# Masader: Metadata Sourcing for Arabic Text and Speech Data Resources

Zaid Alyafeai1, Maraim Masoud2, Mustafa Ghaleb1, and Maged S. Al-shaibani1

1 King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
2 Independent Researcher

###### Abstract

The NLP pipeline has evolved dramatically in the last few years. The first
step in the pipeline is to find suitable annotated datasets to evaluate the
tasks we are trying to solve. Unfortunately, most of the published datasets
lack metadata annotations that describe their attributes. Not to mention, the
absence of a public catalogue that indexes all the publicly available datasets
related to specific regions or languages. When we consider low-resource
dialectical languages, for example, this issue becomes more prominent. In this
paper we create Masader, the largest public catalogue for Arabic NLP datasets,
which consists of 200 datasets annotated with 25 attributes. Furthermore, We
develop a metadata annotation strategy that could be extended to other
languages. We also make remarks and highlight some issues about the current
status of Arabic NLP datasets and suggest recommendations to address them.

##  1 Introduction

The emergence of deep learning and its applications in many fields had a great
impact on the development of various natural language processing (NLP) and
speech techniques that were adapted to many languages. Many might correlate
that to the availability of data especially with the existence of social media
and the manufacturing of hardware devices that fostered research in the field,
namely GPUs. Typically, we are referring to the era of deep learning which
started roughly after 2010. Following that, many public Arabic NLP and speech
datasets have been published in conjunction with the recent advances in deep
learning Zaghouani (2017). Currently, there is no online centralized catalogue
for Arabic NLP and speech datasets. It is unclear how many online datasets
there are as well as the metadata describing the datasets’ characteristics,
such as diversity, demographic distribution, ethical considerations, quality,
and so on. This study attempts to identify the publicly available Arabic NLP
datasets and to provide a catalogue of Arabic datasets to researchers. The
catalogue will increase the discoverability and provide some key metadata that
will help researchers identify the most suitable dataset for their research
questions.

We highlight our contributions as the following:

  * •

We create the largest catalogue with 25 attributes for 200 Arabic NLP and
speech datasets.

  * •

We design a metadata schema for annotating the datasets.

  * •

We analyse the current status of the Arabic NLP and speech datasets, discover
issues and recommend solutions.

The paper is structured as follows. Section 2 looks into previous work in the
literature. Section 3 summarizes our approach to develop the catalogue. It
discusses the research methodology, metadata design, and the annotation
process. Section 4 outlines our findings. The results are then inspected, and
issues and recommendations are highlighted in sections 5 and 6 respectively.

##  2 Related Work

Surveying the literature to derive analysis about a specific research field or
topic is a standard practice. It helps to provide an overview on the
directions and trends on subject of interest. A prominent example of such
effort is Ammar et al. (2018). They collected a large corpus of 280 million
nodes. These nodes are diverse entities representing authors, papers, etc. An
application of their work is the connected papers project Eitan et al. (2020).
It aims to construct a graph of related literature based on a given query.
Radev et al. (2016) extends the analysis by accounting for the citation count
and extensive manual annotation on the collected literature. They curated
their dataset from ACL Anthology papers. Their analysis covers various various
attributes including authors impact factor, h-index and collaboration. For
massive analysis reports on the field of NLP, Mohammad (2020) surveyed the
literature with 1.1 million paper information dataset collected from Google
Scholar. Additionally, Sharma et al. (2021) proposed DRIFT, a data analysis
tool that presents an overview of the landscape of a queried topic. They
constructed their dataset from arXiv papers’ abstracts.

As the dataset collection research is vastly growing with the significant
emergence of data on the web, the need to mandate such process becomes a
necessity. This is, in fact, an active branch of research happening across
various domains and disciplines. Consider, for instance, the systematic
literature review protocol developed by Kitchenham (2004) that governs the
data collection process for surveys. Another example is the guidelines
reported by Mbuagbaw et al. (2017) on clinical trials. There are also studies
that propose standardizing the documentation of datasets. Gebru et al. (2018)
proposed datasheets for datasets. Their aim is to accompany datasets with a
descriptive datasheet schema describing diverse attributes about the dataset.
Such attributes include operating characteristics, recommended uses,
motivation, collection process and test results. Similarly, Bender and
Friedman (2018) propose data statement, a similar standardization approach
that overlaps to datasheets in some of its attributes. However, while
datasheets aims to document more general information about the dataset, data
statement is more specific to linguistics and NLP.

In the field of Arabic datasets, there are many studies that attempted to
survey the available data resources. Shoufan and Alameri (2015) reviewed the
NLP literature for dialectical Arabic. Their work can be considered as a quick
reference to locate important contributions for certain Arabic dialects that
address specific NLP features. However, this study reviewed limited literature
as it was concentrated only on Arabic dialects and the research on Arabic
corpora was still infant. A comprehensive approach was implemented by
Zaghouani (2017) where they collected a list of around 80 freely available
Arabic datasets including datasets that are not related to NLP. The also
provide links to the datasets but some of them do not work anymore. There are
also some efforts to survey specific dialects. For example, Younes et al.
(2020) provided a review of various kinds of constructed language resources
(LRs) of Maghrebi Arabic dialects (MADs). They reviewed MAD raw corpora and
divided it into speech corpora, speech transaction, web, and social media
corpora. Recently, Guellil et al. (2021) presented and classified 90 studies
that covered classical Arabic, Modern Standard Arabic, and Arabic Dialects.
Also, they provided links to around 52 NLP datasets. Another survey paper was
published by Darwish et al. (2021) to review the available tools and resources
for Arabic. However, they provided links to a few datasets.

Figure 1: Metadata schema for Arabic NLP resources.

##  3 Methodology

The research method behind this survey follows the keyword-base literature
review process Rowley and Slack (2004) followed by an annotation process to
enrich the filtered results with metadata. Our methodology follows five steps:
(i) searching resources, (ii) filtering using a selection criteria, (iii)
annotating resources with metadata, (iv) validating the resources, and (v)
analysing the results.

Based on a pre-selected keywords, the retrieved Arabic language resources are
added to our preliminary list of data sources. Next, all the resources
collected are filtered according to a set of inclusion criteria. The datasets
that passed the filtering criteria are ported to our final list of datasets,
and then are annotated with a set of metadata, both manually and
automatically. Those which do not pass the criteria are discarded. Following
that, a verification step is performed to ensure the accuracy of the metadata.
Finally, the final set of resources are analysed according to the metadata and
presented in this study.

The initial search and the filtering for all sources were done between July
and August 2021, and the annotation process took place progressively after the
data was collected and concluded on September 2021. The following subsections
describe each step in more detail.

###  3.1 Step 1: Searching Resources

The targeted search is performed using Google Search Engine to identity Arabic
NLP dataset directly. To search specific dataset, we also conduct the search
against the well-known data repositories and indexing websites using a set of
keywords. The selected repositories are GitHub111https://github.com/,
Paperswithcode222https://paperswithcode.com/,
Huggingface333https://huggingface.co/, LREC444The International Conference on
Language Resources and Evaluation http://www.lrec-conf.org/, Google
Scholar555https://scholar.google.com/, and LDC666The Linguistic Data
Consortium https://www.ldc.upenn.edu/. The search combined terms related to
NLP and Arabic language, such as "NLP", "Natural Language Processing" and all
variations of Arabic dialects, as well as terms such as "database", "dataset",
"resource", and "corpus". This step generates our preliminary list of data
sources which consists of around 299 resources.

###  3.2 Step 2: Filtering with Inclusion Criteria

Our search was additionally supplemented by manually screening retrieved
articles and datasets, which we perform using a set of inclusion criteria,
which are as follows:

  * •

the dataset is specific for NLP.

  * •

there is a publication associated with the language resource.

  * •

the resource is created after 2010.

  * •

the resource, in its raw form or annotated format, is suitable for language
modelling and text generation tasks.

  * •

the resource does not serve as NLP tool such as spelling-checking and stop-
words.

From the preliminary list, we ended up with 207 papers. Our inclusion criteria
removed 93 papers initially.

###  3.3 Step 3: The Annotation Process

By applying the inclusion criteria, the datasets search pool is reduced to the
final set of considered resources. At this stage, an annotation process is
applied to manually annotate them with a set of pre-agreed metadata. When
applied to this process, we consider three main goals: 1) designing metadata
specific for Arabic resources, 3) deciding on the annotation format, 2)
setting up an annotation workflow, and finally 3) defining an annotation task.

