# Rule-Based Error Classification for Analyzing Differences in Frequent Errors
††thanks: This work was supported by the Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant Number JP23H03508.

Atsushi Shirafuji  University of Aizu
Aizu-Wakamatsu, Japan
<EMAIL_ADDRESS>Taku Matsumoto  Japan Atomic Energy Agency
Naraha, Futaba, Japan
<EMAIL_ADDRESS>Md Faizul Ibne Amin  University of Aizu
Aizu-Wakamatsu, Japan
<EMAIL_ADDRESS>Yutaka Watanobe  University of Aizu
Aizu-Wakamatsu, Japan
<EMAIL_ADDRESS>
###### Abstract

Finding and fixing errors is a time-consuming task not only for novice
programmers but also for expert programmers. Prior work has identified
frequent error patterns among various levels of programmers. However, the
differences in the tendencies between novices and experts have yet to be
revealed. From the knowledge of the frequent errors in each level of
programmers, instructors will be able to provide helpful advice for each level
of learners.

In this paper, we propose a rule-based error classification tool to classify
errors in code pairs consisting of wrong and correct programs. We classify
errors for 95,631 code pairs and identify 3.47 errors on average, which are
submitted by various levels of programmers on an online judge system. The
classified errors are used to analyze the differences in frequent errors
between novice and expert programmers.

The analyzed results show that, as for the same introductory problems, errors
made by novices are due to the lack of knowledge in programming, and the
mistakes are considered an essential part of the learning process. On the
other hand, errors made by experts are due to misunderstandings caused by the
carelessness of reading problems or the challenges of solving problems
differently than usual.

The proposed tool can be used to create error-labeled datasets and for further
code-related educational research.

###### Index Terms:

error classification, difference analysis, regular expression, programming
education.

##  I Introduction

Programming education has been mandatory for primary education in Japan since
2020, but educational inequality has become an issue. Some students have
difficulty accessing proper education due to problems in the setup of the
environment and a lack of instructors, especially in rural areas. Instructors
also struggle to prepare lecture materials and provide proper advice because
of the variation of knowledge and experience among learners.

Using an online judge (OJ) system has become increasingly popular as an online
education platform [1]. It allows learners in remote areas to learn
programming using the same platform without the setup of the environment. It
is used by various levels of programmers; novices learn programming, and
experts test their skills. Users are asked to write a program to solve the
chosen programming problem in a browser. They can virtually test the input and
output cases without calling any commands on their environment. It does not
require additional knowledge and allows the users to concentrate on learning
the programming language. The OJ system automatically judges the users’
submitted program and returns the verdict. The verdict is not only the binary
correctness; it provides more detailed failure
reasons111https://onlinejudge.u-aizu.ac.jp/judges_replies., e.g., Time Limit
Exceeded, or Memory Limit Exceeded. They often submit the wrong program (WA)
that contains errors222Several terms are often used synonymously: errors,
bugs, defects, faults, and vulnerabilities. We use errors in this paper.. They
fix the errors in the WA and re-submit it as an accepted program (AC).

One of the popular OJ systems, Aizu Online Judge (AOJ) [2], has approximately
8 million programs submitted by 100 thousand users for 3,000 programming
problems [3]. It also provides pre-note and post-note, which are a commentary
of the problem. The pre-note introduces the knowledge required to solve the
problem for struggling learners. The post-note presents an example solution
program and explains the standard approach to solving the problem. The
problems and the submissions on AOJ are often used for research purposes [4,
5, 6, 7, 8, 9] and contained in popular datasets such as CodeNet [10] and
CodeContests [11].

It is helpful for learners and instructors to know what kinds of programmers
make what kinds of errors frequently. Many prior works examined frequent or
common error patterns [12, 13, 14, 15, 8, 9, 16, 17, 18, 19, 20, 21, 22].
However, the differences in the tendencies between the levels of programmers
have yet to be revealed. Once the differences in the tendencies have been
revealed, learners will be able to pay attention to the frequent errors, and
they will be able to prevent making similar mistakes. In addition, instructors
will also be able to provide helpful advice about each problem for each
learner.

In this work, we analyze the differences in frequent errors between novice and
expert programmers. We propose a rule-based error classification tool that can
classify errors based on predefined rules in code pairs consisting of WA and
AC submitted by various levels of programmers on AOJ. We analyze the error
classification results based on the user levels.

The main contributions of our work are as follows.

  * •

We propose a rule-based error classification tool that can classify both
syntax and logic errors.

  * •

We show an analysis of the characteristic differences between novice and
expert programmers.

  * •

We construct an error-labeled dataset containing over 95 thousand code pairs
from 44 introductory programming problems.

The tool and the dataset can be used for further code-related educational
research, such as detecting potential errors or suggesting fixes for incorrect
programs.

The rest of this paper is organized as follows. Section II mentions related
work in this field. Section III introduces the method of classification and
analysis. Section IV presents and discusses the obtained experimental results.
Section V describes the limitations of this research. Finally, Section VI
concludes this paper.

##  II Related Work

There have been several approaches for detecting and classifying errors made
by programmers [13, 14, 15, 8, 9, 16, 17, 18, 19, 20, 21, 22]. In particular,
to promote programming education by revealing the common and frequent errors
made by novice programmers, classifying and analyzing errors for programming
education has been performed for decades [15, 8, 9, 16, 17, 18, 19, 20, 21,
22].

Denny et al. [15] analyzed syntax errors made by students and found common
errors. They identified that the most common syntax errors are cannot resolve
identifier, type mismatch, and missing semicolon. In addition, the cannot
resolve identifier and type mismatch are reported as the most time-consuming
errors; students spent 11.9 minutes and 10.0 minutes resolving the errors on
average, respectively. Osman et al. [13] found frequent bug-fix patterns in
code change histories from 717 open-source projects on GitHub written in Java.
To capture more essential parts of the code changes, the source code is
anonymized and normalized. They discovered that more than 70% of bug fixes are
less than four lines of code, and about 40% of the bug-fixing code changes are
recurrent. They reported that the most frequent patterns are missing null
checks, missing invocation, wrong name, and undue invocation. Kawabayashi et
al. [8] used K-means clustering to find frequent error patterns from code
pairs consisting of WA and AC programs written in C++ on AOJ. For example,
they identified that, in a problem that requires reversing the given string
and output in a line, the most frequent error is missing line feeds.

While automated tools to classify errors have been developed, human analysis
is still reliable for some purposes. Alzahrani et al. [20] manually analyzed
incorrect programs submitted by students and found common coding errors that
cause struggle.

Recently, error classification using machine learning and deep learning has
been examined. Li et al. [21] developed a compilation error classification
model using TextCNN [23].

Different from the mentioned prior work, our proposed rule-based tool
particularly aims to cover any syntax and logic errors in Python programs by
defining comprehensive rules.

##  III Method

The main phases to analyze the differences in frequent errors are divided into
the following: (1) data collection, (2) code normalization, (3) changes
extraction, (4) error classification, and (5) difference analysis.

Several prior works used compiler error messages to analyze errors [22, 16,
15, 21]. However, the error message analysis only focused on syntax errors,
which fail in compilation and output error messages. In this work, we aim to
analyze not only syntax errors but also logic errors, which work incorrectly
without crashing (e.g., incorrect algorithm or output format). To identify the
cause of these logic errors, we directly analyze the source code rather than
the error messages.

###  III-A Data Collection

In the data collection phase, we collect the Python 3 programs submitted on
AOJ and make code pairs consisting of WA and AC programs, as shown in Figure
1. We target Python in this work because recent work on code-related tasks
primarily targets Python. In this work, WA includes wrong answers, logic
errors, syntax errors, and time or memory limit exceeding.

