# ClimateNLP: Analyzing Public Sentiment Towards Climate Change using Natural
Language Processing

Ajay Krishnan T. K.
Applied NLP Research Lab
School of Digital Sciences
Kerala University of Digital Sciences-
Innovation and Technology
Thiruvananthapuram, India
<EMAIL_ADDRESS>
V. S. Anoop
Applied NLP Research Lab
School of Digital Sciences
Kerala University of Digital Sciences-
Innovation and Technology
Thiruvananthapuram, India
<EMAIL_ADDRESS>

###### Abstract

Climate change’s impact on human health poses unprecedented and diverse
challenges. Unless proactive measures based on solid evidence are implemented,
these threats will likely escalate and continue to endanger human well-being.
The escalating advancements in information and communication technologies have
facilitated the widespread availability and utilization of social media
platforms. Individuals utilize platforms such as Twitter and Facebook to
express their opinions, thoughts, and critiques on diverse subjects,
encompassing the pressing issue of climate change. The proliferation of
climate change-related content on social media necessitates comprehensive
analysis to glean meaningful insights. This paper employs natural language
processing (NLP) techniques to analyze climate change discourse and quantify
the sentiment of climate change-related tweets. We use ClimateBERT, a
pretrained model fine-tuned specifically for the climate change domain. The
objective is to discern the sentiment individuals express and uncover patterns
in public opinion concerning climate change. Analyzing tweet sentiments allows
a deeper comprehension of public perceptions, concerns, and emotions about
this critical global challenge. The findings from this experiment unearth
valuable insights into public sentiment and the entities associated with
climate change discourse. Policymakers, researchers, and organizations can
leverage such analyses to understand public perceptions, identify influential
actors, and devise informed strategies to address climate change challenges.

_Keywords_ Climate change $\cdot$ Sentiment analysis $\cdot$ ClimateBERT
$\cdot$ Public discourse $\cdot$ Natural language processing

##  1 Introduction

According to the World Health Organization (WHO), the greatest health threat
to people in the twenty-first century is climate change. Between 2030 and
2050, this risk is expected to cause an additional 250,000 fatalities annually
and express itself in various ways. Many of these health concerns can be
decreased or avoided with prompt and effective adaptation, but doing so
necessitates in-depth research and policies that are multi-sectoral, multi-
system, and collaborative at several scales[Fard et al., 2022]. One of the
most important issues that needs significant attention is climate change. The
majority of scientists agree that human activity is accelerating the Earth’s
climate change, which is having a disastrous effect on the world and its
population. The consequences of climate change are more clear. In recent
years, extreme weather occurrences like hurricanes, tornadoes, hail,
lightning, fires, and floods have increased in frequency and intensity. As the
world’s ecosystems change quickly, access to the natural resources and
agricultural methods that support humanity is in danger. [Ardabili et al.,
2020]. The problem of climate change is complicated, and there is no quick
fix. But to identify solutions, it’s critical to comprehend the issue.

Large volumes of text data can be analyzed using natural language processing
which may unearth interesting patternsAnoop [2023]. The natural language
processing approaches can be applied to the climate change domain as well for
finding the causes and leveraging patterns such as public sentiment and
discourse towards this global issue. Recent years have witnessed many people
using social media to share their views, concerns, and public opinions on any
topic under the skyAnoop et al. [2023]Jickson et al. [2023]. This has caused a
huge amount of unstructured but dynamic data to be generated in such
platforms, which are goldmines for social science researchersJohn et al.
[2023]Anoop and Sreelakshmi [2023]. Collecting, curating, and analyzing such
data is crucial for finding public perceptions and viewpoints on socially
relevant discussionsVarghese and Anoop [2022]Lekshmi and Anoop [2022].
Similarly, understanding public opinions and sentiments on climate change is
crucial for public policymakers, governments, and other administrators to
devise better policies and intervention measures to address the challenges.

In this research, natural language processing is used to examine tweets that
discuss climate change. We use ClimateBERTWebersinke et al. [2021], a pre-
trained language model trained on a large set of climate change-related
documents, and fine-tune the same for sentiment classification tasks. The
findings may be used better to understand the public’s understanding of
climate change, and it can also be used to identify the key stakeholders in
the climate change debate. The interesting insights from this project will
provide a foundation for informed decision-making and policy formulation
regarding climate change. Additionally, the findings will contribute to
advancing NLP techniques and their application in climate change analysis.

###  1.1 Effects of Climate Change

The consequences of climate change are extensive and have a significant impact
on many facets of our planet. Several important aspects that shed information
on the effects of climate change have been highlighted in research articles.
Climate change is already manifesting globally, with a notable increase in
extreme weather events. The frequency and intensity of hurricanes, floods, and
droughts have amplified, causing widespread destruction and loss of life.
Coastal areas are also seriously threatened by increasing sea levels, which
might result in massive population displacement, increased erosion, and
flooding. As glaciers continue to melt, water supplies diminish, affecting
regions dependent on glacial meltwater for agricultural, industrial, and
domestic purposes. Climate change-induced shifts in environmental conditions
are causing profound changes in plant and animal life. Species are forced to
adapt or face extinction as they grapple with altered ecosystems and changing
habitats. This disruption to biodiversity has cascading effects on ecosystem
functioning and services, with implications for food security, ecosystem
stability, and human well-being.

Another consequence of climate change is the heightened risk of diseases. As
temperatures rise, disease-carrying organisms, such as mosquitoes, expand
their geographic range, exposing previously unaffected regions to vector-borne
illnesses. This poses a significant public health challenge, necessitating the
development of effective strategies for disease prevention, control, and
surveillance. The impacts of climate change extend beyond the natural
environment, affecting societies and economies globally. Disruptions to
ecosystems and weather patterns have severe social and economic repercussions,
with vulnerable communities being disproportionately affected. Climate-induced
events, such as extreme heatwaves, prolonged droughts, and intense storms,
lead to the displacement of populations, loss of livelihoods, and increased
socioeconomic inequality. Consequently, countries face significant challenges
in managing the economic and social ramifications of climate change, including
the need for adaptation measures and the transition to sustainable practices.
In summary, climate change is causing a range of effects that reverberate
across multiple dimensions of our planet. The effects are widespread and
provide serious difficulties for human society and ecosystems, from extreme
weather events and increasing sea levels to melting glaciers and biodiversity
loss. Addressing climate change requires concerted global efforts to mitigate
greenhouse gas emissions, enhance resilience, and foster sustainable
development practices to safeguard the future of our planet and its
inhabitants.

###  1.2 NLP for Climate Change Analysis

Natural Language Processing (NLP) is an emerging discipline in computer
science that concentrates on the creation of algorithms and models to
facilitate computers in comprehending, analyzing, and producing human
language. In the context of climate change analysis, NLP techniques have
proven to be invaluable in extracting meaningful insights from vast amounts of
textual data, offering a new perspective on this critical global issue. One
application of NLP in climate change analysis is the ability to analyze public
opinion on the topic. By leveraging sentiment analysis techniques, researchers
can gauge the prevailing sentiments, attitudes, and beliefs surrounding
climate change. This understanding of public opinion is crucial for
policymakers, as it helps them tailor communication strategies, design
effective interventions, and foster public engagement in addressing climate
change challenges.