#### Metadata Selection

The main motivation behind designing metadata for Arabic NLP resources is to
increase the discoverability and reusability of such resources. The metadata
are chosen to represent different aspects of the language resource. Park et
al. (2021)’s work serves as an example for identifying the appropriate
metadata for our Arabic NLP use case. Following several revisions, the final
agreed-upon metadata is represented by a taxonomy in Figure 1. It consists of
five subcategories as follows:

  * •

Publication: This subcategory concerns about metadata relating to author,
publisher and other publication details for the dataset referenced
publication. It includes attributes such as the title, the link, the year of
publication for the referenced paper, and the venue title, name, and type.

  * •

Content: This subcategory is concerned with the content of the dataset in
terms of the size, the representation and the quality. The size tag indicates
the quantity of the dataset by specifying the unit of measurement (tokens,
sentences, documents, MB, GB, TB, hours, others), as well as the number of
units using the volume tag. The representation dimension describes the
contextual information about the dataset. For example, the tokenized flag
specifies whether the dataset is tokenized. This is useful since various
tokenizers project different behaviour, and when this is not specified, it
impacts downstream tasks. The form tag, on the other hand, defines the form of
the content, being written, or spoken language, while the script tag describes
the writing system used in the dataset (Arab,Latn,Arab-Latn,Other). The third
dimension, quality, describes elements related to the data collection. It
covers, for example, the collection style used for building the dataset (e.g
crawling, translation, etc), the ethical risk associated with utilizing the
data set (low, medium, and high), and the domain of the dataset (social media,
etc ).

  * •

Accessibility: This subcategory concerns about the timeliness and the
reliability of access to the data. Its associated metadata includes: the name
of the data _provider_ , the name of the data _host_ , the _link_ to download
the data from the host, the _licence_ and the _cost_ to obtain the data.

  * •

Diversity: This metadata subclass is used to capture the linguistic and
culture diversity within Arabic language. It covers the _language_ tag to
represent the language of the dataset, either being Arabic (_ar_ or
_multilingual_ to denote a dataset that contains several languages), as well
as the subsets tag to denote the sub-datasets that are contained inside this
dataset. The last tag in this class is dialect. To capture the linguistic
variety of Arabic, we adapted five high-level categories of dialect
variations, resulting in a total of 29 dialect categories. These categories
are as follows: i) MSA for Modern Standard Arabic, ii) CLS for Classic Arabic
and Qura’anic text, ii) Regional dialects for the four regions (GLF, LEV, EGY,
NOR)777GLF (Gulf region), LEV (Levant region), EGY (Egypt and Sudan) and NOR
(North Africa region)., iii) country-based dialects which cover the 22
dialects spoken in Arabic-speaking countries, and finally iv) the other which
includes mixed dialects and code-switched script.

  * •

Evaluation: The metadata within this subcategory describes the process of
using the dataset in relation to the evaluation phase of the NLP pipeline.The
first tag is the test split, which is deployed as a boolean flag to signal if
the dataset is prepared for evaluation task by having a distinct split between
the training and the test sets. The tasks tag defines the list of tasks to
which the dataset is applied, whilst the related datasets attribute, lists
what dataset(s) intersect with the current dataset.

#### Annotation Formats

Based on the chosen metadata, we figured out that different annotation
procedures can be applied to insert the metadata. Hence, two formats of
annotation are adapted in this work; (i) manual curation, and (ii) automatic
annotation of the metadata. The manual curation is performed by the human
annotators via manually inspecting the dataset link, and it’s referenced
paper. We basically use this format to extract metadata that is hard to
automate or not implicitly mentioned. The second format is auto-annotation.
For this format, we rely on APIs from academic publishers, such as Semantic
Scholar API (Python Library) 888https://api.semanticscholar.org/graph/v1. As
such, most of the metadata is manually annotated, except for the publication
information which was retrieved using the API.

#### Annotation Workflow

We used Google Sheets to set up our annotation workflow, where the metadata is
specified as Google Sheet columns. The datasets were annotated with a set of
metadata by the four authors, who are fluent Arabic speakers and researchers
in the field of natural language processing.

We define the manual annotation task as follows. For each link of Arabic
language resource in our filtered pool of resources, the main goal of the task
is to annotate the filtered datasets against the metadata. The annotators were
instructed to follow the following set of guidelines:

  1. 1.

Examine the resource link.

  2. 2.

Examine the referenced paper.

  3. 3.

Fill the metadata entries on Google Sheets.

  4. 4.

If a particular attribute of the dataset is not mentioned, log it in the
notes’ column.

  5. 5.

If a conflict is observed between the reported metadata from the resource link
and the actual published paper, mark this entry in the sheet.

###  3.4 Step 4: Verification of metadata

Following the completion of the annotation, a verification step is performed
to confirm the accuracy of the information given. This step was deemed
necessary in order to explain the notes from the prior stage. It is done
manually and involved active communication amongst the annotators. At this
stage, we removed 7 extra papers, 2 of them were duplicated datasets and 5 had
wrong annotations for the year attribute (before 2010). In Figure 2, we show
an example of the metadata annotations of a chosen dataset.

###  3.5 Step 5: Analysis

After verifying the final collection of annotated datasets, we conduct the
analysis on various metadata. The findings of the analysis will be described
in the next sections.

Figure 2: Example demonstrates the annotation of the metadata on the Shami
dataset Abu Kwaik et al. (2018). The subsets tag represents the dialects and
each subset (For example, Jordanian) inherits all the metadata from the
superset Shami, except the volume.

##  4 Findings and Representation

In this section we describe our findings in collecting all the data resources
related to Arabic NLP published between 2010 and September 2021. We describe
driven statistics of the datasets in addition to how we represent our data in
a user friendly format.

###  4.1 Data Statistics

The total number of datasets included in this catalogue is 200. More than 90 %
of the datasets’ written format is text while the remaining is speech data.
Table 1 summarizes the overall statistics of the catalogue in terms of volume.
We mainly used the reported numbers in the paper and validated the numbers by
downloading the dataset. If the size is different we report the numbers from
the downloaded dataset. If the number can’t be validated, because of the size
of the dataset for example, we report the numbers from the paper. Mostly, we
use tokens to represent datasets that tackle token-based tasks like named
entity recognition (NER), sentences to represent datasets that are related to
sentence-based tasks like sentiment analysis and documents if the size of the
dataset is too large.

Table 1: Summary of the 200 Arabic NLP datasets in the Masader project in terms of volume. Unit | Volume
---|---
Tokens | 451,370,314
Sentences | 1,236,350
Documents | 51,701
Hours | 3,104.1
# Datasets | 200
# Datasets with Dialect Subsets | 23
# Total Subsets | 375

###  4.2 Masader Interface

To easily navigate the sources we created a website
999https://arbml.github.io/masader/ that is connected directly to the Google
Sheets, allowing any updates in the sheets to be reflected immediately on the
website. The website’s primary interface only displays nine
attributes101010index, name, link, year, volume, unit, paper link, access, and
tasks.. These are deliberately chosen for trail, and testing their relevance
for academic search. The interface supports discoverability by including the
following features: 1) a clickable association between the dataset and its
published paper, 2) a direct link to the most recent hosted version of the
dataset, 3) a clickable link on the dataset name, which leads to dataset card
displaying the remaining metadata of a dataset, and finally 4) filtering and
sorting based on each attribute.

##  5 Examining the Arabic NLP Landscape

This section provides an analysis on the surveyed datasets. We mainly focus on
discussing the current trend of publishing Arabic resources and drawing some
remarks about the overall status of the landscape.

###  5.1 Publications development

Figure 3 depicts the evolution of Arabic NLP in the light of publicly
available data resources from various venues. The graph demonstrates a general
growth in the number of published resources, with a particular increase in
even years. This can be attributed to the large number of datasets published
at the bi-annual LREC conference. We also anticipate a significant increase,
particularly in 2020, with the emergence of pretrained language models
language models namely AraBERT Antoun et al. (2020a), Multi-dialect BERT
Talafha et al. (2020) and Araelectra Antoun et al. (2020b). We can also
observe that most of the datasets are published in conferences and workshops.

###  5.2 Data Accessibility

Data accessibility is an important aspect of fostering open research. Making
the data source available extends its lifespan and allows it to be utilized in
the way the dataset authors intended. In our initial data sources collection,
we observed that more than half of the 93 of the discarded datasets had no
online presence, nor an explicit means of accessing any version of the data.
Based on the examination of the data sources, there is general trend of making
the data available through open source repositories such as GitHub, GitLab and
Mendeley Data. In the last three years, more than 80 % of the data can be
accessed freely on different data hosters. This trend is very promising as it
shows an increased interest in making the datasets available online. In the
recent years, there is a small portion of datasets that needs authentication
to access either through email or registration forms. We also observed some
papers that suggested contacting the corresponding author to access the data
privately. We didn’t include such papers on our final list.