Figure 1: Illustration of collecting code pairs consisting of wrong and
correct programs from the same user. AC indicates accepted (correct) programs.
WA, RE, and TLE indicate wrong programs, such as wrong answer, runtime error,
and time limit exceeded, respectively.

We only use the code pairs whose edit distance is less than 100. The code
pairs with a large edit distance are more likely to be submitted to a wrong
problem by mistake, or the changes in the code pair are not only the error
fixes but also additional implementations or major changes in algorithms. To
analyze the frequent errors more accurately, we exclude code pairs that
include such major changes. We employ the Levenshtein
library333https://github.com/maxbachmann/Levenshtein. to calculate the edit
distance of code pairs.

To analyze the user levels who submitted the program, we also collect
metadata, including submission date, total submissions count at the time,
attempts count to the problem, and whether it is the first acceptance.

###  III-B Code Normalization

Rule | Regular Expression | Rule Type | Description
---|---|---|---
missing output |  ^print s ( s.+? s )$|^{p}rin$ s ( s ) | INSERT | Add print statement
needless output | DELETE | Delete print statement
wrong output | LINE-REPLACE | Fix print statement
wrong value |  ^ |  |
d+ |  |  |
d+ |  |  |
. |  |  |
d* |  |  |
. |  |  |
d+ | TOKEN-REPLACE | Fix numeric literal (int or float) |
wrong convert list |  slist |  |
s |  |  |
( |  |  |
s.+? |  |  |
s |  |  |
)— |  |  |
smap |  |  |
s |  |  |
( |  |  |
s |  |  |
w+ |  |  |
s, |  |  |
s.+? |  |  |
s |  |  |
) | WITHIN-REPLACE | Fix conversion of list |

Table I: Examples of classification rules.

In the code normalization phase, to capture the abstract structure of the
source code to classify the errors more accurately, we normalize the source
code by removing unnecessary tokens. While several prior works [24, 25] used
the abstract syntax tree (AST) to capture a more abstract structure than the
normalized source code, the normalized source code can still capture
superficial edits.

For normalization, we first tokenize the source code into a token sequence. We
employ the tokenizer tool provided by the CodeNet Project [10]. By
tokenization, unnecessary tokens are removed, such as comments, white lines,
and white spaces. In addition, to keep the structure of the source code, the
tokens include control tokens, which are not visible to the user, such as
NEWLINE, INDENT, and DEDENT.

Variable and function names are often anonymized in the code normalization
phase [13, 6, 26]. However, in this work, we omit the anonymization to detect
the changes more accurately. Anonymizing variable and function names lose
significant information about variable or function names, and it will be
difficult to detect the changes in the definition or invocation of a variable
or function.

Figure 2: Example of code normalization. Comments, white lines, and white
spaces are removed after tokenization. Variable and function names are
retained (i.e., not anonymized). Control tokens (e.g., NEWLINE) are in italic.

As shown in Figure 2, the comments in the original program are removed, and
white lines and white spaces are excluded in the token sequence. The variable
$x$ defined in the original program also appears as $x$ without anonymization
in the token sequence. In addition, while the token sequence contains control
tokens (e.g., NEWLINE), these tokens are not visible to users.

###  III-C Changes Extraction

In the changes extraction phase, we compare the WA and AC of normalized code
pairs to extract the changed tokens. This phase also involves labeling the
changed type for each token. We employ the difflib
library444https://docs.python.org/3/library/difflib.html. to extract the
changes.

We first compare each line between the WA and AC. If the lines are the same,
all tokens of the lines are labeled EQUAL. Similarly, the added lines are
labeled INSERT, and the deleted lines are labeled DELETE. To identify which
tokens are changed in the line, we further compare each token if the lines are
different and likely to have been changed, and the added or deleted tokens are
labeled REPLACE.

Figure 3: Example of changes extraction

Figure 3 shows an example of the changes extraction. This change includes an
addition of the function invocation to convert a string to an integer. Since
this addition is an inline replacement of tokens, these tokens are labeled
REPLACE.

###  III-D Error Classification

In the error classification phase, we classify errors in the changed tokens by
applying the 55 rules we defined. We employ the regex
library555https://github.com/mrabarnett/mrab-regex. to use the regular
expression. This library supports additional features that the standard re
library does not support, such as the negative lookbehind with the different
lengths of alternatives.

The rules are divided into (1) INSERT, (2) DELETE, (3) LINE-REPLACE, (4)
WITHIN-REPLACE, and (5) TOKEN-REPLACE. The rules are used to detect what kind
of lines of code are added, deleted, or replaced for what purpose. INSERT and
DELETE rules are applied to the lines labeled INSERT or DELETE in the changes
extraction phase, LINE-REPLACE and WITHIN-REPLACE rules are applied to the
lines containing REPLACE tokens, and TOKEN-REPLACE rule is applied to the
tokens labeled REPLACE. Table III-B shows some examples of the rules and the
corresponding regular expressions.

The difference between LINE-REPLACE and WITHIN-REPLACE is that when a regular
expression matches a line, WITHIN-REPLACE requires an additional check whether
the REPLACE token is contained within the matched range. The defined regular
expressions of LINE-REPLACE rules contain ^ at the beginning and $ at the end
to match the whole line. Therefore, the regular expressions of LINE-REPLACE
rules already ensure that at least one REPLACE token is contained in the
matched range. In contrast, as for WITHIN-REPLACE rules, we further check
whether the matched range contains at least one REPLACE token because the
regular expressions of WITHIN-REPLACE rules can match the token sequence only
containing EQUAL tokens.

Figure 4: Example of error classification, which is classified as wrong
variable declaration (LINE-REPLACE) and wrong convert value (WITHIN-REPLACE)
because of missing int conversion of input value.

Figure 4 shows the example of error classification for LINE-REPLACE and
WITHIN-REPLACE. In this example, the wrong variable declaration, one of the
LINE-REPLACE rules, matches the whole line because it contains the = operator
to represent the variable declaration. In contrast, the wrong convert value,
one of the WITHIN-REPLACE rules, matches only a few tokens directly associated
with the int() conversion function.

We also define 21 summarized rules to analyze the summary of the errors. Table
II shows the complete list of summarized rules and the original rules that
belong to the summarized rule. For example, the addition, deletion, or fix of
print() invocation can be summarized as an output error. All the classified
rules are summarized after the classification while the original
classification results are also retained.

Summarized Rule | List of Classified Rules
---|---
output | missing output
needless output
wrong output
input | missing input
needless input
wrong input
convert variable | wrong join list
wrong convert list
wrong convert value
other function invocation | missing function invocation
needless function invocation
wrong function invocation
conditional statement | missing if statement
needless if statement
wrong if statement
missing else elif
needless else elif
wrong else elif
loop statement | missing for statement
needless for statement
wrong for statement
missing while statement
needless while statement
wrong while statement
for range | wrong range
break continue | missing break continue
needless break continue
wrong break continue
literal | wrong string
wrong value
| wrong boolean value
import | missing import
needless import
wrong import
variable declaration | missing variable declaration
needless variable declaration
wrong variable declaration
function definition | missing function definition
needless function definition
wrong function definition
missing return
needless return
wrong return
pass | missing pass
needless pass
wrong pass
comparison operator | wrong comparison operator
logical operator | wrong logical operator
arithmetic operator | wrong arithmetic operator
unpack operator | wrong unpack operator
other operator | wrong in operator
wrong assignment operator
index | wrong list index
list comprehension | wrong list comprehension
indent | wrong indent
Table II: List of summarized rules.

###  III-E Difference Analysis

In the difference analysis phase, we define novice and expert programmers and
analyze the difference in the frequent errors between the levels. We define
novice and expert programmers by the following criteria.

  * •

Novices: Only solved introductory problems, and the number of WA is more than
five times the number of AC (i.e., $|\text{WA}|>|\text{AC}|\times 5$).

  * •

Experts: Solved more than ten non-introductory problems.