Furthermore, NLP techniques allow for identifying key stakeholders involved in
the climate change debate.Through the extraction and examination of written
information from various outlets, including news articles, social media
platforms, and scientific journals, scholars can discern the key individuals,
organizations, and institutions influencing the conversation surrounding
climate change. This knowledge provides valuable insights into the various
perspectives, interests, and motivations shaping climate change discussions,
facilitating informed decision-making and targeted engagement with relevant
stakeholders. NLP also enables the tracking of the progress of climate change
negotiations. By analyzing texts from international agreements, policy
documents, and meeting transcripts, researchers can monitor the evolution of
climate change discussions, assess the effectiveness of existing frameworks,
and identify areas of convergence or divergence among different stakeholders.
This monitoring capability helps policymakers and negotiators evaluate the
efficacy of climate change policies, identify potential barriers to progress,
and inform future negotiations and policy development.

Additionally, NLP techniques can be applied to monitor the impact of climate
change on different regions of the world. By analyzing textual data from
scientific reports, environmental assessments, and socio-economic surveys,
researchers can gain insights into the specific vulnerabilities, risks, and
adaptation strategies associated with climate change in different geographic
areas. This information is crucial for policymakers and local communities to
prioritize resources, implement targeted interventions, and build resilience
against the impacts of climate change. In summary, NLP techniques offer a
powerful toolkit for analyzing climate change-related textual data, enabling
researchers to gain valuable insights into public opinion, identify key
stakeholders, track the progress of climate change negotiations, and monitor
the impact of climate change on different regions. By harnessing the potential
of NLP, policymakers and researchers can enhance their understanding of
climate change dynamics and develop evidence-based strategies for mitigation,
adaptation, and effective decision-making in the face of this global
challenge. The major contributions of this research may be summarized as
follows:

  * •

Conducts a detailed study on different approaches reported in the natural
language processing literature on sentiment analysis using social media data.

  * •

Explores the potential of using ClimateBERT - a pre-trained model on climate
data, for the sentiment analysis of tweets on climate change.

  * •

Conducts extensive experiments and reports the experimental comparisons with
different machine learning algorithms on sentiment analysis using ClimateBERT

##  2 Related Studies

This section provides an overview of recent and influential research papers in
machine learning and natural language processing, specifically on climate
change analysis. It also discusses relevant studies that explore sentiment
text classification approaches that are pertinent to the proposed project. The
reviewed studies have demonstrated the effectiveness of sentiment analysis and
named entity recognition techniques in the context of climate change analysis.
NLP models such as BERT and attention mechanisms have shown promising results
in capturing contextual information and improving performance. These studies
provide valuable insights and methodologies to guide our approach in
implementing sentiment analysis and named entity recognition on climate
change-related tweets and texts using the ClimateBERT pre-trained model.

A study was conducted to assess the effectiveness of ML algorithms in
predicting long-term global warming. The research examined algorithms such as
LR, SVR, lasso, and ElasticNet to connect average annual temperature and
greenhouse gas factors. By analyzing a dataset spanning 100-150 years, the
study found that carbon dioxide (CO2) had the most significant impact on
temperature changes, followed by CH4, N2O, and SF6. Using this information,
the researchers were able to forecast temperature trends and greenhouse gas
levels for the next decade, providing valuable insights for mitigating the
consequences of global warming. The research analyzes public sentiments
regarding climate change by studying Twitter data. The study aims to tackle
the problems of polarization and misinformation that often arise during
climate change discussions on social media platforms. To achieve this, the
researchers introduce a multi-task model named MEMOCLiC, which combines stance
detection with additional tasks like emotion recognition and offensive
language identification. By employing various embedding techniques and
attention mechanisms, the proposed framework effectively captures specific
characteristics and interactions related to different modalities. Experimental
findings highlight the superior performance of the MEMOCLiC model in enhancing
stance detection accuracy compared to baseline methods.

This research paper examines the issue of polarization and belief systems
prevalent in climate change discussions on Twitter. The paper proposes a
framework that aims to identify statements denying climate change and classify
tweets into two categories: denier or believer stances. [Sham and Mohamed,
2022]The framework focuses on two interconnected tasks: stance detection and
sentiment analysis. Combining these tasks, the multi-task model utilizes
feature-specific and shared-specific attention frameworks to acquire
comprehensive features. Experimental results demonstrate that the proposed
framework enhances stance detection accuracy by leveraging sentiment analysis,
outperforming uni-modal and single-task approaches. This research paper
utilizes the BERT model and convolutional neural network (CNN).[Lydiri et al.,
2022] The study analyzes public opinions on climate change by examining
Twitter data. The results indicate that the proposed model surpasses
conventional machine learning methods, accurately identifying climate change
believers and deniers. The authors suggest this model has significant
potential for monitoring and governance purposes, particularly in smart city
contexts. Additionally, future work involves investigating alternative deep
learning algorithms and expanding the analysis to encompass other social media
platforms.

This research paper [Ceylan, 2022]investigates the application of AI and NLP
models to analyze extensive unstructured data concerning climate change. The
study primarily aims to develop an information management system capable of
extracting pertinent information from diverse data sources, particularly
technical design documentation. By utilizing pre-trained AI-based NLP models
trained on textual data and integrating non-textual graphical data, the
researchers showcase the system’s effectiveness in swiftly and efficiently
retrieving precise information. The ultimate objective is to promote knowledge
democratization and ensure the accessibility of information to a broad user
base. This research paper examines people’s emotions and opinions concerning
the conflict between Russia and Ukraine by employing ML and DL techniques.
[Sirisha and Bolem, 2022]The study introduces a novel hybrid model combining
sequence and transformer models, namely ROBERTa, ABSA, and LSTM. To conduct
the analysis, a large dataset of geographically tagged tweets related to the
Ukraine-Russia war is collected from Twitter, and sentiment analysis is
performed using the proposed model. The findings indicate that the hybrid
model achieves a remarkable accuracy of 94.7, surpassing existing approaches
in sentiment analysis. The study underscores the significance of social media
platforms such as Twitter in gaining insights into public sentiment and
opinions regarding global events.

This research paper aims to overcome the limitations of general language
models in effectively representing climate-related texts. The authors
introduce CLIMATEBERT, a transformer-based language model that undergoes
pretraining on a vast corpus of climate-related paragraphs extracted from
diverse sources such as news articles, research papers, and corporate
disclosures.[Webersinke et al., 2021] Comparative evaluations reveal that
CLIMATEBERT surpasses commonly used language models, exhibiting a substantial
48 enhancement in a masked language model objective. The improved performance
of CLIMATEBERT contributes to lower error rates in various climate-related
downstream tasks. To encourage further research at the intersection of climate
change and natural language processing, the authors provide public access to
the training code and weights of CLIMATEBERT. This research paper uses ML
algorithms to analyze and predict climate change. The authors emphasize the
significance of comprehending and adapting to the impacts of climate change on
both human society and the environment. The study discusses the application of
ML methods in analyzing historical temperature data and carbon dioxide
concentrations dating back to the 18th century. It emphasizes the potential
advantages of employing machine learning and artificial intelligence in
interpreting and harnessing climate data for simulations and predictions.
Multiple machine learning algorithms, such as DT, RF, and ANN, are examined
for climate change risk assessment and prediction. The authors conclude that
integrating machine learning techniques can enhance climate modeling, enabling
informed decision-making concerning climate change mitigation and adaptation
strategies.