###  5.3 Data Providers and Licensing

Data providers are important to collect, annotate, distribute, and perhaps
host the datasets. Another responsibility of the data provider is to select
the appropriate licence for the datasets. In fact, having a proper licence is
a key component of any dataset for both data providers and researchers. In
terms of data providers, our findings show that the majority of the data
sources we collected were created in virtue of collaboration of multiple
institutions. Institutions such as QCRI, Qatar university, NYU Abu Dhabi, and
Nile University are the top four providers of Arabic NLP datasets. While
datasets from these prominent providers are typically accompanied by clear
licensing. Unfortunately, about 50 % of the datasets lack licences. Among
those with explicit licence, there is a wide range of used licences. Some
examples include several variations of Common Creative licences, Apache, MIT,
GPL and BSD.

Figure 3: The count of publications across conferences, journals, preprints
and workshops.

###  5.4 Dialects Diversity

As we can see from Figure 4, there were more than 20 entries out of the 200
with annotations of the dialects. These datasets are primarily intended for
dialect identification tasks. The scope of the datasets we collected across
the Middle East and North Africa is depicted in Figure. The Egyptian dialect
is the most prevalent, followed by Algerian, Moroccan, and Saudi dialects.
Somali, Djibouti, and Mauritanian dialects are underrepresented in the
surveyed datasets, with only three resources for each.

Figure 4: Dialects representation across datasets.

###  5.5 Tasks Coverage

Figure 5 illustrates the distribution of tasks that appeared in more than one
dataset. The graph shows that machine translation and sentiment analysis are
the most popular tasks within Arabic NLP community. Machine translation has
received an increasing attention in the literature across many languages,
particularly in multilingual datasets, which explains the high frequency of
publications in that area. Sentiment analysis, on the other hand, is heavily
researched for a variety of reasons. Partly because the datasets for this task
are primarily derived from social media sites with minimum effort, and partly
because it serves as a suitable representation of everyday language that
displays more sentiments. Other tasks that have presence within the community
include dialect identification, topic classification, named entity recognition
and speech recognition. There are also several low resource tasks that
appeared at most once but are not displayed in the figure, such as poetry
classification, word disambiguation, grammar checking, to name a few. Each of
these sources only contains a single dataset addressing a distinct task. In
contrast, datasets like KALIMAT El-Haj and Koulali (2013), contains
annotations for multiple tasks, or evaluation suites such as ALUE which is an
aggregation of multiple datasetsSeelawi et al. (2021).

Figure 5: Tasks’ histogram. We only show the tasks that appeared more than
once in papers

##  6 Arabic NLP Datasets Challenges and Recommendations

This section highlights some issues of Arabic NLP data related to their
legitimization, the haphazard collection, annotation, and the documentation
practices.

#### Data Availability.

It is encouraging that our review identified 200 datasets that were publicly
available, yet discoverability seems, by all accounts, to be an issue. While a
few datasets are well recognized in the field, many are not, which might
potentially lead to missed research opportunities and might result in bias
because of an overuse of a few potentially non-representative datasets.
Further considerations in this regard, arising from our survey, include the
sustainability (persistence) of the dataset URLs. As there is not a dedicated
platform to host Arabic NLP datasets, some datasets’ links appear to be
inaccessible due to URLs invalidity (orphan datasets). We identified one
obvious/clear cause of orphaned datasets, which is the termination of academic
affiliation and so the broken dataset link when the dataset is published as
part of the researcher’s academic webpage. One possible solution to address
this issue is to host the datasets on public repositories like GitHub, Gitlab,
Mendeley Data, SourceForge, to name a few.

#### Data Documentation.

Data documentation refers to the process that describes the collected data and
aims at facilitating cataloguing and discoverability of the data. One key form
of data documentation is metadata, which are characteristics describing the
data. For a dataset to be truly reusable, adequate documentation is important
to offer the necessary insights into the potential usage of the dataset. For
researchers, providing such insights saves time and resources, and it suggests
a reliable dataset for reuse Perrier et al. (2020).

In this work, we analyse the documentation of Arabic NLP datasets in relations
to the proposed metadata in Section 3. When we examine these datasets, we
recognize that a few of them are accompanied by documentation. The majority
appears to report inadequate metadata that is insufficient to make a decision
on the dataset reuse. More precisely, there appears to be a pattern in which
some researchers are satisfied with only publishing the direct URL link to
download the dataset, or accompanying the downloadable dataset with a README
file stating the size of the dataset, and a reference to the publishing paper
on the dataset host page. Within the quality metadata, we observed very few
instances reporting the data collection style. Another consideration includes
the absence of clarity around the terms of access and use from some dataset
providers. In most cases, we noticed that datasets are not accompanied by
sufficient information regarding their provenance, and hence it is not
possible for researchers to know if there is an appropriate ethical and
governance framework underpinning the provision of these datasets. As a result
of this research, we conclude that governance information, such as licencing,
is a crucial part of the documentation and that, if not specified, the
dataset’s potential reuse may be limited. Regardless, as noted earlier, ease
of access and good documentation is an important driver for researchers.
Therefore, deploying a framework for documenting NLP data, such as those
proposed by Bender and Friedman (2018) and Gebru et al. (2018) is considered
as a good step towards promoting data sharing.

#### Data Sharing.

Data sharing is positively seen in the NLP community, with even top
conferences are recognizing researchers who have shown a desire to share
datasets. This process is usually volunteered, unless it is enforced
internally by institutions and corporates measures. Within the Arab NLP
community, we observed a high intention to support the research by sharing
datasets. While some datasets are poorly documented and hardly accessible,
others are well-prepared with clear documentation.

We identified some datasets that are never published, or they are
inaccessible, even where there was an intention to make them available, as
declared in the formal publications. In terms of openness, we also observed a
pattern in which some providers require a form of registration prior to
sharing the datasets. Regarding sharing dataset links, as it was detailed in
the data documentation, the unsustainability of dataset link poses a
challenge, and hence data repository such as Github and Gitlab are usually
adopted as a platform for sustainable data sharing.

#### Evaluation.

NLP models are usually evaluated by training on specific tasks. In the
literature, the test split is provided as an approach to evaluate models after
training on the training split. In our metadata collection process, we
observed that more than 60 % of the datasets do not have predefined test
splits. To mitigate that, researchers replicate the experiments by evaluating
the old models again on a chosen random split of the data. As a result, a
dataset’s results will be incomparable across different NLP models.

#### Data Collection or Curation.

Having stated annotation protocols and clear justification behind inter-rater
agreement increases the reliability of the data. In the surveyed datasets, we
have the following observations about the collection style. First, some
crawling driven datasets lack any consideration of ethics and legal frameworks
imposed by the platform from which the data is scraped, and the country of the
data subjects. It also imposes an ethical risk by stating personal identified
information. Secondly, when using machine translation to drive an Arabic
version of a non-Arabic dataset, we observed missing information such as the
translation models, verification process by native speakers, the reported
errors, to name a few. Given the current quality of machine translation
models, this approach in creating Arabic datasets opens many questions about
the quality of the dataset and its potential usage. While this approach can be
used to help create datasets for some tasks, it is a risky approach if used to
drive benchmark datasets for tasks such as common sense. Thirdly, as it was
highlighted in previous points, not having an indication of the ethical risk
of using datasets is a weak point in the Arabic NLP datasets. While each
Arabic-speaking country has its own legislations and data protection acts Abu-
Ghazaleh (2000), it is important to flag any potential risk of using the
datasets for future usage.

#### Ethical Concerns and Privacy.

Social media data, such as Twitter data, composed the greatest proportion of
Arabic NLP datasets, particularly for dialect representations. Typically, such
data is associated with the risk of exploiting personal information. In fact,
this raises the concerns of considering data subject’s right and the ethics
behind using such data. Likewise, datasets acquired from publications and
human-produced literature pose concerns regarding the incorporation of
copyright consideration in the derived Arabic NLP datasets. In either type of
the datasets, we found no explicit risk indications at any point of the NLP
pipeline: collection, modelling, evaluation, or deployment. In this context,
we encourage data providers to state information about the ethical risks
associated with their released datasets, as well as the appropriate approach
to mitigate them, in order to enrich the Arabic NLP landscape.

##  7 Limitations and Future Work

We recognize some limitations to our study. Firstly, given the nature of our
search strategy, only datasets that are probably indexed with metadata, and
whose publication contains one of our search key-phrases are likely to have
been retrieved. Secondly, there exists some additional data resources
available that are either with open access or regulated access (e.g LDC), but
they were not explored in this study since they do not conform to our
inclusion criteria. As a future work, we plan to keep the catalogue updated by
adding new datasets and also support community-based contributions where
authors can submit the metadata of their datasets to our online catalogue.