Note that we set the number of errors classified in a code pair to up to one
for each type of error. Our early experiment has revealed that indent fixes
are sometimes made over many lines in a code pair, and function definition
fixes rarely appear many times in a code pair. Therefore, it may cause a bias
in the number of fixes. In addition, even if the problem tended to include
remarkable errors, there is a concern that the tremendous number of indent
fixes would drown them out. For these reasons, we exclude duplicated errors in
this analysis phase.

We first use the chi-square test to examine the differences in the errors. The
chi-square test is denoted as Formula 1, where $O$ is an observed frequency,
and $E$ is an expected frequency.

$\chi^{2}=\sum_{k=1}^{n}\frac{(O_{k}-E_{k})^{2}}{E_{k}}$ (1)

Our null hypothesis is that “no difference of occurrence exists in the errors
between novices and experts.” Therefore, our alternative hypothesis is that
“there exist differences of occurrence in the errors between novices and
experts.” We set the significance level $\alpha=0.05$. We reject the null
hypothesis and accept the alternative hypothesis if the p-value of the
$\chi^{2}$ is less than $\alpha$.

After the chi-square test, we use residual analysis to reveal the different
errors. The chi-square test can only test whether there exist differences.
Therefore, when we find a significant difference in the chi-square test, we
perform a residual analysis to test for the specific error that makes the
difference.

Standardized Pearson residual $r_{ij}$ is denoted as Formula 2 [27], where $i$
is a row variable representing the classified errors, $j$ is a column variable
representing novices or experts, $n_{i}$ is the row total number of
frequencies, $n_{j}$ is the column total number of frequencies, and $N$ is the
total number of frequencies.

$r_{ij}=\frac{O_{ij}-E_{ij}}{\sqrt{E_{ij}\cdot(1-n_{i}/N)\cdot(1-n_{j}/N)}}$
(2)

The standardized Pearson residuals play the same role as z-scores. Thus, we
perform a two-tailed test and calculate the p-values for each cell. In this
analysis, our null hypothesis is that “no difference of occurrence exists in
the error between novices and experts.” Therefore, our alternative hypothesis
is that “there exists a difference of occurrence in the error between novices
and experts.” When we find a significant difference in the residual analysis,
we conclude that the error causes the difference between novices and experts.

##  IV Results & Discussion

We conduct two types of experiments. The first experiment is to manually
analyze the classification results and check the validity of the
classification tool. The second experiment is to test the differences in the
errors for each introductory problem.

In the experiments, we use
ITP1666https://onlinejudge.u-aizu.ac.jp/courses/lesson/2/ITP1/all., which has
44 introductory problems available on AOJ, and the target source code is
written in Python 3.

The total number of submitted programs on ITP1 is 515,769, and the final
number of constructed valid code pairs is 95,631. Table III and Table IV show
the statistics of the constructed error-labeled dataset.

Name | Value
---|---
Source | AOJ ITP1
Language | Python 3
#Problems | 44
#Pairs | 95,631
#Users | 10,361
Avg. #Errors | 3.47 ($\pm$ 2.69)
Avg. Char-based Edit Distance | 13.54 ($\pm$ 15.23)
Avg. Token-based Edit Distance | 5.49 ($\pm$ 5.85)
Avg. Char-based Similarity | 90.94% ($\pm$ 11.13%)
Avg. Token-based Similarity | 89.44% ($\pm$ 12.51%)
Table III: Basic statistics of the error-labeled dataset. Name | WA Value | AC Value
---|---|---
Avg. #Lines | 8.02 ($\pm$ 7.78) | 8.04 ($\pm$ 7.76)
Avg. #Chars | 222.79 ($\pm$ 326.77) | 227.25 ($\pm$ 326.77)
Avg. #Tokens | 80.76 ($\pm$ 105.67) | 82.14 ($\pm$ 105.73)
Avg. Cyclomatic Complexity | 3.89 ($\pm$ 4.13) | 3.84 ($\pm$ 4.03)
Table IV: Additional statistics of the error-labeled dataset. WA Value and AC
Value indicate the value on average among the WA programs and AC programs,
respectively.

###  IV-A Manual Evaluation

As the evaluation of the classification tool, we randomly select 1,000 unique
code pairs and manually validate the classified errors. The selected 1,000
code pairs are classified as 3,704 errors (i.e., each code pair has 3.7 errors
on average). From our manual evaluation, 3,397 classified errors out of 3,704
are correct. Therefore, the classification accuracy of the proposed tool is
91.71%.

###  IV-B Differences

Problem |  Problem Specification | P-value | Classified Rules | P-value | Novices | Experts
---|---|---|---|---|---|---
ITP1_1_A |  Print ”Hello World”. | $<.001$ | other function invocation | $<.001$ | 1.09% | 3.85%
ITP1_1_B |  Calculate the cube of a given integer. | .0390 | input | .0245 | 10.70% | 7.02%
literal | .0339 | 14.11% | 10.17%
ITP1_4_B |  Calculate the area and circumference of a given circle radius. | $<.001$ | arithmetic operator | .0084 | 7.53% | 4.04%
literal | .0025 | 22.34% | 15.41%
convert variable | $<.001$ | 12.21% | 24.55%
ITP1_4_C |  Given two integers $a$ and $b$, and an operator $op$, print the value of $a\,op\,b$. | .0047 | arithmetic operator | .0261 | 19.59% | 13.28%
comparison operator | $<.001$ | 2.03% | 8.07%
ITP1_10_A |  Calculate the Euclidean distance between two points $P1(x_{1},y_{1})$ and $P2(x_{2},y_{2})$. | $<.001$ | arithmetic operator | $<.001$ | 16.67% | 3.09%
convert variable | .0400 | 30.95% | 47.46%
Table V: Results of the analysis for each problem that showed significant
differences. The third column indicates the p-value from the chi-square test,
and the fifth column indicates from the residual analysis. The Novices and
Experts indicate the ratio of the error frequency in the problem. The greater
ratio is in bold.

Table V shows the results of the analysis for each problem. The table only
includes problems and the summarized rule names that showed significant
differences between novices and experts in the chi-square test and the
residual analysis. In the table, the third column represents the p-value from
the chi-square test, and the fifth column represents the p-value from the
residual analysis. The sixth and seventh columns in the table represent the
frequency ratio in the problem. The greater ratio cell is marked as bold.

####  IV-B1 ITP1_1_A

In ITP1_1_A, we identify that experts tend to make errors of other function
invocation than novices. The other functions include functions that are not
listed in other classification rules, e.g., string manipulation and
mathematical functions. In this problem, invoking functions other than print()
is unnecessary because it only asks to output a specific string, “Hello
World”. The reason for this result is that some experts try to solve the
problem differently than usual by implementing their own print methods, such
as calling system functions. Moreover, since the problem is placed first and
easy to access, some submissions contain completely unrelated programs, such
as examining the execution environment of the OJ system, although the Terms of
Use777https://onlinejudge.u-aizu.ac.jp/term_of_use. prohibit such submissions.

####  IV-B2 ITP1_1_B

In ITP1_1_B, input error and literal error are more common among novices. It
results from trial and error by novices unfamiliar with the input() function
because it is the first problem novices need to receive a value from standard
input. Also, instead of receiving input, some novices directly write a numeric
literal given in the example case. The submission will receive a verdict of
the wrong answer since the value will be changed in the evaluation phase.

####  IV-B3 ITP1_4_B

In ITP1_4_B, arithmetic operator error and literal error are common among
novices, and convert variable error is common among experts. The problem
requires using $\pi$ to calculate the area of a given circle, and the error of
the result digit is allowed up to 0.00001.

As for novices, they often define $\pi$ as a numeric literal rather than using
the $\pi$ provided by the standard library (math.pi) due to their
unfamiliarity with the standard library. In many cases, the $\pi$ defined by
novices does not have enough digits (e.g., $\pi=3.14$), leading to incorrect
results due to the large error in the calculation.