This research paper introduces the ClimaText dataset[Varini et al., 2020],
specifically developed to detect sentence-level climate change topics within
textual sources. The authors emphasize the significance of automating the
extraction of climate change information from media and other text-based
materials to facilitate various applications, including content filtering,
sentiment analysis, and fact-checking. Through a comparative analysis of
different approaches for identifying climate change topics, they find that
context-based algorithms like BERT outperform simple keyword-based models.
However, the authors also identify areas that require improvement,
particularly in capturing the discussion surrounding the indirect effects of
climate change. The authors anticipate this dataset will be a valuable
resource for further research in natural language understanding and climate
change communication. [Upadhyaya et al., 2022]It underscores the importance of
comprehending public perception and acceptance of climate change policies. The
study examines diverse data sources, such as social media, scientific papers,
and news articles, to perform sentiment analysis. ML techniques, specifically
SVM, are evaluated for extracting valuable insights from these data sources.
The paper concludes that supervised machine learning techniques exhibit
effectiveness in sentiment analysis, highlighting that ensemble and hybrid
approaches yield superior outcomes compared to individual classifiers.

##  3 Materials and Methods

###  3.1 Label Studio

Label Studio is an open-source data annotation tool (available at
https://labelstud.io/) that provides a user-friendly interface for creating
labeled datasets by annotating data for machine learning and artificial
intelligence tasks. The tool supports various annotation types, including text
classification, NER, object detection, image segmentation, and more. Label
Studio allows users to import data from various sources, such as CSV files,
JSON, or databases, and annotate them using a customizable interface. It
provides a collaborative environment where multiple annotators can collaborate
on a project, with features like task assignment, annotation review, and
inter-annotator agreement measurement. One of the key features of Label Studio
is its extensibility. It provides a flexible architecture that allows users to
customize the annotation interfaces and incorporate custom labeling functions
using JavaScript and Python. This enables the tool to adapt to different
annotation requirements and integrate with existing machine-learning
workflows. Label Studio also supports active learning, where the tool can
suggest samples to be annotated based on a model’s uncertainty, helping to
optimize the annotation process and improve model performance.

###  3.2 snscrape

snscrape is a Python library and command-line tool (available at
https://github.com/JustAnotherArchivist/snscrape) for scraping social media
content. It lets you retrieve public data from various social media platforms,
including Twitter, Instagram, YouTube, Reddit, etc. With snscrape, you can
fetch posts, comments, likes, followers, and other relevant information from
social media platforms. It provides a flexible and customizable way to search
for specific keywords, hashtags, usernames, or URLs and extract the desired
content. The library supports scraping recent and historical data from social
media platforms, enabling you to gather insights, perform analysis, monitor
trends, and conduct research based on social media content. snscrape offers a
command-line interface that allows you to search for and scrape social media
data interactively. You can specify various parameters, such as the number of
results, date range, and output format, to customize your scraping process. In
addition to the command-line interface, snscrape provides a Python API that
allows you to integrate social media scraping into your own Python scripts and
applications. The API offers more advanced functionalities, giving you fine-
grained control over the scraping process and allowing you to process the
scraped data programmatically. One of the key advantages of snscrape is its
ability to work with multiple social media platforms, providing a unified
interface for scraping different types of content. It handles the intricacies
of each platform’s APIs and HTML structures, making it easier for developers
to extract data without needing to learn the specific details of each
platform. It’s important to note that snscrape respects the terms of service
and usage restrictions of each social media platform. It is primarily intended
for scraping publicly available content and should be used responsibly and in
compliance with the platform’s policies.

###  3.3 Newspaper 3k

Newspaper3k is a Python library and web scraping tool (available at
https://newspaper.readthedocs.io/) that allows you to extract and parse
information from online news articles. It provides a simple interface to
automate the fetching and processing of news articles from various online
sources. With Newspaper3k, you can retrieve article metadata such as the
title, author, publish date, and article text from news websites. It also
supports extracting additional information like keywords, summaries, and
article images. The library uses advanced NLP techniques to extract relevant
information from the HTML structure of the news articles. Newspaper3k is
designed to handle various complexities of news websites, including different
article formats, pagination, and content extraction. It has built-in
functionality to handle newspaper-specific features like multi-page articles,
article pagination, and RSS feeds. One of the advantages of Newspaper3k is its
ease of use. It abstracts away the complexities of web scraping and provides a
clean and intuitive API. It also handles various encoding and parsing issues
that often arise when dealing with news articles from different sources.
Newspaper3k is widely used for various applications, including content
analysis, sentiment analysis, and data mining. It offers a convenient way to
gather news data for research, data analysis, and machine learning projects.

###  3.4 ClimateBERT

ClimateBERT is a specialized variant of the BERT model specifically trained
and tailored for addressing climate change-related language tasks. Building
upon the foundation of BERT, ClimateBERT is pre-trained on a large corpus of
climate change-related documents and text sources, enabling it to capture the
nuances and domain-specific knowledge relevant to climate science. This fine-
tuning process equips ClimateBERT with a deep understanding of climate-related
concepts, terminology, and contextual dependencies. [Iqbal et al., 2023] By
leveraging ClimateBERT, researchers and practitioners in climate change
analysis can effectively tackle various NLP tasks, such as sentiment analysis
on climate-related tweets or named entity recognition on climate change
articles. Integrating domain-specific knowledge into the pre-training process
makes ClimateBERT a powerful tool for extracting insights, identifying
patterns, and extracting valuable information from climate-related text data.
Its application in climate change analysis can aid in improving decision-
making, facilitating research, and enhancing our understanding of the complex
challenges climate change poses.

##  4 Proposed Approach

This section deals with the proposed methodology for the sentiment analysis of
climate-related tweets from Twitter using ClimateBERT embeddings and Random
Forest Classifier. The overall workflow of the proposed approach is given in
Figure 1.

Figure 1: Overall workflow of the sentiment analysis on climate Change tweets

###  4.1 Dataset

The methodology begins with data collection from Twitter using the snscrape
library. The collected data is loaded into a pandas DataFrame for further
processing. The dataset consists of climate change-related tweets, valuable
for sentiment analysis. The tweets gathered between 1 January 2022 and 2
February 2023. However, it’s important to note that the collected data may
contain class imbalance, where certain sentiment categories are
overrepresented while others are underrepresented. This could potentially bias
the model’s predictions. Initially, the dataset consisted of 4410 data points.
After data augmentation, the final dataset consists of 5506 data points, with
three labels, Positive, Negative and Neutral. The data set is available at
https://github.com/appliednlp-duk/nlp-climate-change. Table 1 shows the
example of labeling for each tweet, whether it is Positive, Negative, or
Neutral.

