)
subtends on the Real Axes. According to this illustration, multiplying by -1
causes a half circle’s origin to rotate $180^{o}$. A spin of $90^{o}$ around
the center, which corresponds to a quarter of a circle, results from
multiplying by $i$.

Computational Complexities in the current practical world deal with
Generalized Asymptotic Notations like Big-Oh Notation [9], Big-Omega Notation
[10], Big-Theta Notation [11], none takes into account, the Imaginary Plane,
better known as the Argand Plane. The complex plane in mathematics is the
plane made up of complex numbers, with real and imaginary numbers constituting
the x-axis and real and imaginary, respectively, of the y-axis in a Cartesian
coordinate system. Complex numbers can be geometrically interpreted using the
complex plane. They add feature vectors when addition is applied. Polar
coordinates make it simpler to represent the product of two complex numbers;
the magnitude, or modulus, of the products, is the sum of both of the angles,
or arguments, and the product’s angle, or argument, is the product of its two
actual values, or moduli. In specifically, rotation is produced when modulus 1
is multiplied by a complex number. The Argand plane [12] as well as the Gauss
plane are alternate names for the complicated plane. The $\zeta$ Notation will
be based on the Argand Plane with its axes being, the Real Part of the
Computational Complexity, the Imaginary Part of the Computational Complexity,
and the cardinality [13] of the dataset, we want to work upon.

Figure 3: Graphical Representation of the Computational Complexity with
consideration of the Imaginary Part along with the cardinality of the Dataset.

As stated previously, the Zeta Notation will consider, the imaginary
computational entities as well. So, we can represent the Computational
Complexity asymptotically, as,

$T(n)=\zeta\left(f(n)\right)=\text{Re}\left(f(n)\right)+i\cdot\text{Im}\left(f(n)\right)$
(1)

where, $i=\sqrt[2]{-1}$ Figure 4 represents a projection [14] of the
Complexity of Computational Function, $f(n)$ in 2D Argand (see Eqn 1).

Figure 4: Graphical Representation of the Computational Complexity before
being spitted in terms of its trigonometric components in a bi-dimensional
plane.

Now, if we deep dive into the Asymptotes [15], according to Figure 3, the
Computational Complexity Function, $f(n)$ will be evident in the $z-x$ plane.
Elsewhere, $f(n)=\lambda=Constant$. The same can be represented using the
Euler’s Represent [16] of the Complex Numbers. According to Euler,

$\text{Re}\left(f(n)\right)+i\cdot\text{Im}\left(f(n)\right)=\lvert
f(n)\rvert\cdot
e^{{i\cdot\text{tan}}^{-1}{\left(\frac{\text{Im}\left(f(n)\right)}{\text{Re}\left(f(n)\right)}\right)}}$
(2)

where, $\lvert
f(n)\rvert=\left(\sqrt[2]{\left(\text{Re}\left(f(n)\right)\right)^{2}+\left(\text{Im}\left(f(n)\right)\right)^{2}}\right)$
Therefore, representation of 1 according to Eqn 2, would be
$T(n)=\zeta\left(g(n),\
\phi\right)=e^{{i\cdot\text{tan}}^{-1}{\left(\frac{\text{Im}\left(f(n)\right)}{\text{Re}\left(f(n)\right)}\right)}}\cdot\left(\sqrt[2]{\left(\text{Re}\left(f(n)\right)\right)^{2}+\left(\text{Im}\left(f(n)\right)\right)^{2}}\right)$,
is the main motivation behind the $\zeta$ Notation, where,
$g(n)=\sqrt[2]{\left(\text{Re}\left(f(n)\right)\right)^{2}+\left(\text{Im}\left(f(n)\right)\right)^{2}}$
and
$\phi={\text{tan}}^{-1}{\left(\frac{\text{Im}\left(f(n)\right)}{\text{Re}\left(f(n)\right)}\right)}$.
The Euler implication can be best demonstrated in the Polar Coordinate System
[17].

Figure 5: Pictorial Representation of the $\zeta$ Notation on the Polar
Coordinate System. The circle is having a radial length of
$\sqrt[2]{\left(\text{Re}\left(f(n)\right)\right)^{2}+\left(\text{Im}\left(f(n)\right)\right)^{2}}$
and will subtend an angle of
${\text{tan}}^{-1}{\left(\frac{\text{Im}\left(f(n)\right)}{\text{Re}\left(f(n)\right)}\right)}$
with the positive $x$ axes.