As for experts, because many of the problems on AOJ require conversions from
string to an integer, some experts write an int conversion out of habit.
However, the input constraint is a real number in this problem, and they need
to write a float conversion instead.

####  IV-B4 ITP1_4_C

In ITP1_4_C, arithmetic operator error is common among novices, and comparison
operator error is common among experts.

As for novices, the arithmetic operator error is because of the difference
between the / and // operators. The difference between the two division
operators in Python is that (1) / is to divide as a float and retain the
fractional part, and (2) // is to divide as an integer and truncate the
fractional part. The / operator is typical to divide values in mathematics and
many programming languages, and novices use it. However, because the problem
requires truncating any fractional part, using the / operator will result
wrong.

As for experts, they misunderstand the terminal condition of the input, and it
results in the high frequency of fixing the comparison operator. The problem
provides the terminal condition in the constraints section, but the condition
can be estimated by referring to the input/output examples in the problem.
However, the actual condition is slightly different from the condition
estimated from the examples, and it results in the fixing comparison operator.

####  IV-B5 ITP1_10_A

In ITP1_10_A, arithmetic operator error is common among novices, and convert
variable error is common among experts.

As for novices, the arithmetic operator error is because of the difference
between the ^ and ** operators. The problem requires calculating Euclidean
distance, which is denoted as $d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$.
The square of $x$ can be denoted as `x^2` in mathematics, and the novices
write the same notation in Python code. However, the ^ operator is a bit XOR
operator in Python, and the ** operator is the correct operator.

The reason for the convert variable error among experts is the same as the
ITP1_4_B; experts write an int conversion out of habit.

##  V Limitation

The set of problems used in this work, ITP1, is mainly solved by novice
programmers as it is an introductory course. There is a difference in the
number of samples between novices and experts because novices submit more
programs to introductory problems than experts, and experts make fewer errors
in such introductory problems than novices. However, for the same reason, the
number of submissions will be larger for experts when we choose difficult
problems. Although AOJ is used by various levels of programmers, there are not
so many problems equally solved by all levels of users because the users
choose the problems that match their capability. For future work, asking
several levels of users to solve the same problem can make more significant
differences to observe.

##  VI Conclusion

In this paper, we have proposed a rule-based error classification tool and
analyzed its results. The proposed tool can classify errors with 91.71%
accuracy according to our manual evaluation of the randomly selected
classified errors. The analyzed results showed that the errors made by novices
are due to their fundamental lack of knowledge about the existence, usage, or
specification of the functions. On the other hand, the errors made by experts
are due to misunderstandings caused by not reading the problems carefully. For
future work, we expect to be used for further code-related tasks, such as
detecting potential errors and suggesting fixes for programmers using the
proposed tool or the labeled dataset.

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###### Abstract

Our focus is on a specific type-N space-time that exhibits closed time-like
curves in general relativity theory within the framework of Ricci-inverse
gravity model. The matter-energy content is solely composed of a pure
radiation field, and it adheres to the energy conditions while featuring a
negative cosmological constant. One of the key findings in this investigation
is the non-zero determinant of the Ricci tensor ($R_{\mu\nu}$), which implies
the existence of an anti-curvature tensor ($A^{\mu\nu}$) and, as a
consequence, an anti-curvature scalar ($A\neq R^{-1}$). Furthermore, we
establish that this type-N space-time serves as a solution within modified
gravity theories via the Ricci-inverse model, which involves adjustments to
the cosmological constant ($\Lambda$) and the energy density ($\rho$) of the
radiation field expressed in terms of a coupling constant. As a result, our
findings suggest that causality violations remain possible within the
framework of this Ricci-inverse gravity model, alongside the predictions of
general relativity.

Cosmological constant Petrov type-N space-time in Ricci-inverse gravity

F<EMAIL_ADDRESS>
Department of Physics, University of Science & Technology Meghalaya,Ri-Bhoi,
793101, India
J. C. R. de<EMAIL_ADDRESS>and A. F.
<EMAIL_ADDRESS>(Corresp. author)
Instituto de Física, Universidade Federal de Mato Grosso, Cuiabá, Mato Grosso
78060-900, Brazil

Keywords: Exact solutions; modified gravity theories; null dust ; cosmological
constant

PACS number(s): 04.20.Jb; 04.50.Kd; 98.80.Es;

##  1 Introduction

The solutions of Einstein’s field equations that exhibit intriguing features
have garnered acceptance within the scientific community. These solutions are
systematically categorized using the Petrov classification scheme. Since
curvature is a localized property of space-time, the Petrov type provides
insight into the local algebraic characteristics of the space-time geometry.
Following this scheme [1, 2] involves first establishing a set of null tetrad
vector fields denoted as $(\mathbf{k},\mathbf{l},\mathbf{m},\bar{\mathbf{m}})$
for a given space-time and subsequently determining the Weyl scalars
$\Psi_{i}$ ($i=0,1,2,3,4$) [3, 4]. In the context of Petrov type-N space-time,
the only non-zero Weyl scalar is $\Psi_{4}\neq 0$, and the Weyl tensor
satisfies the condition $C_{\mu\nu\rho\sigma}\,k^{\sigma}=0$, where
$k^{\sigma}$ is the quadruple principal null direction (PND) along which the
gravitational radiation propagates. The Petrov classification of the Weyl
scalars and the asymptotic forms of radiative fields originating from
spatially bounded sources underscores the fundamental role played by type-N
solutions in gravitational radiation theory (for a comprehensive review, see
[5, 6]). All solutions of the vacuum Einstein equations, including those with
a non-zero cosmological constant $\Lambda$, falling under type N and featuring
a non-twisting null geodesic congruence, are well-established [3, 4, 7, 8, 9,
10, 11, 12, 13, 14]. Additional non-vacuum solutions, both without and with a
cosmological constant, within the framework of Petrov type-N space-times with
non-twisting, non-expanding, and shear-free geodesic null congruence, are
constructed in [15, 16, 17]. Recent investigations have extended to include
twisting type-N vacuum solutions with a nonzero cosmological constant [18,
19]. Furthermore, type N universal space-time has also attracted attention in
recent times (see, refs. [20, 21, 22]). In type-N space-time without twisting,
a non-expanding and shear-free geodesic null vector field emerges as the
conduit for gravitational wave propagation. These gravitational wave space-
times are either plane-fronted gravitational waves with parallel rays (called
pp-wave for non-vacuum solutions of the field equations) or plane-wave (vacuum
solutions) space-times, belonging to the Kundt class. The confirmation of
gravitational waves, or ripples in space and time, resulting from the merger
of black holes by the LIGO scientific collaboration in 2015, validated the
success of general relativity in contemporary physics. Thus, Petrov type-N
space-time holds significant importance in the realm of gravitational wave
theory.

One of the biggest problems in modern cosmology is the current accelerated
expansion of the universe. There is strong observational data confirming this
phenomenon [23, 24, 25]. One possibility to understand this accelerated phase
is add a new exotic component, called dark energy. This component has a
negative pressure that causes gravity to behave repulsively on large
cosmological scales [26]. The best-known candidate for dark energy is the
cosmological constant $\Lambda$. Taking this component and assuming the
existence of dark matter, the $\Lambda$CDM model has been proposed. Although
it is a very popular model and can successfully explain the observational
results, it is not without problems. The main problem is the famous
cosmological constant problem [27]. Another way to explain the cosmological
observations is to propose modifications to Einstein’s general relativity [28,
29, 30, 31, 32]. The simplest manner to construct alternative theories of
general relativity is to include an additional term in the Einstein-Hilbert
Lagrangian or to modify the structure of the Lagrangian which implies
modifying the Ricci scalar itself. Among the various modified gravity theories
existing in the literature, there is a recent model, called Ricci-inverse
gravity, which has attracted attention. In this paper, this model is
considered.