##  8 Conclusion

In this research, we created an online catalogue of 200 Arabic NLP datasets
with metadata annotations. We analyzed our findings, discovered some issues
and suggested some resolutions. Mainly, we recognise that the NLP field is
rapidly evolving, and that both Arabic NLP researchers and practitioners
recognise the value of incorporating Arabic into language technologies,
particularly beyond Modern Standard Arabic. As a result, while this research
provides a comprehensive analysis, it is only a snapshot in time and extra
efforts are required to drive the field more in that direction.

## Acknowledgements

This research was conducted under the BigScience
111111https://bigscience.huggingface.co/ initiative for open research, a one-
year-long research initiative targeting the study of large models and
datasets. It was conducted as part of the data sourcing group for collecting
datasets for different languages. We would like to thank the members of the
data sourcing group for the insightful discussions.

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##  Appendix A Extra Analysis

#### Repositories

In Figure 6, we highlight the most used repositories to host datasets. More
than 50 % of the datasets are hosted on GitHub. While around 23 % are hosted
on arbitrary websites. We notice that two main university resources are used
which are QCRI (Qatar Computing Research Institute) and CAMeL (Computational
Approaches to Modeling Language Lab). On the other hand most of the paid
resources are on LDC (Linguistic Data Consortium). There are also other free
websites for data hosting including SourceForge, GitLab and Mendeley Data. A
few percentage of the datasets are also hosted in Google Drive and Dropbox.

Figure 6: Most used repositories to host datasets.

#### Accessibility

In Figure 8, we breakdown the three types of accessibility for datasets. We
highlight that most of the datasets are free with a small percentage that
either requires registration or a fee (mostly LDC). We observe a general trend
of mainly publishing free data (around more than 80 % in the last few years).
In the repositories that host data, we observe that more than 50 % of the data
providers don’t declare the type of the license for the datasets as stated in
Figure 9. On the other hand around 10 % use custom licenses. Typically the
most used standard licenses are variations of Creative Common, followed by
Apaache, GPL and MIT.

#### Venues

Figure 10 breakdowns the venues that are used to publish the datasets across
the different years. We observe variations in the venues with around 70 unique
venues across conferences, journals, workshops and preprints. As we observe
from the figure the most used ones are LREC, WANLP followed by preprints
(including arXiv and others). The top venues are mainly conferences and
workshops.

#### Abstract Projection

In Figures 12 and 11, we highlight all the datasets that we collected as
projected embeddings. The embeddings were created by extracting the abstracts
of all the datasets using Semantic Scholar API then project the sentence
embeddings extracted from sentence-transformers
121212https://github.com/UKPLab/sentence-transformers. The embeddings are of
shape (200, 384), were projected to the 2D space using the t-SNE algorithm Van
der Maaten and Hinton (2008). We separate the positive and negative embeddings
of the x-axis for better visualization and analysis. We manually highlight 11
clusters for datasets that share common attributes. The clusters could be
grouped by task (like sentiment analysis), format (like speech) or type (like
parallel or multilingual datasets).

Figure 7: Distribution of datasets in terms of ethical risks.

#### Ethical Risks

Figure 7 highlights the distribution of datasets in terms of ethical risks.
Datasets that could potentially contain personal information are labeled as
medium. On the other hand, datasets that might contain, additionally, toxic
information or hate speech text are considered as high ethical risks. All the
remaining datasets are considered low. As we can observe from the Figure, most
of the collected resources have low ethical risks with around 30 % having
medium or high ethical risks.

#### Domain Representation

In language modelling, it is important to include datasets that are
representative not just of language diversity, but also of the domains or
topics covered by the datasets. In Table 2, we highlight the various domains
presented in the surveyed datasets. The majority of the datasets cover a
variety of domains, because they could be scrapped from the web, Wikipedia, or
collected manually. Around 30 % of the datasets are from social media, with
the remaining 12 % coming from news articles. Books and reviews account for
just a minor fraction of the genres in the datasets.

Table 2: Summary of domains in the surveyed datasets.  Domain | Count
---|---
social media | 61
news articles | 24
transcribed audio | 16
reviews | 9
books | 7
other | 83
Figure 8: Breakdown of the cost associated with the datasets from 2010 to
2021. Figure 9: Distribution of licenses across the datasets. Figure 10: The
count of venues within each year.  Figure 11: Positive projected embeddings of
all datasets’ abstracts.  Figure 12: Negative projected embeddings of all
datasets’ abstracts.


# Classical Mereology is Axiomatizable Using
Primitive Fusion in Two-Sorted Logic

Marcin Łyczak

(The work contains my axiomatization of Classical Mereology with a primitive
notion of mereological fusion. I presented this work, along with all the
necessary proofs, to Achille Varzi in a series of seminars in November 2022
during my scholarship at Columbia University, financed by the University of
Warmia and Mazury (POWR.03.05.00-00-Z310/17).)

###### Abstract

Mereological fusion, also known as composition and sum, was originally used by
me as a primitive notion to axiomatize Extensional Mereology with atoms in
[8]. Here, I extend this idea to axiomatize General Extensional Mereology,
also called Classical Mereology, which is neutral regarding the existence of
atoms. I give a proof that classical mereology is axiomatizable using
primitive mereological fusion within the framework of two-sorted logic, as I
announced in [8]. The use of the primitive notion of fusion instead of
primitive notion of part was considered by G. Leibniz [1, 50], S. Leśniewski
[7, CCLXIV, CCLXIV], K. Fine [3], J. Ketland, and T. Schindler [5], and S.
Kleishmid [6]. C. Lejewski formulated mereology using a single axiom with
primitive mereological sum [13, 222]. However, Lejewski’s theory is formulated
in non-classical language of Leśniewski’s ontology and contains complicated
quantification over function symbols, including quantification on part of and
sum of. Lejewski’s approach is not expressible in modern mereologies. The
presented theory is the first contemporary axiomatization of classical
mereology in two-sorted logic with primitive mereological fusion.

##  1 Two-sorted logic framework for both mereologies

The language of logic consists of variables of two sorts. The first for
individuals ($x,y,z,...$), and the second for pluralities ($zz,yy,zz,...$).
The predicate $\prec$, read as is one of, is applied to individuals and
pluralities, respectively. We use first-order identity $=$ between variables
of the first sort. We use notation from [4] because it is often used in
mereological considerations that cannot be expressed in first-order logic. The
considered theories are expressed within a two-sorted logic framework (see
[9]) that includes axioms for first-order identity, for the first sort, and
the unrestricted comprehension schema [10, 285]. Thus, we allow for the
existence of an empty name in the case of the second sort. We take the
following instantiations of the comprehension schema, and definitions:

$\displaystyle x\prec\mathsf{I}y$ $\displaystyle\leftrightarrow x=y,$
($\mathsf{I}$) $\displaystyle x\prec yy\cup zz$ $\displaystyle\leftrightarrow
x\prec yy\lor x\prec zz,$ ($\cup$) $\displaystyle x\prec yy\cap zz$
$\displaystyle\leftrightarrow x\prec yy\land x\prec zz,$ ($\cap$)
$\displaystyle xx\preccurlyeq yy$ $\displaystyle\leftrightarrow\forall
z(z\prec xx\to z\prec yy),$ ($\preccurlyeq$) $\displaystyle xx\approx yy$
$\displaystyle\leftrightarrow xx\preccurlyeq yy\land yy\preccurlyeq x.$
($\approx$)

##  2 Axiomatisation of Classical Mereology with primitive fusion

The only primitive notion of the theory is a predicate $F$ representing
mereological fusion, which we also call mereological composition. We use the
defined function symbol $\mathsf{U}$ read components of. The second sort term
$\mathsf{U}xx$ will be defined using the composition as all objects that form
wholes that are $xx$. We take the following specific axioms for $F$:

$\displaystyle\exists x(x\prec zz)\to\exists yF_{zz}y,$ ($\exists_{F}$)
$\displaystyle F_{xx}z\land xx\approx yy\to F_{yy}z,$ ($\approx_{F}$)
$\displaystyle F_{zz}x\land F_{yy}x\land F_{uu\cup zz}v\to F_{uu\cup yy}v,$
($\mathsf{ext}_{F}$) $\displaystyle F_{\mathsf{I}{y}}x\to x=y,$
($\mathtt{id}_{F}$) $\displaystyle F_{zz\cup\mathsf{I}x}y\land
F_{zz}y\to\exists_{vv\preccurlyeq\mathsf{U}zz}(F_{vv}x\land\exists z(z\prec
vv)),$ ($\mathtt{comp}_{F}$) $\displaystyle
F_{\mathsf{I}x\cup\mathsf{I}y}y\land
x\not=y\to\exists_{z\prec\mathsf{U}\mathsf{I}y}\neg\exists
u(F_{\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}z}u),$

where $\mathsf{U}zz$ is defined by an instantiation of the comprehension
schema as