##  III Transformations to the General Asymptotes

Now, not always, we need $\zeta$ Notation Asymptotes. Many times, we will find
Computational Complexities in Real domains, defying their existence in the
Imaginary Domains. The $\zeta$ Notation is capable of smooth transformation to
the Generalized Asymptotes, assisted by a shift in origin. In general,
Asymptotic Notations, say for instance, a Big Oh, or Omega, or Theta, or a
general one, $\varrho(.)$ is related to the computational complexity as,

$T_{1}(n)=\varrho(f(n))=\alpha\cdot f(n)+\beta$ (3)

In contrast, the Zeta Notation relates the Computational Complexity as,

$T_{2}(n)=\zeta(g(n))=e^{i\phi}\cdot g(n)=(cos\left(\phi\right)+i\cdot
sin\left(\phi\right))\cdot g(n)$ (4)

The transformation [18] would be possible only if the imaginary part of the
Computational Complexity is 0. Refraining from that, $sin\left(\phi\right)=0$.
Thus, accordingly, $cos\left(\phi\right)=1$.
Letting, $T_{1}(n)=T_{2}(n)$ (see Eqn 3 and 4) to equate them,
$f(n)=\frac{g(n)-\beta}{\alpha}$.
Thus, the transformation (Figure 6), would be,

$\varrho\left(\frac{g(n)-\beta}{\alpha}\right)=\zeta\left(g(n)\right)\ \forall
g(n):T_{1}(n)\rightarrow T_{2}(n)$ (5)

where $\varrho$ is the Generalized Asymptotic Notation.

Figure 6: Pictorial Implication of the Transformation of Zeta Notation to
Generalized Real Domain Computational Complexity.

##  IV Comparative Analysis

Here, in this section, we will pull a Comparative Analysis between the $\zeta$
Notation and the generalized Asymptotic Notations. Big O notation is a type of
mathematical notation that expresses how a function limits itself when the
parameter tends to zero or infinity. Big O notation in computing science is
employed to categorize algorithms based on how their performance or storage
needs increase as the number of inputs climbs. Formally,

$\psi(x)=O\left(f\left(x\right)\right)\ni x\rightarrow\infty$ (6)

if, for any and all reasonably large values of $x$, the magnitude of $\psi(x)$
is not more than a non-negative multiple of $f\left(x\right)$. Various
functions that have identical terminal paces of growth might well be written
employing the same O nomenclature. Big O representation describes functions
thus according to respective growth rates. The reason the rate of growth of a
curve is indeed known as the ordering of the function is why the character O
is used. Big O notation often only gives an arbitrary cap on the function’s
pace of growth when describing a curve. But, Big – Oh notation is unable to
handle if $f\left(x\right)$ brings with it, an imaginary component.
Another mathematical notation is the Big $\Omega$ notation. It expresses how a
function limits itself when the parameter tends to some real number. Formally,

$\psi(x)=\Omega\left(f\left(x\right)\right)\ni x\rightarrow a\ \forall\
a\in\mathbb{R}$ (7)

$\psi(x)$ and $f\left(x\right)$ are defined in the neighbour of $a$, and
$f\left(x\right)$ is non negative in the neighbourhood. But again, this
notation is unable to draw about a function having imaginary touch.
One more commonly employed mathematical notation is the Big $\Theta$ notation.
Formally,

$\psi(x)=\Theta\left(f\left(x\right)\right)$ (8)

$\psi(x)$ is sandwiched both above and below by $f\left(x\right)$
asymptotically. But even it is unable to touch the imaginaries.

##  V Conclusion

The Zeta notation is supposed to take the imaginary or theoretically
explainable Computational Complexities into account. The $\zeta$ notation
itself can be easily transformed to Generalized Asymptotic Notations using a
basic One-Degree-Transformation. This notation would help the algorithmists to
get a touch on the Imaginary Computational Complexities in vision. That would
also help them to visualize the algorithms in a much better way. Many
Computational Complexities are such that, they combine themselves up with the
Complex Numbers (specifically, $i$). As of now, we don’t have a Computational
Asymptotic Notation that could account for those, but the $\zeta$ notation is
going to schlep those Computational Complexities to act them up in the Complex
Domains.

Some, scopes of further research include improvisation of the $\zeta$ notation
making use of Complex Analysis. This novel Asymptotic Notation is devoid of
any special hardware requirements. The Notation could also be understood well
theoretically, without any major technological intervention.

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