Ricci-inverse gravity [33] is an alternative theory to general relativity that
modifies the Einstein-Hilbert action by introducing a geometrical object
called anti-curvature scalar $A$. The anti-curvature scalar is the trace of
the anti-curvature tensor denoted as $A^{\mu\nu}$, which is defined as the
inverse of the Ricci tensor $R^{\mu\nu}$. The anti-curvature tensor satisfies
the condition $A^{\mu\sigma}R_{\sigma\nu}=\delta^{\mu}_{\nu}$. It is important
to note that the anti-curvature scalar $A$ is not the inverse of the Ricci
scalar $R$. There are two classes that define this model: class I,
characterized by the function $f(R,A)$ that depends on Ricci and anti-
curvature scalars, and class II, given by the function $f(R,A^{2})$ which
depends on the square of the anti-curvature tensor. Here it is considered
class I. In recent years, some investigations have been developed using this
gravitational theory. For example, no-go theorem for inflation has been
investigated [34, 35], cosmic structure has been studied [36], evolution from
matter-dominated epoch to accelerated expansion epoch has been analyzed [37],
the matter-antimatter asymmetry through baryogenesis in the realm of $f(R,A)$
theory of gravity has been discussed [38], the causality issue using an
axially symmetric space-time has been investigated [39], anisotropic stellar
structures has been explored [40], among others applications.

In this paper, we aim to investigate a type-N space-time in general
relativity, possessing a non-twisting, non-expanding, and shear-free geodesic
null congruence within the framework of Ricci-inverse gravity. For that we
consider a specific example: pure radiation field space-time characterized by
a negative cosmological constant that was investigated in [17]. This space-
time represents pp-wave non-vacuum solution of the field equations satisfying
the energy conditions. In general relativity, this solution represents an
exact solution of the field equations and exhibits closed time-like curves at
an instant of time, thus violating the causality condition. Our study reveals
that this also serve as a solution within the framework of Ricci-inverse
gravity, where the cosmological constant $\Lambda$ is replaced by a modified
value $\Lambda_{\mbox{mod}}$, and the energy-density of the radiation field,
$\rho$, is altered to $\rho_{\mbox{mod}}$, which is connected to a coupling
parameter. Consequently, we demonstrate that the violation of causality is
permitted in this modified gravity theory as well. It is worthwhile mentioning
that this particular type-N space-time is a solution in Ricci-inverse gravity
since the determinant of the Ricci tensor is nonzero, $\det(R_{\mu\nu})\neq 0$
which ensures the existence of an anti-curvature tensor $A^{\mu\nu}$ which is
symmetric in nature analogue to the metric tensor or the Ricci tensor.

This paper is organized as following. In Section 2, the cosmological constant
type-N space-time with causality violation is introduced. Considering pure
radiation as matter content, it is shown that this metric is the solution in
general relativity obtaining a negative cosmological constant and a positive
energy density. Next, this metric is investigated in Ricci-inverse gravity.
Some conclusions and remarks are presented in Section 3.

##  2 Cosmological constant type-N space-time with causality violation in
Ricci-inverse gravity

In this section, we examine an example of a type-N pure radiation field
solution derived from Einstein’s field equations in the context of general
relativity. The matter content in this scenario consists of a pure radiation
field that satisfies the energy conditions and includes a negative
cosmological constant. Notably, this type-N space-time permits the existence
of closed time-like curves within time-like regions, thus violating the
causality condition. We take this type-N space-time and investigate it within
the framework of modified gravity theories using the Ricci-inverse approach.

The line-element that describes this type-N space-time in cylindrical
coordinates ($t,r,\phi,z$) is given by the following expression (assuming
$c=1$ and $8\pi G=1$) [17]:

$ds^{2}=g_{rr}\,dr^{2}+2\,g_{t\phi}\,dt\,d\phi+g_{\phi\phi}\,d\phi^{2}+2\,g_{z\phi}\,dz\,d\phi+g_{zz}\,dz^{2},$
(1)

with different components of the metric tensor $g_{\mu\nu}$ given by

$\displaystyle g_{t\phi}=-\frac{1}{2}\,\cosh t\,\sinh^{2}(\alpha\,r),$
$\displaystyle g_{rr}=\mbox{coth}^{2}(\alpha\,r),$ $\displaystyle
g_{\phi\phi}=-\sinh t\,\sinh^{2}(\alpha\,r),$ $\displaystyle
g_{zz}=\sinh^{2}(\alpha\,r),$ $\displaystyle
g_{z\phi}=\beta\,z\,\sinh^{2}(\alpha\,r),$ (2)

where $\alpha>0,\beta>0$ are arbitrary positive constants.

To convert this metric (1) with (2) into a standard form, we perform
transformations as follows

$t\to\sinh^{-1}(\tau),\quad
r\to\frac{1}{\alpha}\,\sinh^{-1}(\alpha\,\varrho),$ (3)

into the metric (1), we obtain the following line-element in the chart
$(\tau,\varrho,\phi,z)$ given by

$ds^{2}=\frac{d\varrho^{2}}{\alpha^{2}\,\varrho^{2}}+\alpha^{2}\,\varrho^{2}\,\Big{(}-d\tau\,d\phi-\tau\,d\phi^{2}+2\,\beta\,z\,dz\,d\phi+dz^{2}\Big{)}.$
(4)

Our primary objective is to investigate this metric (4) within the context of
Ricci-inverse gravity. This choice is motivated by the fact that the
determinant of the Ricci tensor for the space-time described by the metric (4)
is non-zero, as we will discuss in detail below. Before that, we first review
this metric in the context of general relativity and then will investigate
within the frame of modified theories of gravity.

The covariant metric tensor $g_{\mu\nu}$ and its contravariant form
$g^{\mu\nu}$ for the metric (4) are given by

$\displaystyle
g_{\mu\nu}=\begin{pmatrix}0&0&-\frac{\alpha^{2}\,\varrho^{2}}{2}&0\\\
0&\frac{1}{\alpha^{2}\,\varrho^{2}}&0&0\\\
-\frac{\alpha^{2}\,\varrho^{2}}{2}&0&-\alpha^{2}\,\varrho^{2}\,\tau&\beta\,z\,\alpha^{2}\,\varrho^{2}\\\
0&0&\beta\,z\,\alpha^{2}\,\varrho^{2}&\alpha^{2}\,\varrho^{2}\end{pmatrix},$
$\displaystyle
g^{\mu\nu}=\begin{pmatrix}\frac{4\,(\tau+\beta^{2}\,z^{2})}{\alpha^{2}\,\varrho^{2}}&0&-\frac{2}{\alpha^{2}\,\varrho^{2}}&\frac{2\,\beta\,z}{\alpha^{2}\,\varrho^{2}}\\\
0&\alpha^{2}\,\varrho^{2}&0&0\\\ -\frac{2}{\alpha^{2}\,\varrho^{2}}&0&0&0\\\
\frac{2\,\beta\,z}{\alpha^{2}\,\varrho^{2}}&0&0&\frac{1}{\alpha^{2}\,\varrho^{2}}\end{pmatrix}.$
(5)

The covariant Ricci tensor $R_{\mu\nu}$ and its contravariant form for the
metric (4) are given