Table 1: A snapshot of the dataset used for the experiment Content | Labels
---|---
Researchers use deep learning to simulate chlorophylla & phycocyanin with an internet of things system to detect & quantify #cyanobacteria, to improve #eutrophication management schemes for freshwater reservoirs. #algae #microbiology #environment #iot eeer.org/journal/view.pâ€¦ | Positive
Why is our @Conservatives government so evil? #RishiSunak #climateChange #Conservatives #FuckingThieves https://t.co/cCGyylmYlf | Negative
Sierra snowpack 205% of its historical average | Climate Change … - San Francisco Examiner dlvr.it/ShpGVN #ClimateChange | Neutral

###  4.2 Data Preprocessing

To prepare the data for sentiment analysis, several preprocessing steps are
applied. Special characters and digits are removed, and the text is converted
to lowercase to ensure consistency. Tokenization is performed, which involves
splitting the words into individual units. Stopwords (common words with little
contextual meaning) are removed, and stemming or lemmatization techniques may
be applied to normalize the words. This preprocessing step ensures the text
data is cleaned and ready for further analysis.

###  4.3 Experimental setup

This section describes the experiment for implementing the proposed approach
detailed in section 3. All the experiments were executed on NVIDIA A100 with
80 GB GPU memory and 1,935 GB/Second bandwidth. The ChatGPT Sentiment tweets
were pre-processed to make them ready for experimentation. All scripts were
written in Python 3.9, and the Machine Learning Models were used from the
Scikit-Learn library available at https://scikitlearn.org/stable/.

##  5 Results and Discussions

This section presents the results obtained from the experiment using the
proposed approach outlined in Section 4. The results, along with a detailed
discussion, are given in this section.

Table 2: Precision, Recall, Accuracy, and F-Measure values for the TF-IDF feature encoding Model | TF-IDF |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 87.11 | 86.98 | 87.11 | 87.01 |
SVM | 84.39 | 84.34 | 84.39 | 84.14 |
DT | 73.23 | 72.63 | 73.23 | 72.71 |
LR | 90.10 | 90.13 | 90.10 | 90.04 |
Table 3: Precision, Recall, Accuracy, and F-Measure values for word2Vec Model | word2Vec |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 72.59 | 72.91 | 72.59 | 72.69 |
SVM | 40.29 | 40.59 | 40.29 | 36.92 |
DT | 62.34 | 61.24 | 62.34 | 61.59 |
LR | 39.56 | 40.25 | 39.56 | 36.53 |
Table 4: Precision, Recall, Accuracy, and F-Measure values for CountVectorizer Model | CountVectorizer |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 89.56 | 89.53 | 89.56 | 89.50 |
SVM | 88.11 | 88.02 | 88.11 | 87.98 |
DT | 79.67 | 79.50 | 79.67 | 79.58 |
LR | 79.21 | 79.09 | 79.21 | 78.76 |
Table 5: Precision, Recall, Accuracy, and F-Measure values for TF-IDF + CountVectorizer Model | TF-IDF + CountVectorizer |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 86.93 | 86.96 | 86.93 | 86.87 |
SVM | 81.94 | 81.92 | 81.94 | 81.75 |
DT | 74.77 | 74.30 | 74.77 | 74.46 |
LR | 81.30 | 81.11 | 81.30 | 80.99 |
Table 6: Precision, Recall, Accuracy, and F-Measure values for TF-IDF + word2Vec Model | TF-IDF + word2Vec |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 79.21 | 79.45 | 79.21 | 79.29 |
SVM | 85.39 | 85.34 | 85.39 | 85.22 |
DT | 62.43 | 61.28 | 62.43 | 61.60 |
LR | 74.77 | 74.33 | 74.77 | 74.28 |
Table 7: Precision, Recall, Accuracy, and F-Measure values for CountVectorizer + word2Vec Model | CountVectorizer + word2Vec |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 79.03 | 79.45 | 79.03 | 79.18 |
SVM | 81.30 | 81.10 | 81.30 | 81.09 |
DT | 62.06 | 61.01 | 62.06 | 61.37 |
LR | 81.21 | 81.04 | 81.21 | 80.92 |
Table 8: Performance evaluation of SVM, LR, RF, and DT algorithms using BERT Model | BERT |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 76.78 | 77.46 | 76.78 | 76.93 |
SVM | 64.35 | 63.65 | 64.35 | 63.70 |
DT | 68.89 | 67.13 | 68.89 | 67.89 |
LR | 63.81 | 63.48 | 63.81 | 63.60 |
Table 9: Performance evaluation of SVM, LR, RF, Naive Bayes, and DT algorithms using ClimateBERT Model | ClimateBERT |
---|---|---
| Accuracy | Precision | Recall | F-measure |
RF | 85.22 | 85.73 | 85.22 | 83.33 |
SVM | 75.66 | 76.20 | 75.66 | 75.07 |
DT | 80.62 | 79.88 | 78.62 | 77.47 |
LR | 73.84 | 72.92 | 73.84 | 75.69 |

(a) Performance evaluation of SVM, LR, RF, and DT algorithms with TF-IDF

(b) Performance evaluation of SVM, LR, RF, and DT algorithms with TF-IDF

(a) Performance evaluation of SVM, LR, RF, and DT algorithms with BERT

(b) Performance evaluation of SVM, LR, RF, and DT algorithms with ClimateBERT

(a) Performance evaluation of SVM, LR, RF, and DT algorithms with
CountVectorizer

(b) Performance evaluation of SVM, LR, RF, and DT algorithms with Combination
of TF-IDF and CountVectorizer

(a) Performance evaluation of SVM, LR, RF, and DT algorithms with Combination
of TF-IDF and Word2Vec

(b) Performance evaluation of SVM, LR, RF, and DT algorithms with Combination
of CountVectorizer and Word2Vec

Table 1 to 8 shows the accuracy, precision, recall, and f-measure values for
RF, SVM, DT, and LR algorithms. For BERT embeddings, RF has 76.78%, 77.46%,
76.78%, and 76.93%, SVM has 64.35%, 63.65%, 64.35%, and 63.70%, DT has 68.89%,
67.13%, 68.89%, and 67.59%, and LR has 63.81%, 63.48%, 63.81%, and 63.60% for
the A, P, R, and F values. Table 4 shows the Accuracy, Precision, Recall, and
F-measure values for RF, SVM, DT, and LR algorithms. For ClimateBERT
embeddings, RF has 85.22%, 85.73%, 85.22%, and 83.33%, SVM has 75.66%, 76.20%,
75.66%, and 75.07%, DT has 80.62%, 79.88%, 78.62%, and 77.47%, and LR has
73.84%, 72.92%, 73.84%, and 75.69% for the A, P, R, and F values. After
training, the model’s performance is evaluated on the test set to assess its
ability to predict sentiment. The model is switched to evaluation mode, and
predictions are made on the test set. Accuracy, precision, recall, and
F1-score are calculated to measure the model’s performance. The results
obtained from the evaluation metrics are reported. Accuracy provides an
overall measure of correctness, precision measures the proportion of correctly
predicted positive sentiments, recall captures the ability to identify all
positive sentiments, and the F1-score provides a balanced measure between
precision and recall. These metrics provide insights into how well the model
predicts sentiment on climate change-related tweets. By following this
experimental setup, the methodology ensures that the collected data is
cleaned, balanced, and used effectively to train a sentiment analysis model.
The results and discussions provide valuable insights into the model’s
performance and its ability to analyze sentiment in climate change discussions
on Twitter.