$x\prec\mathsf{U}zz\leftrightarrow\exists_{z\prec zz}\exists yy(F_{yy}z\land
x\prec yy).$ ($\mathtt{df}\mathsf{U}_{F}$)

We briefly comment on axiomatization with the primitive predicate $F$.
($\exists_{F}$) expresses Leśniewski’s idea that there is a mereological
fusion of any objects. ($\approx_{F}$) describes a indiscernibility of
identicals by $F$. ($\mathsf{ext}_{F}$) is a version of extensionality, which
states that if some $x$ is a fusion of $zz$ and fusion of $yy$ ($zz$ and $yy$
are possibly different), then both $zz$ and $yy$ can be substituted for each
other when creating various fusions, as long as at least one of them is
required. ($\mathtt{id}_{F}$) expresses the idea that the fusion of all
objects identical to $y$ does not create a new individual; in other words, the
fusion is identical to $y$. In mereology with part as primitive,
($\mathtt{id}_{F}$) is an obvious consequence of basic properties of being a
part, and definition of fusion, which was noted already by Leśniewski.
($\mathtt{comp}_{F}$) describes the constructiveness of composition: if $y$ is
the fusion of $zz$ and $x$, but $x$ is unnecessary, then $x$ can be composed
of some $yy$ that forms some wholes belonging to $zz$. An informal version of
(2) was considered by S. Kleinschmidt in [6, 13]. (2) is closely connected
with the so-called weak supplementation principle, which we will introduce
shortly.
We denote the set of mereology with primitive fusion as

$\mathsf{GEM}^{\mathsf{pl}}_{F}=\eqref{esum}+\eqref{F=}+\eqref{fs}+\eqref{ext}+\eqref{comp}+\eqref{WSP}.$

In $\mathsf{GEM}^{\mathsf{pl}}_{F}$, we define the notion of being a part as

$Pxy\leftrightarrow\exists zz(F_{zz}y\land x\prec zz).$ ($\mathtt{df}P_{F}$)

The counterpart of this definition was a theorem in the original Leśniewski’s
mereology. Leśniewski proved it and noted that this shows that the
mereological fusion may be a primitive notion of mereology, instead of the
primitive notion of mereological part [7]:

> THEOREM CCLXIV. $P$ is an ingredient of object $Q$, when and only when, for
> some $a$, $Q$ is the sum of objects $a$, and $P$ is $a$.
> Theorems $[...]$, CCLXIV, $[...]$ show that instead of such terms as ‘part’
> and ‘ingredient’ I could choose as the fundamental term of my ‘general
> theory of sets’, any of the terms: $[...]$, ‘class’, $[...]$.

##  3 Axiomatization of Classical Mereology with primitive part

We use nomenclature from [14]. Axioms of the so-called Classical Mereology
(see [2]), called also General Extensional Mereology, we express in the two-
sorted logic are:

$\displaystyle PPxy\to\neg PPyx,$ ($\mathtt{as}_{PP}$) $\displaystyle
PPxy\land PPyz\to PPxz,$ ($\mathtt{trans}_{PP}$) $\displaystyle\exists
x(x\prec zz)\to\exists yF_{zz}y,$ ($\exists_{F}$) $\displaystyle F_{zz}x\land
F_{zz}y\to x=y,$ ($\mathtt{fun}_{F}$)

where

$\displaystyle F_{zz}x\leftrightarrow\forall_{y\prec zz}(Pzx)\land\forall
y(Pyx\to\exists_{v\prec zz}(Ovy)),$ ($\mathtt{df}F_{P}$) $\displaystyle
Pxy\leftrightarrow PPxy\lor x=y,$ ($\mathtt{df}P_{PP}$) $\displaystyle
Oxy\leftrightarrow\exists z(Pzx\land Pzy).$ ($\mathtt{df}O$)

To characterize mereology it is often chosen primitive notion of being a part,
in an inclusive sense, with different axioms:

$\displaystyle Pxx,$ ($\mathtt{ref}_{P}$) $\displaystyle Pxy\land Pyx\to x=y,$
($\mathtt{antis}_{P}$) $\displaystyle Pxy\land Pyz\to Pxz,$
($\mathtt{trans}_{P}$)

and $PP$ is defined as

$PPxy\leftrightarrow Pxy\land x\not=y.$ ($\mathtt{df}PP_{P}$)

It as a well-known fact that

$\eqref{ring}+\eqref{antising}+\eqref{transing}+\eqref{defpring}=\eqref{aspr}+\eqref{transpr}+\eqref{ing/pr}.$

Thus, from a formal perspective, the choice of the two primitives doesn’t make
much difference. However, for convenience, $P$ is often preferred.
We denote the set of theses of classical mereology by

$\mathsf{GEM}^{\mathsf{pl}}_{P}=\eqref{ring}+\eqref{antising}+\eqref{transing}+\eqref{esum}+\eqref{fsum}.$

##  4 Definitional equivalence of $\mathsf{GEM}^{\mathsf{pl}}_{P}$ and
$\mathsf{GEM}^{\mathsf{pl}}_{F}$

######  Lemma 1.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}$ the following formulas are provable:

$\displaystyle F_{\mathsf{I}x}x,$ ($F_{\mathsf{I}x}x$) $\displaystyle
Pxy\leftrightarrow F_{\mathsf{I}x\cup\mathsf{I}y}y.$ ($P_{F}2$)

###### Proof.

To prove ($F_{\mathsf{I}x}x$) we use the fact that $x\prec\mathsf{I}x$ and so
from axiom ($\exists_{F}$) we have $\exists yF_{\mathsf{I}x}y$. Thus, using
classical logic and ($\mathtt{id}_{F}$), we obtain $\exists
y(F_{\mathsf{I}x}y\land x=y)$, and so $F_{\mathsf{I}x}x$. To prove
‘$\leftarrow$’ of ($P_{F}2$) we assume $F_{\mathsf{I}x\cup\mathsf{I}y}y$ and
then we just use the fact that $x\prec\mathsf{I}x\cup\mathsf{I}y$ and
definition of ($\mathtt{df}P_{F}$). To prove ‘$\to$’ of ($P_{F}2$) we assume
$Pxy$ and from definition ($\mathtt{df}P_{F}$) we have $F_{aa}y\land x\prec
aa$, for some $aa$. Having $x\prec aa$ we obtain from logic
$aa\approx\mathsf{I}x\cup aa$, so using $F_{aa}y$ and ($\approx_{F}$) we get
$F_{\mathsf{I}x\cup aa}y$. So we have $F_{\mathsf{I}x\cup aa}y$, $F_{aa}y$,
and $F_{\mathsf{I}y}y$ from ($F_{\mathsf{I}x}x$), thus we use
($\mathsf{ext}_{F}$) and obtain $F_{\mathsf{I}x\cup\mathsf{I}y}y$. ∎

######  Lemma 2.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$ formulas
($\mathtt{ref}_{P}$), ($\mathtt{antis}_{P}$), and ($\mathtt{trans}_{P}$) are
provable.

###### Proof.

To prove ($\mathtt{ref}_{P}$) we just use ($F_{\mathsf{I}x}x$) and by
($\mathtt{df}P_{F}$) and $x\prec\mathsf{I}x$ we obtain $Pxx$. To prove
($\mathtt{antis}_{P}$) we assume $Pxy$ and $Pyx$. From this using ($P_{F}2$)
and ($\approx_{F}$) we obtain $F_{\mathsf{I}x\cup\mathsf{I}y}x$ and
$F_{\mathsf{I}x\cup\mathsf{I}y}y$. From the latter with the use of logical
thesis
$\mathsf{I}x\cup\mathsf{I}y\approx\mathsf{I}x\cup\mathsf{I}x\cup\mathsf{I}y$
and ($\approx_{F}$) we obtain
$F_{\mathsf{I}x\cup\mathsf{I}x\cup\mathsf{I}y}y$. Now we take
($\mathsf{ext}_{F}$) with:
$\mathsf{I}x\cup\mathsf{I}y/zz,\mathsf{I}x/yy,\mathsf{I}x/uu,y/v$ and we
obtain as $\mathsf{GEM}^{\mathsf{pl}}_{F}$ thesis the following

$(\star)\colon F_{\mathsf{I}x\cup\mathsf{I}y}x\land F_{\mathsf{I}x}x\land
F_{\mathsf{I}x\cup\mathsf{I}x\cup\mathsf{I}y}y\to
F_{\mathsf{I}x\cup\mathsf{I}x}y.$