$\displaystyle
R_{\mu\nu}=\begin{pmatrix}0&0&\frac{3\,\alpha^{4}\,\varrho^{2}}{2}&0\\\
0&-\frac{3}{\varrho^{2}}&0&0\\\
\frac{3\,\alpha^{4}\,\varrho^{2}}{2}&0&\beta+3\,\tau\,\alpha^{4}\,\varrho^{2}&-3\,\alpha^{4}\,\beta\,z\,\varrho^{2}\\\
0&0&-3\,\alpha^{4}\,\beta\,z\,\varrho^{2}&-3\,\alpha^{4}\,\varrho^{2}\end{pmatrix},$
$\displaystyle
R^{\mu\nu}=\begin{pmatrix}\frac{4}{\varrho^{4}}\,\big{(}\frac{\beta}{\alpha^{4}}-3\,\varrho^{2}\,(\tau+\beta^{2}\,z^{2})\big{)}&0&\frac{6}{\varrho^{2}}&-\frac{6\,\beta\,z}{\varrho^{2}}\\\
0&-3\,\alpha^{4}\,\varrho^{2}&0&0\\\ \frac{6}{\varrho^{2}}&0&0&0\\\
-\frac{6\,\beta\,z}{\varrho^{2}}&0&0&-\frac{3}{\varrho^{2}}\end{pmatrix}.$ (6)

Finally, the Ricci scalar for metric (4) is given by

$R=g_{\mu\nu}\,R^{\mu\nu}=-12\,\alpha^{2}.$ (7)

The non-zero components of the Einstein tensor $G_{\mu\nu}$ are

$\displaystyle
G_{\tau\phi}=-\frac{3\,\alpha^{4}\,\varrho^{2}}{2}=3\,\alpha^{2}\,g_{\tau\phi},$
$\displaystyle
G_{\varrho\varrho}=\frac{3}{\varrho^{2}}=3\,\alpha^{2}\,g_{\varrho\varrho},$
$\displaystyle
G_{\phi\phi}=-3\,\tau\,\alpha^{4}\,\varrho^{2}+\beta=3\,\alpha^{2}\,g_{\phi\phi}+\beta,$
$\displaystyle G_{\phi
z}=3\,\alpha^{4}\,\varrho^{2}\,\beta\,z=3\,\alpha^{2}\,g_{\phi z},$
$\displaystyle G_{zz}=3\,\alpha^{4}\,\varrho^{2}=3\,\alpha^{2}\,g_{zz}\,.$ (8)

The Einstein field equations with a cosmological constant and the energy-
momentum tensor $\mathcal{T}^{\mu\nu}$ are given by

$G_{\mu\nu}+\Lambda\,g_{\mu\nu}=\mathcal{T}_{\mu\nu},$ (9)

where the right hand side is given by

$\mathcal{T}_{\mu\nu}=\rho\,k_{\mu}\,k_{\nu},\quad\mathcal{T}^{\mu}_{\,\,\mu}=0,$
(10)

with $\rho$ being the energy-density of a pure radiation field, and
$k_{\mu}=\delta^{\phi}_{\mu}=(0,0,1,0)$ is a null vector that satisfies the
relation $k^{\mu}\,k_{\mu}=0$. The non-zero component of the energy-momentum
tensor is $\mathcal{T}_{\phi\phi}=\rho$.

Using the metric tensor $g_{\mu\nu}$ given by (5), the Einstein tensor
$G_{\mu\nu}$ by (8), and the energy-momentum tensor (10) into the equations
(9), one will find the following physical quantities given by

$\displaystyle\Lambda=-3\,\alpha^{2},$ $\displaystyle\rho=\beta>0.$ (11)

Thus, space-time (4) is an exact solution of the field equations in general
relativity with matter content a pure radiation field having constant energy-
density and a negative cosmological constant. To determine type of the chosen
space-time (4) using the Petrov classification scheme, one can use the
Newman–Penrose formalism and construct a set of null tetrad vectors $(\bf
k,l,m,\bar{m})$ [3]. One can easily show that the only Weyl scalars
$\Psi_{4}\neq 0$ and the rest are all equal to zero, which indicates that the
chosen metric is of the Petrov type-N. The null vector field ${\bf k}$
satisfies the geodesic null congruence, that is, $k_{\mu;\nu}\,k^{\nu}=0$.
Thus, the space-time we have chosen here admits a non-expanding, non-twisting,
and shear-free geodesic congruence.

Now, we show that this space-time admits closed causal curves. For that, let
us consider curves defined by

$\varrho=\varrho_{0},\quad z=z_{0},$ (12)

where $\varrho_{0},z_{0}$ are constants. Therefore, the space-time (4) reduces
to 2D Misner-like space [42, 43] given by

$ds^{2}_{conf-Misner}=\Omega_{0}\,(-d\tau\,d\phi-\tau\,d\phi^{2}),$ (13)

where $\Omega_{0}=\alpha^{2}\,\varrho^{2}_{0}$ is the conformal constant
factor.

The Misner space is a $2D$ space-time with the metric [44]

$ds^{2}_{Misner}=-2\,dT\,d\psi-T\,d\psi^{2},$ (14)

where $-\infty<T<+\infty$ and the coordinate $\psi$ is periodic, that is,
$\psi\to\psi+2\,n\,\pi$ with $n=0,\pm\,1,\pm\,2,....$. The curves
$T=const=T_{0}$ are all closed due to the periodicity of $\psi$. The curves
$T_{0}<0$ are spacelike and $T_{0}>0$ are time-like. It then follows that all
points at $T_{0}>0$ rest on closed time-like curves (CTCs) but those at
$T_{0}<0$ do not. Hence, the Misner space admits closed time-like at an
instant of time, $T=const=T_{0}>0$.

In our case, for the chosen space-time (4), the closed curves defined by
$\\{\varrho,\phi,z\\}\to\\{\varrho_{0},\phi+2\,n\,\pi,z_{0}\\}$ is spacelike
for $\tau<0$ and time-like for $\tau>0$. Hence, our four-dimensional space-
time (4) admits closed time-like curves at an instant of time analogue to the
two-dimensional Misner space. These closed time-like curves evolve from an
initial spacelike hypersurface in a causally well-behaved manner [43]. This
point is clear from the following discussion. The metric component
$g^{\tau\tau}$ for the metric (4) is given by

$g^{\tau\tau}=\frac{4\,(\tau+\beta^{2}\,z^{2})}{\alpha^{2}\,\varrho^{2}}\,.$
(15)

In the constant $z$-plane defined by $z=z_{0}=0$, we obtain

$g^{\tau\tau}=\frac{4\,\tau}{\alpha^{2}\,\varrho^{2}}\,.$ (16)

Thus, the hypersurface $\tau=const$ is spacelike for $\tau>0$ since
$g^{\tau\tau}>0$ and time-like for $\tau<0$. The curve $\tau=0$ is null and
serve as chronology horizon, that is, the hypersurface separating the causal
and non-causal parts of space-time.

Thus, the Petrov type-N space-time (4) which is a non-vacuum solution of the
field equations admits closed time-like curves analogue to the Misner space.
Now, we study this line-element (4) in the context of modified theories of
gravity via the Ricci-inverse gravity by introducing an anti-curvature tensor
into the Lagrangian of the system, as discussed in Refs. [33, 37].