##  6 Conclusions

Climate change, a pressing global concern, necessitates thorough analysis and
understanding across diverse domains to mitigate its impacts effectively. In
recent years, the fusion of NLP techniques and machine learning algorithms has
emerged as a promising approach for comprehending the complexities and nuances
of climate change through the lens of textual data. This paper utilized the
advancements in domain-specific large language models to harness the potential
of NLP in addressing the challenges posed by climate change through sentiment
analysis. By leveraging advanced NLP methodologies, we could identify climate
change discourse that may enable uncovering valuable insights and facilitate
informed decision-making.

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# On Control Networks Over Finite Lattices

Zhengping Ji, Daizhan Cheng This work is supported partly by the National
Natural Science Foundation of China (NSFC) under Grants 62073315 and
61733018.Zhengping Ji is with the Key Laboratory of Systems and Control,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, P. R. China, and School of Mathematical Sciences, University
of Chinese Academy of Sciences, Beijing 100049, P. R. China (e-mail:
jizhengping@amss.ac.cn).Daizhan Cheng is with the Key Laboratory of Systems
and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of
Sciences, Beijing 100190, P. R. China (e-mail: dcheng@iss.ac.cn).

###### Abstract

The modeling and control of networks over finite lattices are investigated via
the algebraic state space approach. Using the semi-tensor product of matrices,
one can obtain the algebraic state space representation of the dynamics of a
(control) network over a finite lattice. Then some properties involving
networks over sublattices and product lattices are investigated, which shows
the advantages of networks over finite lattices in analysis and control
design. Finally, algorithms are developed to recover the lattice structure
from the structural matrix of a network over a lattice, and construct
comparability graphs over a finite set to verify whether a multiple-valued
logical network is defined over a lattice. Numerical examples are also
demonstrated.

###### Index Terms:

Multiple-valued control networks, semi-tensor product of matrices, finite
lattices

##  I Introduction

The Boolean networks were firstly introduced by Kauffman to formulate genetic
regulatory networks in 1969 [17]. The theory of Boolean networks is based on
Boolean algebra proposed by G. Boole [3], where the value of a logical node is
quantized to zero or one. To improve the accuracy of the logical variables,
multiple-valued logical networks were then considered. The theory of
$n$-valued logic was introduced by Post in 1921 [24]. Later in 1942 Rosenbloom
[28] considered the system of operations in this kind of logic fo variables to
generalize the notion of Boolean algebra to $n$-valued cases, called Post
algebra, which, by revealing itself as a special class of finite lattices,
provided a way of treating multiple-valued algebra from the perspective of
lattice theory [14, 32].

Lattices are common objects in combinatorics. Since multiple-valued logical
networks can be considered as defined over Post algebras which are special
cases of lattices, one can in general consider a network where each node takes
value in a finite lattice, and the evolution depends on the $\sup$ and $\inf$
operations of the states. For example, distant data verification and breakdown
restoration in some multi-agent systems with logically linked list of entries
and the distributed ledger can be modelled using Allen-Givone algebra, which
can be viewed as a special class of lattices [4, 5]; The asynchronous dynamics
of a gene network with multiple expression levels can be modelled through some
basic operators defined on a finite lattice [27]. On the other hand, networks
over lattices can have simple logical expressions due to the properties such
as commutativity and idempotency of the generating operators, together with
many other advantages, for example, restricting such a network to a sublattice
provides an invariant sub-network; the control properties of networks over
product lattices are completely determined by their decomposed subnetworks
over sub-lattices, etc. These properties reduce the complexity of analyzing
and control design of such systems.

However, to the authors’ best knowledge, there have been few investigations on
general (control) networks over lattices. There are two problems as possible
reasons for this. First, given the input-output relation of a finite-valued
network, we are not always able to recognize whether it is defined over a
lattice; secondly, when a network is indeed expressed through the composition
of basic operators on a lattice, the expression of the nonlinear dynamics is
not easy to be simplified for deriving a transition law, impeding the analysis
of the network.

Consider the first problem. Since networks over lattices may encode
information of the underlying partial order structure in its algebraic
expression, it is possible that from the dynamics of a given multiple-valued
logical network one may recognize the basic operators serving as “bricks” or
“building blocks” that generates the system, and hence determine if it is
actually defined over a lattice. The problem of verifying whether a network
can be considered as one defined over a lattice (and further reconstruct this
lattice structure if possible) is of practical importance since it is a
special case of constraint satisfaction problems [29]. Expressing a $k$-valued
network over a lattice is not only practically useful but also theoretically
challenging. It is well known that even expressing a $3$-valued logical
networks into $3$-valued logical form is not an easy job, not to mention
$k$-valued case [1]. There have been investigations on the generating systems
of function algebras [18], but as far as the authors are concerned, few
research has been done for designing a lattice structure on a finite set to
make a (control) network be generated by the basic operators on the lattice,
which is an important issue, since sometimes we only have the desired input-
output relation of a system and may wonder how to realize it through simple
generating functions [26]. For example, if the desired transition law of a
network can be written explicitly by some operators on a lattice, then the
system design will be much easier. This can be viewed as an discrete-state
analogue of the circuit realization problem [16, 31].

As for the second problem, we introduce the semi-tensor product (STP) of
matrices as the tool. Since 2009, it has been applied to the study of Boolean
networks [10, 11], providing a convenient way to represent the dynamics and
promoting the development of the theory of finite-valued networks.
Investigations have been merged mainly on two directions. One is the control
problems of Boolean networks, such as controllability [10], observability
[15], stabilization [19], tracking [33], etc. Another direction is the values
of states in the networks have been extended from the Boolean case (where they
are binary) to $k$-valued, mixed [20], and finite field cases [23]. The STP
allows one to simplify the expression of a finite-valued control network to a
unified bilinear from, making the analysis much easier.

This paper aims to provide a framework for (control) networks over finite
lattices to solve the above problems. Using the STP, the algebraic state space
representation (ASSR) of such networks is derived, allowing us to analyze the
system with existing results in multiple-valued logical networks. We show that
when the lattice is the product of some finite lattices, the controllability
and observability properties are determined by the subnetworks defined over
factor lattices. Further, from a network over a lattice, we recover the
underlying lattice structure from its ASSR; when the existence of underlying
partial order relations is unknown for an arbitrarily given network, we give
necessary conditions for it to be a network over a lattice, and try to
construct such a partial order to make the network be generated by some
classes of basic operators over a lattice.

The rest of this paper is organized as follows. Section II provides
preliminaries about STP and lattice theory; Section III treats the networks
over finite lattices in a framework of algebraic state space and vector
expressions: first it gives criteria for a finite set endowed with an operator
to be a lattice, and then investigates some properties of networks over
product lattices; Section IV deals with constructing and recovering the
underlying lattice structures for finite-valued networks, designing algorithms
to construct partial order so that a given network can be viewed as generated
by basic operators of a lattice. These results are illustrated by numerical
examples in Section V. Section VI is a brief conclusion.