We have all predecessors of ($\star$) so we get
$F_{\mathsf{I}x\cup\mathsf{I}x}y$, i.e $F_{\mathsf{I}x}y$ by
$\mathsf{I}x\cup\mathsf{I}x\approx\mathsf{I}x$ and ($\approx_{F}$). Having
$F_{\mathsf{I}x}y$ we just use ($\mathtt{id}_{F}$) and have $x=y$. To prove
($\mathtt{trans}_{P}$) we assume $Pxy\land Pyz$. Using ($P_{F}2$) we obtain
$F_{\mathsf{I}x\cup\mathsf{I}y}y\land F_{\mathsf{I}y\cup\mathsf{I}z}z$. Now we
use ($\mathsf{ext}_{F}$) with: $\mathsf{I}y/zz$,
$\mathsf{I}x\cup\mathsf{I}y/yy$, $\mathsf{I}z/uu$, $y/x$, $z/v$ and we obtain
as $\mathsf{GEM}^{\mathsf{pl}}_{F}$ thesis the following

$(\star\star)\colon F_{\mathsf{I}y}y\land F_{\mathsf{I}x\cup\mathsf{I}y}y\land
F_{\mathsf{I}z\cup\mathsf{I}y}z\to
F_{\mathsf{I}z\cup\mathsf{I}x\cup\mathsf{I}y}z.$

$F_{\mathsf{I}x\cup\mathsf{I}y}y\land F_{\mathsf{I}y\cup\mathsf{I}z}z$ we
already have and $F_{\mathsf{I}y}y$ we have as earlier from
($F_{\mathsf{I}x}x$), so we have all predecessors of $(\star\star)$, and thus
we obtain $F_{\mathsf{I}z\cup\mathsf{I}x\cup\mathsf{I}y}z$, so $Pxz$ directly
from ($\mathtt{df}P_{F}$) and
$z\prec\mathsf{I}z\cup\mathsf{I}x\cup\mathsf{I}y$. ∎

Next we show uniqueness of fusion in $\mathsf{GEM}^{\mathsf{pl}}_{F}$.

######  Lemma 3.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$ formula ($\mathtt{fun}_{F}$)
is provable.

###### Proof.

We assume $F_{zz}x$ and $F_{zz}y$. From $F_{zz}x$, $F_{\mathsf{I}x}x$, and
$F_{zz\cup zz}y$ (taken from $F_{zz}y$, $zz\approx zz\cup zz$ and
($\approx_{F}$)) using ($\mathsf{ext}_{F}$) we have $F_{zz\cup\mathsf{I}x}y$
thus $Pxy$ by ($\mathtt{df}P_{F}$). Analogically from $F_{zz}y$,
$F_{\mathsf{I}y}y$, and $F_{zz\cup zz}x$ using ($\mathsf{ext}_{F}$) we have
$F_{zz\cup\mathsf{I}y}x$ thus $Pyx$ by ($\mathtt{df}P_{F}$). So $Pxy$ and
$Pyx$. Therefore, using proved ($\mathtt{antis}_{P}$) we get $x=y$. ∎

Now we show that fusion axiomatized in $\mathsf{GEM}^{\mathsf{pl}}_{F}$ is
fusion in Leśniewski’s sense.

######  Lemma 4.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$ formula
$F_{zz}x\to\forall_{y\prec zz}(Pyx)\land\forall y(Pyx\to\exists_{v\prec
zz}(Ovy))$ is provable.

###### Proof.

We assume $F_{zz}x$. From this and ($\mathtt{df}P_{F}$) we directly have
$\forall_{y\prec zz}Pyx$. We assume $Pyx$, for any $y$ and by ($P_{F}2$) we
obtain $F_{\mathsf{I}y\cup\mathsf{I}x}x$. From this, ($F_{\mathsf{I}x}x$), and
assumed $F_{zz}x$ we obtain $F_{zz\cup\mathsf{I}y}x$ by ($\mathsf{ext}_{F}$).
Thus we have $F_{zz}x$ and $F_{zz\cup\mathsf{I}y}x$. We use
($\mathtt{comp}_{F}$) and there is $aa\preccurlyeq\mathsf{U}zz$ such that
$F_{aa}y$ and $a\prec aa$ for some $a$. From $aa\preccurlyeq\mathsf{U}zz$ and
$a\prec aa$ we have $a\prec\mathsf{U}zz$ thus applying
($\mathtt{df}\mathsf{U}_{F}$) for some $v\prec zz$ and $yy$ we have
$F_{yy}v\land a\prec yy$. So using the latter and ($\mathtt{df}P_{F}$), we
have $Pav$. We also know that $F_{aa}y$ and $a\prec aa$, so $Pay$. Thus
applying ($\mathtt{df}O$) we obtain $Ovy$, and we know that $v\prec zz$, which
ends the proof. ∎

To prove the converse implication we need the following Lemma

######  Lemma 5.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$ formula
$F_{\mathsf{U}\mathsf{I}x}x$ is provable.

###### Proof.

First, we show that $F_{\mathsf{U}\mathsf{I}x}y\to x=y$ is a thesis of
$\mathsf{GEM}^{\mathsf{pl}}_{F}$. We assume that $F_{\mathsf{U}\mathsf{I}x}y$
and $x\not=y$. From assumed $F_{\mathsf{U}\mathsf{I}x}y$, and the fact that
$\mathsf{U}\mathsf{I}x\approx\mathsf{U}\mathsf{I}x\cup\mathsf{I}x$ using
($\approx_{F}$) we obtain that $F_{\mathsf{U}\mathsf{I}x\cup\mathsf{I}x}y$.
The latter with $F_{\mathsf{U}\mathsf{I}x}y$, and $F_{\mathsf{I}y}y$ by
($\mathsf{ext}_{F}$) gives $F_{\mathsf{I}x\cup\mathsf{I}y}y$. Having
$F_{\mathsf{I}x\cup\mathsf{I}y}y$ and assumed $x\not=y$ we use (2) and we
obtain that there is $a$ such that
$a\prec\mathsf{U}\mathsf{I}y\land\neg\exists
u(F_{\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}a}u)$.
$a\prec\mathsf{U}\mathsf{I}y$ from definitions ($\mathtt{df}\mathsf{U}_{F}$)
and ($\mathsf{I}$) gives that for some $aa$ we have $F_{aa}y\land a\prec aa$,
so with the use of $aa\approx aa\cup\mathsf{I}a$ and ($\approx_{F}$) we get
$F_{aa\cup\mathsf{I}a}y$. Having $F_{\mathsf{U}\mathsf{I}x}y$ and $F_{aa}y$
and $F_{aa\cup\mathsf{I}a}y$ we use ($\mathsf{ext}_{F}$) and obtain
$F_{\mathsf{U}\mathsf{I}x\cup\mathsf{I}a}y$. So we have
$F_{\mathsf{U}\mathsf{I}x\cup\mathsf{I}a}y$ and $F_{\mathsf{U}\mathsf{I}x}y$
thus we apply ($\mathtt{comp}_{F}$) and we obtain that there are $bb$ and $b$
such that $b\prec bb$ and $bb\preccurlyeq\mathsf{U}\mathsf{U}\mathsf{I}x$ and
$F_{bb}a$. Thus we have $b\prec\mathsf{U}\mathsf{U}\mathsf{I}x$ and $(i)\colon
Pba$ from ($\mathtt{df}P_{F}$), $F_{bb}a$, and $b\prec bb$. From
$b\prec\mathsf{U}\mathsf{U}\mathsf{I}x$ we have that there are $cc$ and $c$
such that $b\prec cc$, $c\prec\mathsf{U}\mathsf{I}x$, and $F_{cc}c$, so
$(ii)\colon Pbc$ using ($\mathtt{df}P_{F}$). Now from
$c\prec\mathsf{U}\mathsf{I}x$ we have that there is $dd$ and such that
$F_{dd}x\land c\prec dd$, so $(iii)\colon Pcx$ by ($\mathtt{df}P_{F}$). Thus
from $(ii)$ and $(iii)$ using ($\mathtt{trans}_{P}$) we obtain $Pbx$. So, we
have $Pbx$ and $Pba$ from (i) and using definitions ($\mathtt{df}P_{F}$) and
($\mathtt{df}\mathsf{U}_{F}$) we get $b\prec\mathsf{U}\mathsf{I}x\land
b\prec\mathsf{U}\mathsf{I}a$, thus
$b\prec\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}a$. Now we can use the
axiom of the existence of fusion ($\exists_{F}$) which gives $\exists
u(F_{\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}a}u)$ which is false what we
showed earlier, so we finally have $\forall y(F_{\mathsf{U}\mathsf{I}x}y\to
x=y)$. We know that $x\prec\mathsf{U}\mathsf{I}x$ so we again use
($\exists_{F}$) and we have $F_{\mathsf{U}\mathsf{I}x}d$ for some $d$, but
then we use already proved thesis and get $x=d$, so finally
$F_{\mathsf{U}\mathsf{I}x}x$. ∎

######  Lemma 6.

In $\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$ formula $\forall_{y\prec
zz}(Pyx)\land\forall y(Pyx\to\exists_{v\prec zz}(Ovy))\to F_{zz}x$ is
provable.