The action that describes the Ricci-inverse gravity is given as

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[(R+\kappa\,A-2\,\Lambda)+{\cal
L}_{m}\right],$ (17)

where $g$ is the metric determinant, $\kappa$ is the coupling constant,
$R=g_{\mu\nu}\,R^{\mu\nu}$ is the Ricci scalar, $A=g_{\mu\nu}\,A^{\mu\nu}$ is
anti-curvature scalar, $\Lambda$ is the cosmological constant, and ${\cal
L}_{m}$ is the matter Lagrangian. Varying the action (17) with respect to the
metric, the field equations that describe this gravitational theory is [39]

$\displaystyle
R^{\mu\nu}-\frac{1}{2}\,R\,g^{\mu\nu}+\Lambda\,g^{\mu\nu}+M^{\mu\nu}=\mathcal{T}^{\mu\nu}.$
(18)

Using the value of the Ricci scalar given in (7), the last equation becomes

$\displaystyle
R^{\mu\nu}+(6\,\alpha^{2}+\Lambda)\,g^{\mu\nu}+M^{\mu\nu}=\mathcal{T}^{\mu\nu},$
(19)

with $\mathcal{T}^{\mu\nu}$ being the standard energy-momentum tensor and the
tensor $M^{\mu\nu}$ is defined as

$\displaystyle M^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\kappa\,\Big{(}A^{\mu\nu}+\frac{A}{2}\,g^{\mu\nu}\Big{)}$ (20)
$\displaystyle+$
$\displaystyle\frac{\kappa}{2}\,\Big{\\{}2\,g^{\kappa\mu}\nabla_{\iota}\nabla_{\kappa}(A^{\iota}_{\sigma}\,A^{\nu\sigma})-\nabla^{2}(A^{\mu}_{\iota}\,A^{\nu\iota})-g^{\mu\nu}\nabla_{\kappa}\nabla_{\iota}(A^{\kappa}_{\sigma}\,A^{\iota\sigma})\Big{)}\Big{\\}}.$

The determinant of the metric tensor $g_{\mu\nu}$ and the Ricci tensor
$R_{\mu\nu}$ for the space-time (4) are given by

$\displaystyle\mbox{det}\,g_{\mu\nu}=-\frac{\alpha^{4}\,\varrho^{4}}{4},\quad\mbox{det}\,R_{\mu\nu}=-\frac{81\,\alpha^{12}\,\varrho^{4}}{4}.$
(21)

Since the determinant of the Ricci tensor is non-zero, there exist an anti-
curvature tensor $A^{\mu\nu}$ defined by

$\displaystyle
A^{\mu\nu}=R^{-1}_{\mu\nu}=\frac{\mbox{adj}\,(R_{\mu\nu})}{\mbox{det}\,(R_{\mu\nu})}.$
(22)

For the space-time (4), this anti-curvature tensor $A^{\mu\nu}$ and its
covariant form $A_{\mu\nu}$ are given by

$\displaystyle
A^{\mu\nu}=\begin{pmatrix}-\frac{4}{9\,\alpha^{8}\,\varrho^{4}}\Big{(}\beta+3\,\alpha^{4}\,\varrho^{2}\,(\tau+\beta^{2}\,z^{2})\Big{)}&0&\frac{2}{3\,\alpha^{4}\,\varrho^{2}}&-\frac{2\,\beta\,z}{3\,\alpha^{4}\,\varrho^{2}}\\\
0&-\frac{\varrho^{2}}{3}&0&0\\\ \frac{2}{3\,\alpha^{4}\,\varrho^{2}}&0&0&0\\\
-\frac{2\,\beta\,z}{3\,\alpha^{4}\,\varrho^{2}}&0&0&-\frac{1}{3\,\alpha^{4}\,\varrho^{2}}\end{pmatrix},$
$\displaystyle A_{\mu\nu}=\begin{pmatrix}0&0&\frac{\varrho^{2}}{6}&0\\\
0&-\frac{3}{\alpha^{4}\,\varrho^{2}}&0&0\\\
\frac{\varrho^{2}}{6}&0&\frac{t\,\varrho^{2}}{3}-\frac{\beta}{9\,\alpha^{4}}&-\frac{\beta\,z\,\varrho^{2}}{3}\\\
0&0&-\frac{\beta\,z\,\varrho^{2}}{3}&-\frac{\varrho^{2}}{3}\end{pmatrix}.$
(23)

At last, the anti-curvature scalar is given by

$A=g_{\mu\nu}\,A^{\mu\nu}=-\frac{4}{3\,\alpha^{2}}.$ (24)

Thereby, substituting the metric tensor $g^{\mu\nu}$ from (5), the anit-
curvature tensor $A^{\mu\nu}$ from (23), and the anti-curvature scalar from
(24) into the relation (20), we obtain the nonzero components of the symmetric
tensor $M^{\mu\nu}=M^{\nu\mu}$ (20) given by

$\displaystyle M^{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{4\,\kappa\,\left(27\,\alpha^{4}\,\varrho^{2}\,(\beta^{2}\,z^{2}+\tau)+7\beta\right)}{27\,\alpha^{8}\,\varrho^{4}}\,,$
$\displaystyle M^{\tau\phi}$ $\displaystyle=$
$\displaystyle-\frac{2\,\kappa}{\alpha^{4}\,\varrho^{2}}\,,$ $\displaystyle
M^{\tau z}$ $\displaystyle=$
$\displaystyle\frac{2\,\beta\,\kappa\,z}{\alpha^{4}\,\varrho^{2}}\,,$
$\displaystyle M^{\varrho\varrho}$ $\displaystyle=$
$\displaystyle\kappa\,\varrho^{2}\,,$ $\displaystyle M^{zz}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{\alpha^{4}\varrho^{2}}\,.$ (25)

The nonzero components of the covariant tensor $M_{\mu\nu}$ are

$\displaystyle M_{\tau\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\,\varrho^{2}\,,$ $\displaystyle
M_{\varrho\varrho}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{\alpha^{4}\,\varrho^{2}}\,,$ $\displaystyle
M_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{7\,\beta\,\kappa}{27\,\alpha^{4}}-\kappa\,\varrho^{2}\,\tau\,,$
$\displaystyle M_{\phi z}$ $\displaystyle=$
$\displaystyle\beta\,\kappa\,\varrho^{2}\,z\,,$ $\displaystyle M_{zz}$
$\displaystyle=$ $\displaystyle\kappa\,\varrho^{2}\,.$ (26)

The trace of the tensor $M^{\mu\nu}$ is given by
$M=g_{\mu\nu}\,M^{\mu\nu}=\frac{4\,\kappa}{\alpha^{2}}$.

In order to simplify the modified field equations (19), let’s define

$J^{\mu\nu}=R^{\mu\nu}+(6\,\alpha^{2}+\Lambda)\,g^{\mu\nu}+M^{\mu\nu}\,.$ (27)

Then, the non-zero components of the tensor $J^{\mu\nu}$ using the metric
tensor (5), the Ricci tensor (6), and the tensor $M^{\mu\nu}$ (25) are given
by

$\displaystyle J^{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{4\,\left[27\,\alpha^{4}\,\varrho^{2}\,(\beta^{2}\,z^{2}+\tau)\,(3\,\alpha^{4}+\alpha^{2}\,\Lambda+\kappa)+(27\,\alpha^{4}+7\,\kappa)\,\beta\right]}{27\,\alpha^{8}\,\varrho^{4}}\,,$
$\displaystyle J^{\tau\phi}$ $\displaystyle=$
$\displaystyle-\frac{2\left(3\,\alpha^{4}+\alpha^{2}\,\Lambda+\kappa\right)}{\alpha^{4}\,\varrho^{2}}\,,$
$\displaystyle J^{\tau z}$ $\displaystyle=$
$\displaystyle\frac{2\,\beta\,z\left(3\,\alpha^{4}+\alpha^{2}\,\Lambda+\kappa\right)}{\alpha^{4}\,\varrho^{2}}\,,$
$\displaystyle J^{\varrho\varrho}$ $\displaystyle=$
$\displaystyle\varrho^{2}\left(3\,\alpha^{4}+\alpha^{2}\,\Lambda+\kappa\right)\,,$
$\displaystyle J^{zz}$ $\displaystyle=$
$\displaystyle\frac{3\alpha^{4}+\alpha^{2}\,\Lambda+\kappa}{\alpha^{4}\,\varrho^{2}}\,.$
(28)

Taking $J^{\mu\nu}$ into account, a possible solution will be checked for
$\Lambda$ with $\mathcal{T}^{\mu\nu}=0$ and
$\mathcal{T}^{\mu\nu}=\rho\,k^{\mu}\,k^{\nu}$, i. e., a vacuum and pure
radiation field, respectively. One can see from the above set of equation (28)
that for vacuum case, where $\mathcal{T}^{\mu\nu}=0=J^{\mu\nu}$, there is a
solution for $\Lambda$ provided the parameter $\beta$ in the space-time (4)
must be zero, that is, $\beta=0$. Otherwise there is no such solution for
$\Lambda$ if the parameter $\beta>0$.