Before ending this introduction we give a list for notations, which are used
in the sequel:

  1. 1.

${\mathcal{M}}_{m\times n}$: set of $m\times n$-dimensional real matrices.

  2. 2.

$\operatorname{Col}(A)$ ($\operatorname{Row}(A)$): the set of columns (rows)
of $A$; $\operatorname{Col}_{i}(A)$ ($\operatorname{Row}_{i}(A)$): the $i$-th
column (row) of $A$.

  3. 3.

$\delta_{n}^{i}$: $i$-th column of the identity matrix $I_{n}$.

  4. 4.

${\mathcal{D}}_{k}:=\\{0,\cdots,k-1\\}$.

  5. 5.

$\Delta_{k}:=\operatorname{Col}(I_{k})$.

  6. 6.

${\mathcal{L}}_{m\times n}$: the set of logical matrices. (A matrix
$L\in{\mathcal{M}}_{m\times n}$ is called a logical matrix, if
$\operatorname{Col}(L)\subset\Delta_{m}$.)

  7. 7.

$\delta_{m}[i_{1},\cdots,i_{n}]$: a brief notation for logical matrices, that
is,
$\delta_{m}[i_{1},\cdots,i_{n}]:=\left[\delta_{m}^{i_{1}},\cdots,\delta_{m}^{i_{n}}\right]$.

##  II Preliminaries

###  II-A STP and Algebraic Expression of Multiple-Valued Networks

We first give a brief review on STP of matrices which is the main tool in this
paper. We refer to [13] for details.

###### Definition II.1

Let $A\in{\mathcal{M}}_{m\times n}$, $B\in{\mathcal{M}}_{p\times q}$ be real
matrices, and the least common multiple of $n$ and $p$ be
$t=\operatorname{lcm}\\{n,p\\}$. The STP of $A$ and $B$, denoted by $A\ltimes
B$, is defined as

$\left(A\otimes I_{t/n}\right)\left(B\otimes I_{t/p}\right).$ (1)

where $I_{k}$ is the $k\times k$ identity matrix, and $\otimes$ is the
Kronecker product.

Throughout this paper, all products are assumed to be semi-tensor products and
the symbol $\ltimes$ is usually omitted.

When the dimensions of two matrices are compatible, the STP is the same as the
conventional matrix product, and the laws concerning associativity,
distributivity, transpose and inverse hold for it as well. Further, it has the
following “commutative” properties:

######  Proposition II.2 ([13])

Let $X\in{\mathbb{R}}^{m}$ be a column vector and $M$ be a matrix. Then

$\displaystyle X\ltimes M=\left(I_{m}\otimes M\right)\ltimes X.$

Given two column vectors $X\in{\mathbb{R}}^{m}$, $Y\in{\mathbb{R}}^{n}$, then

$\displaystyle W_{[m,n]}\ltimes X\ltimes Y=Y\ltimes X.$ (2)

Here, $W_{[m,n]}\in{\mathcal{M}}_{mn\times mn}$ is the $(m,n)$-order swap
matrix defined as

$\displaystyle\begin{array}[]{ccl}W_{[m,n]}&=\delta_{mn}[&1,m+1,\cdots,(n-1)m+1;\\\
{}\hfil&{}\hfil&2,m+2,\cdots,(n-1)m+2;\\\
{}\hfil&{}\hfil&\cdots;m,2m,\cdots,nm].\end{array}$

###### Definition II.3

  1. 1.

Let $x_{i}\in{\mathcal{D}}_{k_{i}}$, $i=1,\cdots,n$. A map
$f:\prod_{i=1}^{n}{\mathcal{D}}_{k_{i}}\rightarrow{\mathcal{D}}_{k_{0}}$ is
called a multiple-valued logical function. If
$k_{1}=k_{2}=\cdots=k_{n}=k_{0}$, $f$ is called a $k_{0}$-valued logical
function.

  2. 2.

Let $x_{i}(t)\in{\mathcal{D}}_{k_{i}}$,
$f_{i}:\prod_{i=1}^{n}{\mathcal{D}}_{k_{i}}\rightarrow{\mathcal{D}}_{k_{i}}$,
$i=1,\cdots,n$. The system

$\displaystyle\begin{cases}x_{1}(t+1)=f_{1}(x_{1}(t),\cdots,x_{n}(t))\\\
\vdots~{}~{}\\\ x_{n}(t+1)=f_{n}(x_{1}(t),\cdots,x_{n}(t)),\\\ \end{cases}$
(3)

is called a multiple-valued logical dynamic system.

If $k_{1}=k_{2}=\cdots=k_{n}=k_{0}$, the system is called a $k_{0}$-valued
logical dynamic system.

Identify $i\sim\delta_{k}^{i},~{}i=1,\cdots,k-1$, and $0\sim\delta_{k}^{k}$,
then $x\in{\mathcal{D}}_{k}$ can be expressed as $x\in\Delta_{k}$. The latter
is called the vector expression of a logical variable. Using this and the
above properties of STP, one can express a multiple-valued logical function
and further a network in the algebraic state space.

######  Theorem II.4 ([12])

Given a multiple-valued logical function
$f:\prod_{i=1}^{n}{\mathcal{D}}_{k_{i}}\rightarrow{\mathcal{D}}_{k_{0}}$
denoted as $y=f(x_{1},\cdots,x_{n})$, there exists a unique
$M_{f}\in{\mathcal{L}}_{k_{0}\times k}$, where $k=\prod_{i=1}^{n}k_{i}$, such
that in vector expression we have

$\displaystyle y=M_{f}\ltimes_{i=1}^{n}x_{i}.$

$M_{f}$ is called the structure matrix of $f$.

Applying Theorem II.4 to each equation of (3), we have

$\displaystyle\begin{cases}x_{1}(t+1)=M_{1}\ltimes_{i=1}^{n}x_{i}(t)\\\
\vdots~{}~{}\\\ x_{n}(t+1)=M_{n}\ltimes_{i=1}^{n}x_{i}(t),\\\ \end{cases}$ (4)

where $M_{i}$ is the structure matrix of $f_{i}$, $i=1,\cdots,n$.

######  Theorem II.5 ([12])

Denote by $x(t)=\ltimes_{i=1}^{n}x_{i}(t)$. Then (4) can be expressed as

$\displaystyle x(t+1)=Mx(t),$ (5)

where $M=M_{1}*\cdots*M_{n}$, and $*$ is the Khatri-Rao product of matrices.
(5) is called the algebraic state space representation (ASSR) of the multiple-
valued logical network (3), and $M$ is called its structure matrix.

###  II-B Finite Lattices

Next, we review the basic notions of lattice theory.

A lattice is a partially ordered set $(L,\leqslant)$ where each pair of
elements $x,y\in L$ has a least upper bound (denoted by $x\vee y$ or
$\sup(x,y)$) and a greatest lower bound (denoted by $x\wedge y$ or
$\inf(x,y)$). For example, define a partial order on $\mathbb{N}$ as
$a\leqslant b\Leftrightarrow a|b$, and let $a\vee b$ be the least common
multiple and $a\wedge b$ the greatest common divisor of two integers $a$, $b$,
then $(\mathbb{N},\leqslant)$ is a lattice. Let $L:=\\{0,1\\}$, then $L$ is a
lattice under natural order if we define the operators $\vee$, $\wedge$ as
disjunction and conjunction of the Boolean variables respectively.