###### Proof.

We assume $\forall_{y\prec zz}(Pyx)\land\forall y(Pyx\to\exists_{v\prec
zz}(Ovy))$ from $\forall_{y\prec zz}(Pyx)$ and definitions
($\mathtt{df}P_{F}$) and ($\mathtt{df}\mathsf{U}_{F}$) we obtain
$zz\preccurlyeq\mathsf{U}\mathsf{I}x$, thus $\mathsf{U}\mathsf{I}x\approx
zz\cup\mathsf{U}\mathsf{I}x$. From Lemma 5 we know that
$F_{\mathsf{U}\mathsf{I}x}x$, thus using $\mathsf{U}\mathsf{I}x\approx
zz\cup\mathsf{U}\mathsf{I}x$ and ($\approx_{F}$) we obtain
$F_{zz\cup\mathsf{U}\mathsf{I}x}x$. From $\forall y(Pyx\to\exists_{v\prec
zz}(Ovy))$ and proved reflexivity of $P$ ($\mathtt{ref}_{P}$) we obtain that
$\exists v(v\prec zz)$. From the latter and axiom of the existence of fusion
($\exists_{F}$) we have that $F_{zz}a$ for some $a$. So, we have $F_{zz}a$ and
$F_{\mathsf{I}a}a$ and $F_{zz\cup\mathsf{U}\mathsf{I}x}x$ thus using
($\mathsf{ext}_{F}$) we obtain $F_{\mathsf{I}a\cup\mathsf{U}\mathsf{I}x}x$ and
we have $F_{\mathsf{U}\mathsf{I}x}x$ and $F_{\mathsf{I}x}x$ so again using
($\mathsf{ext}_{F}$) we have $F_{\mathsf{I}a\cup\mathsf{I}x}x$. Now, we
additionally assume indirectly that $x\not=a$. From (2) for some $b$ we obtain
that $b\prec\mathsf{U}\mathsf{I}x\land\neg\exists
u(F_{\mathsf{U}\mathsf{I}b\cap\mathsf{U}\mathsf{I}a}u)$. From
$b\prec\mathsf{U}\mathsf{I}x$ and definition ($\mathtt{df}\mathsf{U}_{F}$) we
obtain $\exists yy(F_{yy}x\land b\prec yy)$, i.e. $Pbx$ by
($\mathtt{df}P_{F}$). Now we take assumption $\forall y(Pyx\to\exists_{v\prec
zz}(Ovy))$ with $b/y$ and using $Pbx$ for some $c$ we obtain $c\prec zz\land
Ocb$. From $Ocb$ we obtain that for some $d$ we have $Pdc\land Pdb$. From
$Pdc$ and $c\prec zz$ using ($\mathtt{df}P_{F}$) we have $d\prec\mathsf{U}zz$.
Having $d\prec\mathsf{U}zz$ with the use of $F_{zz}a$ and ($\mathsf{ext}_{F}$)
we get $d\prec\mathsf{U}\mathsf{I}a$. From $Pdb$ we obtain
$d\prec\mathsf{U}\mathsf{I}b$ by definitions ($\mathtt{df}P_{F}$) and
($\mathtt{df}\mathsf{U}_{F}$). Thus, we have $d\prec\mathsf{U}\mathsf{I}b$ and
$d\prec\mathsf{U}\mathsf{I}a$ and next
$d\prec\mathsf{U}\mathsf{I}b\cap\mathsf{U}\mathsf{I}a$. The latter using
($\exists_{F}$) gives $\exists
u(F_{\mathsf{U}\mathsf{I}b\cap\mathsf{U}\mathsf{I}a}u)$, which is false, so
finally we obtain $x=a$. We know that $F_{zz}a$ so using $x=a$ we finally
obtain $F_{zz}x$, what we wanted to prove. ∎

Thus using Lemmas 2-4 and 6 we obtain

######  Theorem 1.

$\mathsf{GEM}^{\mathsf{pl}}_{P}\subseteq\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}$

Now we show converse dependency.

######  Lemma 7.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}$ the following formulas are provable:

$\displaystyle PPxy\to\exists z(PPzy\land\neg Ozx),$ ($\mathtt{WSP}$)
$\displaystyle F_{zz}x\leftrightarrow Mub_{zz}x,$ ($F_{P}=Mub$)

where $Mub$ is defined as

$Mub_{zz}x\leftrightarrow\exists y(y\prec zz)\land\forall_{y\prec
zz}(Pyx)\land\forall y(\forall_{v\prec zz}(Pvy)\to Pxy).$ ($\mathtt{dfMub}$)

Weak supplementation principle $\eqref{WSP}$ was analyzed, and used as
anaxiom, by P. Simons in his minimal extensional mereology [11, 28], but its
stronger versions were also used as axioms by F. Drewnowski and Leśniewski
[12]. Analog of ($F_{P}=Mub$) was proved in Leśniewski’s mereology by A.
Tarski [7, 327–328].

######  Lemma 8.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}$ formula ($\mathtt{id}_{F}$) is provable.

###### Proof.

We assume $F_{\mathsf{I}{y}}x$. We get $Pyx$ and using ($F_{P}=Mub$) we take
$\forall_{v\prec\mathsf{I}y}(Pvy)\to Pxy)$. We take $y/v$ and using
($\mathtt{ref}_{P}$) we obtain $Pxy$. Thus we have $Pyx$ and $Pyx$ so using
($\mathtt{antis}_{P}$) we have $x=y$. ∎

######  Lemma 9.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}$ principle of fusion extensionality
($\mathsf{ext}_{F}$) is provable.

###### Proof.

We assume (i): $F_{zz}x$ (ii): $F_{yy}x$, and (iii): $F_{uu\cup zz}v$. From
(i) and ($F_{P}=Mub$) we have $\forall_{z\prec zz}(Pzv)\to Pxv$. Predecessor,
we obtain from (iii) and ($\mathtt{df}F_{P}$), so $Pxv$. Now from (ii) and
($\mathtt{df}F_{P}$) we have $\forall_{z\prec yy}(Pzx)$, thus by
($\mathtt{trans}_{P}$) and obtained $Pxv$ we have $\forall_{z\prec yy}(Pzv)$.
From the latter using (iii) and ($\mathtt{df}F_{P}$) we get $\forall_{z\prec
uu\cup yy}(Pzv)$. Now for any $u$ such that $Puv$ we assume indirectly that
$\forall_{y\prec uu\cup yy}(\neg Ouy)$, i.e. ($\star$): $\forall_{y\prec
uu}(\neg Ouy)$ and $(\star\star)\colon\forall_{y\prec yy}(\neg Ouy)$. From
$Puv$, (iii), ($\mathtt{df}F_{P}$), and ($\star$) we obtain $Oua$ for some
$a\prec zz$, thus $Pax$, because $F_{zz}x$. Now using $Pax$ and $Oua$ we have
$Oux$, i.e there is $b$ such that $Pbu\land Pbx$. From $Pbx$, (ii) and
($\mathtt{df}F_{P}$) we have $\exists_{y\prec yy}(Oby)$, and so using $Pbu$ we
get $\exists_{y\prec yy}(Ouy)$ which contradicts with ($\star\star$). So
finally we have $\forall_{z\prec uu\cup yy}(Pzv)\land\forall
u(Puv\to\exists_{z\prec uu\cup yy}(Ozu))$ and $F_{yy\cup uu}v$ by
($\mathtt{df}F_{P}$). ∎

In $\mathsf{GEM}^{\mathsf{pl}}_{P}$ we define symbol $\mathsf{U}$ in the
following way

$x\prec\mathsf{U}zz\leftrightarrow\exists_{y\prec zz}Pxy.$
($\mathtt{df}\mathsf{\mathsf{U}}_{P}$)

######  Lemma 10.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}+\eqref{UP}$ principle of composition
($\mathtt{comp}_{F}$) is provable.