We now focus on the case, where $\mathcal{T}^{\mu\nu}\neq 0$, non-vacuum
solution of the modified theory of gravity. In general relativity theory, we
have shown that the space-time (4) satisfies the field equation with a pure
radiation field as matter content with a negative cosmological constant. In
the Ricci-inverse gravity, we choose the same pure radiation field as matter
content whose energy-momentum tensor is given in equation (10), that is,
$\mathcal{T}^{\mu\nu}=\rho\,k^{\mu}\,k^{\nu}$, where in the chart
$\\{t,\varrho,\phi,z\\}$, we defined the null vector field $k_{\mu}=(0,0,1,0)$
and its contravariant form will be
$k^{\mu}=(-\frac{2}{a^{2}\,\varrho^{2}},0,0,0)$, such that the vector field
satisfies the null condition $k^{\mu}\,k_{\mu}=0$.

Now, using the relation
$J^{\mu\nu}=\mathcal{T}^{\mu\nu}=\rho\,k^{\mu}\,k^{\nu}$ and substituting the
non-zero components (28) leads to the following system of equation

$\displaystyle\rho\left(-\frac{2}{\alpha^{2}\varrho^{2}}\right)^{2}\,$
$\displaystyle=$
$\displaystyle\frac{4\,\left[27\,\alpha^{4}\,\varrho^{2}\,(\beta^{2}\,z^{2}+\tau)\\{\kappa+\alpha^{2}\,(3\,\alpha^{2}+\Lambda)\\}+(27\alpha^{4}+7\,\kappa)\,\beta\right]}{27\alpha^{8}\varrho^{4}},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{2\left(3\alpha^{4}+\alpha^{2}\Lambda+\kappa\right)}{\alpha^{4}\varrho^{2}}\,,$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{2\beta
z\left(3\alpha^{4}+\alpha^{2}\Lambda+\kappa\right)}{\alpha^{4}\varrho^{2}}\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\varrho^{2}\left(3\alpha^{4}+\alpha^{2}\Lambda+\kappa\right)\,,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{3\alpha^{4}+\alpha^{2}\Lambda+\kappa}{\alpha^{4}\varrho^{2}}.$
(29)

The solution to the above system of equations gives us the following physical
quantities

$\displaystyle\Lambda\to\Lambda_{m}=\Big{(}-3\,\alpha^{2}-\frac{\kappa}{\alpha^{2}}\Big{)},$
$\displaystyle\rho\to\rho_{m}=\beta\,\left(1+\frac{7\,\kappa}{27\,\alpha^{4}}\right)\,.$
(30)

Hence, the space-time described by the line-element (4) serves as a solution
in modified gravity theories, specifically in Ricci-inverse gravity. In this
framework, the matter-energy content consists of a pure radiation field,
characterized by a modified energy density $\rho_{m}$ and a modified
cosmological constant $\Lambda_{m}$, as given by equation (30). These
modifications are determined by the coupling constant $\kappa$ along with
other parameters ${\alpha,\beta}$.

It is noteworthy that this new theory also allows for the existence of closed
time-like curves, which are observed within certain regions where $\tau>0$. It
is worth noting that the modified energy-density $\rho_{m}$ satisfies the
energy condition for a positive coupling constant. Notably, the energy
condition is automatically satisfied for a positive coupling constant,
$\kappa>0$ since $\beta>0$. It can be readily demonstrated that when the
coupling constant approaches zero, i. e., $\kappa\to 0$, the results obtained
revert to the original findings in general relativity, as discussed earlier as
well as in Ref. [17].

##  3 Conclusions

General relativity is a classical theory of gravity that has been intensively
tested since it was proposed by Einstein. Although Einstein’s theory has been
successfully verified, it has problems in explaining some observational data.
Then, based on observational motivations, alternative models of gravity have
been proposed. Modified theories of gravity have been a subject of significant
research interest for an extended period. Several alternative theories of
gravity have been proposed by researchers over time. These include the $f(R)$
theory, $f(T)$ theory, $f(R,T)$ theory, $f(R,G)$ theory, and $f(G,T)$ theory
(For references to these theories, see [41]). More recently, a novel gravity
theory called Ricci-inverse theory has been introduced in [33]. The main
characteristic of this theory is that the determinant of the Ricci tensor for
any space-time geometry must differ from zero.

In this study, we have examined an example of Petrov type-N pure radiation
field solution in the backdrop of anti-de Sitter space. This particular
solution allows for the formation of closed time-like curves in specific
regions, thus, serving as a model for a time machine within the framework of
general relativity. Subsequently, we have taken this type-N space-time and
explored it within the context of modified theories using Ricci-inverse
gravity. Our findings reveal that this type-N metric is also a solution in
this novel Ricci-inverse gravity theory, thus, allows the formation of closed
time-like curves, similar to the previous theory. We have observed that the
energy-density of the pure radiation field and the cosmological constant
undergo modifications due to the coupling constant defined in equation (30).
It’s noteworthy that, as long as the parameter $\beta$ remains positive, the
energy-density satisfies the null energy condition as long as the coupling
constant is positive. It’s worth noting that one can explore matter content
other than a pure radiation field, including scalar fields, pressureless
perfect fluids, or anisotropic fluids, within the framework of this modified
theory of gravity. In addition, Ricci-inverse gravity is a new theory that
should be tested in different contexts. Several points have been investigated,
for example, there is no Minkowskian limit for the Ricci-inverse gravity, this
theory cannot explain the cosmic expansion history starting from the
radiation-dominated epoch to the matter-dominated epoch to the dark energy-
dominated epoch, the existence of ghosts must be carefully analyzed, among
others. Some studies on instability and perturbation should be considered for
future investigation. Therefore, the study developed here follows this
necessary line of testing a theory that is proposed as an alternative theory
of gravity. It is important to emphasize that general relativity allows
solutions that present CTCs that lead to violation of causality. Then
verifying an exact solution of general relativity with such a characteristic
in theories of modified gravity is an important test like the others mentioned
previously.

The Einstein tensor of a null dust solution (or null fluid) is expressed as
$G^{\mu\nu}=\Phi\,k^{\mu}\,k^{\nu}$ [3, 4], where $k^{\mu}$ is a null vector
field, and $\Phi$ is a scalar multiplier. Introducing a stress-energy tensor
in the space-time as $T^{\mu\nu}=\Phi\,k^{\mu}\,k^{\nu}$ satisfies Einstein’s
field equation, providing a clear physical interpretation in terms of massless
radiation. Physically, a null dust solution can describe gravitational
radiation or non-gravitational radiation governed by a relativistic classical
field theory, such as electromagnetic radiation. Phenomena modeled by null
dust solutions include (i) a beam of neutrinos, assumed to be massless and
treated using classical physics, (ii) a high-frequency electromagnetic wave,
and (iii) a beam of incoherent electromagnetic radiation. Petrov-type N
regions are associated with transverse gravitational radiation, the type
detected by astronomers using LIGO and Virgo detectors recently [45, 46].
Typically, it experiences decay on the order of $\mathcal{O}(r^{-1})$,
indicating that the long-range radiation field falls under type N.

## Data Availability

No data generated or analyzed in this study.

## Conflict of Interests

Author(s) declares no such conflict of interests.

## Acknowledgments

We sincerely acknowledged the anonymous referee’s for their valuable remarks
and suggestions. F.A. acknowledged the Inter University Centre for Astronomy
and Astrophysics (IUCAA), Pune, India for granting visiting associateship.
This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2. J.
C. R. S. thanks CAPES for financial support.

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