An alternative definition is that a lattice is a tuple $(L,\wedge,\vee)$ where
$\wedge,\vee$ are binary operators on $L$ satisfying commutativity,
associativity, idempotency and the absorption property. Since only finite
lattices are considered in this paper, we adopt the following definition.

######  Definition II.6 ([2])

A finite set $L$ is called a lattice, if there exists a binary operator $\vee$
on $L$, satisfying

  1. (i)

$x\vee x=x$;

  2. (ii)

$x\vee y=y\vee x$;

  3. (iii)

$(x\vee y)\vee z=x\vee(y\vee z)$;

  4. (iv)

$\exists{\bf 0}\in L$, s.t. $w\vee{\bf 0}=w$, $\forall w\in L$,

where $x,y,z$ are arbitrary elements in $L$.

###### Remark II.7

If we define a new binary operator $\wedge$ on $L$ as $a\wedge
b:=\bigvee_{u\in S}u$, where $S:=\\{u\in L|u\leqslant a,u\leqslant b\\}$, then
the triple $(L,\wedge,\vee)$ coincides with the conventional definition of a
lattice, which means there exists a partial order $\leqslant$ on $L$ such that
$\vee$, $\wedge$ are least upper bound and greatest lower bound operators
under this order, and $(L,\leqslant)$ admits unique maximal and minimal
elements. The proofs can be found in [2, 30].

$P_{1}$$P_{4}$$P_{2}$$P_{3}$ Figure 1: Hasse diagram of a 4-element lattice
$L$

A lattice $L$, like other partially ordered sets, can be completely
characterized by its Hasse diagram [30]. It is a graph whose vertices are the
elements in the lattice. There is an edge between two elements $a,b$ if and
only if they are comparable ($a\leqslant b$ or $b\leqslant a$), and if $b$
covers $a$, which means $a\leqslant b$ and there is no element $c\in L$ such
that $a\leqslant c\leqslant b$, then $b$ is drawn “above” $a$ (i.e., with a
higher vertical coordinate).

For example, consider a lattice $L=\\{P_{1},P_{2},P_{3},P_{4}\\}$ whose Hasse
diagram is shown as in Figure 1. One can see that $P_{1}$ is the greatest
element and $P_{4}$ the least, while $P_{2}$ and $P_{3}$ are incomparable,
$P_{2}\vee P_{3}=P_{1}$, $P_{2}\wedge P_{3}=P_{4}$.

##  III Networks Over Finite Lattices

We first use the STP to give an example of how algebraic state space will help
to solve the problems in lattice theory.

Since an operator over a finite set is a logical function (according to
Definition II.3), we can easily adapt the STP framework to networks over
finite lattices. For example, the following proposition gives criteria for a
finite set endowed with a binary operator to be a lattice.

###### Proposition III.1

Let $L$ be a finite set of cardinal $k$ and $f$ a binary operator over $L$
with a structure matrix $M=[M_{1},\cdots,M_{k}]$ where $M_{i}\in L_{k\times
k}$, $i=1,\cdots,k$. Then $(L,f)$ is a lattice if and only if $M$ satisfies
the following conditions:

  1. (i)

$M\mathrm{diag}(\delta_{k}^{1},\cdots,\delta_{k}^{k})=I_{k}$;

  2. (ii)

$MW_{[k,k]}=M$;

  3. (iii)

$M^{2}=M(I_{k}\otimes M)$;

  4. (iv)

$\exists i\in{\mathcal{D}}_{k}$, s.t. $M_{i}=I_{k}$.

The proof simply follows from the basic properties of STP stated in the
previous section. This gives a convenient way to verify through the structure
matrix whether a binary function over a finite set defines a lattice.

###  III-A ASSR of Control Networks Over Lattices

Let $F:L^{n}\rightarrow L$ be an $n$-ary function on $L$. If $F$ is obtained
from the superposition of $\vee$ and $\wedge$ on $L$, we call it a lattice
function. Since lattice functions are special cases of multiple-valued logical
functions, we may apply the STP method to model the networks over finite
lattices.

In general, a network over a finite lattice $L$ has the form of (3) where
$x_{i}(t)\in L$ are state variables and the maps $f_{i}:L^{n}\rightarrow L$
are lattice functions, $i=1,\cdots,n$. When $L={\mathcal{D}}_{2}$, the system
becomes a standard Boolean network.

When there are controls $u_{1}(t),\cdots,u_{m}(t)\in L$ in the network, its
dynamics can be expressed as

$\displaystyle\begin{cases}x_{1}(t+1)=g_{1}(x_{1}(t),\cdots,x_{n}(t);u_{1}(t),\cdots,u_{m}(t))\\\
\vdots\\\
x_{n}(t+1)=g_{n}(x_{1}(t),\cdots,x_{n}(t);u_{1}(t),\cdots,u_{m}(t)),\\\
\end{cases}$ (6)

where $g_{i}$ are lattice functions, $i=1,\cdots,n$.

Assume that $\operatorname{card}(L)=\kappa$. Since the systems over finite
lattices belong to the class of multiple-valued logical systems, by Theorem
II.5, using STP, (3) can be converted into its algebraic state space
representation as

$\displaystyle x(t+1)=Mx(t),$ (7)

where $x(t)=\ltimes_{j=1}^{n}x_{j}(t)$,
$M\in{\mathcal{L}}_{\kappa^{n}\times\kappa^{n}}$, and (6) as

$\displaystyle x(t+1)=Pu(t)x(t),$ (8)

where $u(t)=\ltimes_{s=1}^{m}u_{s}(t)$,
$P\in{\mathcal{L}}_{\kappa^{n}\times\kappa^{n+m}}$.

The STP provides a way to express the nonlinear control system (6) to a
simplified and unified form. We give a pedagogical example to illustrate how
ASSR is derived.

###### Example III.2

Consider a $4$-element lattice $L$ with Hasse diagram as shown in Figure 1.
Setting $P_{i}\sim\delta_{4}^{i}$, $i=1,2,3,4$, we can solve the structure
matrix of the operators $\vee$, $\wedge$ respectively, that is, $\forall
x,y\in\\{P_{1},\cdots,P_{4}\\}$,

$\displaystyle x\vee y=M_{\vee}xy,\quad x\wedge y=M_{\wedge}xy,$
$\displaystyle M_{\vee}=\delta_{4}[1,1,1,1,1,2,1,2,1,1,3,3,1,2,3,4];$
$\displaystyle M_{\wedge}=\delta_{4}[1,2,3,4,2,2,4,4,3,4,3,4,4,4,4,4].$

Here we make no distinction between an element in the lattice and its vector
expression. Next, assume a network over $L$ is defined as

$\displaystyle\begin{cases}x_{1}(t+1)=x_{1}(t)\vee(x_{2}(t)\wedge u(t)),\\\
x_{2}(t+1)=x_{1}(t)\vee x_{2}(t)\vee u(t).\end{cases}$ (9)

Then using the commutative properties in Proposition II.2, it is easy to
construct the component-wise ASSR from the structure matrices of the
operators.