###### Proof.

Proof indirect. We assume $F_{zz\cup\mathsf{I}x}y\land F_{zz}y$ and
$\neg\exists_{yy\preccurlyeq\mathsf{U}zz}(F_{yy}x)$. From
$F_{zz\cup\mathsf{I}x}y$ and the definition of fusion ($\mathtt{df}F_{P}$) we
obtain $Pxy$. Let $u\prec zz^{*}\leftrightarrow Pux\land u\prec\mathsf{U}zz$.
Full comprehension schema guarantees that such a plurality exists. From
$F_{zz}y$, $Pxy$ and ($\mathtt{df}F_{P}$), we know that $\exists_{z\prec
zz}Ozx$ thus $\exists z\exists u(Pux\land Puz\land z\prec zz)$, $\exists
u(Pux\land\exists z(Puz\land z\prec zz))$ and next $\exists u(Pux\land
u\prec\mathsf{U}zz)$ by ($\mathtt{df}\mathsf{\mathsf{U}}_{P}$), i.e. $\exists
u(u\prec zz^{*})$ by definition od $zz^{*}$, thus from ($\exists_{F}$) for
some $a$ we have $F_{zz^{*}}a$. From
$\neg\exists_{yy\preccurlyeq\mathsf{U}zz}(F_{yy}x)$ we have (i): $\neg
F_{zz^{*}}x$, because $zz^{*}\preccurlyeq\mathsf{U}zz$. From $F_{zz^{*}}a$ and
($F_{P}=Mub$) we have $Mub_{zz^{*}}a$, thus $\forall_{z\prec zz^{*}}(Pzx)\to
Pax)$, and so $Pax$, from the definition of $zz^{*}$. Assume furthermore that
$a\not=x$. Then from $PPax$ and ($\mathtt{WSP}$) we have that exists $v$ such
that $Pvx\land\neg Ova$. Now, we additionally indirectly assume that for some
$t\prec zz$ we have $Ovt$. So there is $u$ such that $Put\land Puv$. From
$Puv$ and $Pvx$ we have $Pux$ by ($\mathtt{trans}_{P}$). Having that $Put$,
$t\prec zz$ and $Pux$, from definitions of $zz^{*}$ and
($\mathtt{df}\mathsf{\mathsf{U}}_{P}$) we obtain $u\prec zz^{*}$ but then
$Pua$ and this contradict to $Puv$ and earlier obtained $\neg Ova$. Thus we
have $\forall_{t\prec zz}\neg Ovt$. From assumption we have $F_{zz}y$, so from
definition ($\mathtt{df}F_{P}$) and transposition we have $\forall_{t\prec
zz}(\neg Otv)\to\neg Pvy$, predecessor we already have thus $\neg Pvy$.
However, we have $Pvx$, and $Pxy$ from basic assumption, thus by
($\mathtt{trans}_{P}$) $Pvy$, which gives a contradiction. So, we have $x=a$
and so $\neg F_{zz^{*}}a$ by (i) but we know that $a$ is the sum of $zz^{*}a$
thus contradiction. ∎

######  Lemma 11.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}+\eqref{UP}$ formula (2) is provable.

###### Proof.

We assume $F_{\mathsf{I}x\cup\mathsf{I}y}y\land x\not=y$ and $\forall
z(z\prec\mathsf{U}\mathsf{I}y\to\exists
u(F_{\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}z}u))$. From
$F_{\mathsf{I}x\cup\mathsf{I}y}y$ and definition of fusion
($\mathtt{df}F_{P}$) we obtain $Pxy$, thus using assumed $x\not=y$ we have
$PPxy$. Using the latter and ($\mathtt{WSP}$) we obtain $PPay\land\neg Oax$
for some $a$. From $Pay$ and definition of
($\mathtt{df}\mathsf{\mathsf{U}}_{P}$) we obtain
$a\prec\mathsf{U}\mathsf{I}y$, thus from assumption taking $a/z$ we obtain
$F_{\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}a}b$ for some $b$. The latter
by definition of fusion ($\mathtt{df}F_{P}$) and ($\mathtt{ref}_{P}$) implies
that $\exists z(z\prec\mathsf{U}\mathsf{I}x\cap\mathsf{U}\mathsf{I}a)$. Thus
by definition ($\mathtt{df}\mathsf{\mathsf{U}}_{P}$) we have $\exists
z(Pzx\land Pza)$, i.e. $Oxa$ which gives a contradiction. ∎

######  Lemma 12.

In $\mathsf{GEM}^{\mathsf{pl}}_{P}$ formula ($\approx_{F}$) is provable.

###### Proof.

We assume indirectly that $F_{zz}x\land zz\approx yy$ and $\neg F_{yy}x$ We
obtain (i): $\forall_{z\prec zz}Pzx$, (ii): $\forall y(Pyx\to\exists_{z\prec
zz}Ozy)$, and (iii): $\neg\forall_{z\prec yy}Pzx\lor\neg\forall
y(Pyx\to\exists z(z\prec yy\land Ozy))$. If $\neg\forall_{z\prec yy}Pzx$ then
we take $a/z\colon a\prec yy\land\neg Pax$, so $a\prec zz$ by $zz\approx yy$
but then using (i) we have $Pax$ which is false. Therefore, we have
$\forall_{z\prec yy}Pzx$ so using (iii) we obtain $\neg\forall y(Pyx\to\exists
z(z\prec yy\land Ozy))$ we take $b/y\colon Pbx\land\forall_{z\prec yy}\neg
Ozb$. From $Pbx$ and (ii) we have that $\exists_{z\prec zz}Ozb$ so we take
$c/z\colon$ $c\prec zz\land Ocb$. From $c\prec zz$ and $zz\approx yy$ we have
$c\prec yy$ thus using $\forall_{z\prec yy}\neg Ozb$ and $c/z$ we obtain $\neg
Ocb$ which gives a contradiction. ∎

So, all axioms of $\mathsf{GEM}^{\mathsf{pl}}_{F}$ are theorems of
$\mathsf{GEM}^{\mathsf{pl}}_{P}$. Lastly, what we need to show is that the
definition ($\mathtt{df}P_{F}$) in $\mathsf{GEM}^{\mathsf{pl}}_{F}$ is a
thesis in $\mathsf{GEM}^{\mathsf{pl}}_{P}$, definition of
($\mathtt{df}\mathsf{U}_{F}$) in $\mathsf{GEM}^{\mathsf{pl}}_{F}$ is a thesis
in $\mathsf{GEM}^{\mathsf{pl}}_{P}$ and conversely that definition
($\mathtt{df}\mathsf{\mathsf{U}}_{P}$) in $\mathsf{GEM}^{\mathsf{pl}}_{P}$ is
thesis in $\mathsf{GEM}^{\mathsf{pl}}_{F}$.

######  Lemma 13.

Definition ($\mathtt{df}P_{F}$) in $\mathsf{GEM}^{\mathsf{pl}}_{F}$ is a
thesis in $\mathsf{GEM}^{\mathsf{pl}}_{P}$.

###### Proof.

(“$\leftarrow$”) We assume $\exists zz(F_{zz}y\land x\prec zz)$. From the
definition of fusion ($\mathtt{df}F_{P}$) we know that all $zz$ are parts of
$y$ thus also $x$, i.e. $Pxy$. (“$\to$”). We assume indirectly that $Pxy$ and
$\forall zz(F_{zz}y\to\neg x\prec zz)$. We take
$\mathsf{I}x\cup\mathsf{I}y/zz$ and obtain
$F_{\mathsf{I}x\cup\mathsf{I}y}y\to\neg x\prec\mathsf{I}x\cup\mathsf{I}y$. The
right side of the latter implication is false, thus $\neg
F_{\mathsf{I}x\cup\mathsf{I}y}y$. We have that
$\forall_{z\prec\mathsf{I}x\cup\mathsf{I}y}Pzy$, thus from the definition of
fusion ($\mathtt{df}F_{P}$) and
$\forall_{z\prec\mathsf{I}x\cup\mathsf{I}y}Pzy$ we obtain that for some $a$ we
obtain $Pay$ and $\neg Oay$ which is contradictory. ∎

As an immediate consequence of Lemma 13, we obtain

######  Lemma 14.

Definition ($\mathtt{df}\mathsf{U}_{F}$) in $\mathsf{GEM}^{\mathsf{pl}}_{F}$
is a thesis of $\mathsf{GEM}^{\mathsf{pl}}_{P}+\eqref{UP}$.

Lemmas 8-14 give that

######  Theorem 2.

$\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}\subseteq\mathsf{GEM}^{\mathsf{pl}}_{P}+\eqref{UP}$

Definition ($\mathtt{df}\mathsf{\mathsf{U}}_{P}$) in
$\mathsf{GEM}^{\mathsf{pl}}_{P}$ is thesis of
$\mathsf{GEM}^{\mathsf{pl}}_{F}$, and this is simple consequence of the
definitions ($\mathtt{df}P_{F}$) and ($\mathtt{df}\mathsf{U}_{F}$). Thus,
using this, along with Theorems 1 and 2, we ultimately arrive at the main
result of the work. Our axiomatization of general extensional mereology with
fusion as a primitive notion is definitionally equivalent to general
extensional mereology with the primitive notion of being a part.

######  Theorem 3.

$\mathsf{GEM}^{\mathsf{pl}}_{F}+\eqref{dfing}=\mathsf{GEM}^{\mathsf{pl}}_{P}+\eqref{UP}.$

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