$\begin{cases}x_{1}(t+1)=M_{1}u(t)x(t),\\\
x_{2}(t+1)=M_{2}u(t)x(t),\end{cases}$

where $x(t)=x_{1}(t)x_{2}(t)$, and

$\begin{array}[]{ccl}M_{1}&=&M_{\wedge}(I_{4}\otimes M_{\vee})W_{[4,16]}\\\
{}\hfil&=&\delta_{4}[1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,1,2,1,2,2,\\\
{}\hfil&{}{}\hfil&2,2,2,3,4,3,4,4,4,4,4,1,1,3,3,2,2,4,4,3,3,\\\
{}\hfil&{}{}\hfil&3,3,4,4,4,4,1,2,3,4,2,2,4,4,3,4,3,4,4,4,4,4],\end{array}$
$\begin{array}[]{ccl}M_{2}&=&M_{\vee}(I_{4}\otimes M_{\wedge})W_{[4,16]}\\\
{}\hfil&=&\delta_{4}[1,1,1,1,1,2,1,2,1,1,3,3,1,2,3,4,1,1,1,1,2,\\\
{}\hfil&{}{}\hfil&2,2,2,1,1,3,3,2,2,4,4,1,1,1,1,1,2,1,2,3,3,\\\
{}\hfil&{}{}\hfil&3,3,3,4,3,4,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4].\end{array}$

Then by a construction similar with (5), the ASSR is $x(t+1)=Mu(t)x(t)$, where

$\begin{array}[]{ccl}M&=&M_{1}*M_{2}\\\
{}\hfil&=&\delta_{16}[1,1,1,1,5,6,5,6,9,9,11,11,13,14,15,16,\\\
{}\hfil&{}\hfil&1,5,1,5,6,6,6,6,9,13,11,15,14,14,16,16,1,1,\\\
{}\hfil&{}\hfil&9,9,5,6,13,14,11,11,11,11,15,16,15,16,1,5,\\\
{}\hfil&{}\hfil&9,13,6,6,14,14,11,15,11,15,16,16,16,16].\end{array}$

###### Remark III.3

Note that one advantage of networks over lattices is that they are composed of
basic binary operators so that we may use STP to solve the structure matrix
solely from its operator expression, which cannot be done for arbitrary
multiple-valued networks. The ASSR make the computation and analysis of system
(6) much easier, since it allows one to use existing results in multiple-
valued logical networks to investigate the control properties of the system.
For example, one may consider problems such as controllability and
observability of (9).

###### Example III.4

Consider the controllability problem of the network (9). Denote the matrix $M$
by $[M_{1},\cdots,M_{16}]$ where $M_{i}\in{\mathcal{L}}_{16\times 16}$,
following [12], construct its controllability matrix as

$\displaystyle C=\sum_{j=1}^{16}\big{(}\sum_{i=1}^{16}M_{i}\big{)}^{(j)},$
(10)

Taking the addition and multiplication in (10) to be Boolean type (that is to
say, $1+1=1$), we solve the controllability matrix as

$\displaystyle\tiny
C=\left(\begin{array}[]{cccccccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&1&0&1&1&0&1&0&0&0&0&0&0&0&0&0\\\
0&1&0&1&1&1&1&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&1&1&0&0&0&0&1&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&1&1&0&0&0&0&1&1&1&1&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&1&0&0&1&0&0&1&0&0&0\\\
0&0&0&1&0&0&1&1&0&1&0&0&1&1&0&0\\\ 0&0&0&1&0&0&1&0&0&1&0&1&1&0&1&0\\\
0&0&0&1&0&0&1&1&0&1&0&1&1&1&1&1\end{array}\right).$

Therefore, a state $\delta_{16}^{i}$ is controllable from $\delta_{16}^{j}$ if
and only if the $(i,j)$-th entry of $C$ is nonzero.

Another special property for networks over lattices is that the subnetworks
defined over a sublattice of $L$ are invariant subspaces of the network over
$L$, that is, the dynamics starting from a point in the sublattice will remain
in it.

###### Example III.5

Recall the system (9) and remove the controls in the network. Assume
$S=\\{P_{1},P_{2},P_{4}\\}\subset L$, which is clearly a sub-lattice of $L$.
Re-assign the nodes as $P_{1}\sim\delta_{3}^{1}$, $P_{2}\sim\delta_{3}^{2}$,
$P_{4}\sim\delta_{3}^{3}$. It is easy to figure out that setting $z=x|_{S}$ we
have

$z(t+1)=M|_{S}z(t),$

where $M|_{S}=\delta_{9}[1,1,1,4,5,5,7,8,9]$.

If we add the controls back and restrict them to the sublattice $S$, then it
turns out that the system is a control-invariant subspace which can be viewed
as defined over the sublattice with the dynamics

$z(t+1)=P|_{S}u(t)x(t),$

where

$\begin{array}[]{rl}P|_{S}=\delta_{9}&[1,1,1,2,5,5,3,6,9,1,2,2,5,\\\
{}\hfil&5,5,6,6,9,1,2,3,5,5,6,9,9,9]\end{array}$

###  III-B Control Networks Over Product Lattices

Next we consider the networks over a special class of finite lattices, which
are the Cartesian product of some finite lattices. Suppose
$L=L_{1}\times\cdots\times L_{p}$, where $L_{i}$ is a finite lattice of
cardinal $k_{i}$, $i=1,\cdots,p$. Define the partial order relation
$\leqslant$ on $L$ by

$(x_{1},\cdots,x_{p})\leqslant(y_{1},\cdots,y_{p})\Leftrightarrow
x_{1}\leqslant y_{1},\cdots,x_{p}\leqslant y_{p}.$

Then a network over $L$ with ASSR (8), denoted by $\Sigma$, can be decomposed
into several factors $\Sigma_{1},\cdots,\Sigma_{p}$, where the factor
$\Sigma_{i}$ is a network consisting of the corresponding $i$-th factor of the
state $(x_{1}(t),\cdots,x_{p}(t))$, with the binary operations in $L_{i}$,
$i=1,\cdots,p$.

Then a decomposition theorem follows.

###### Theorem III.6

The system $\Sigma$ is controllable (observable, synchronizable,
stabilizable), if and only if, the factor systems $\Sigma_{i}$, $i=1,\cdots,p$
are controllable (observable, synchronizable, stabilizable).

Proof. We only need to show that a network defined over a product lattice of
$L_{1}$ and $L_{2}$ can be decomposed into two systems $\Sigma_{1}$,
$\Sigma_{2}$, which are defined over $L_{1}$, $L_{2}$ respectively, i.e. any
trajectory $\\{z(t)|t\geqslant 0\\}\subset L$ starting from a point
$(x_{0},y_{0})$ can be decomposed into two trajectories
$\\{x(t,x_{0})|t\geqslant 0\\}$, $\\{y(t,y_{0})|t\geqslant 0\\}$ starting from
points $x_{0}\in L_{1}$, $y_{0}\in L_{2}$ respectively, such that
$z(t)=(x(t,x_{0}),y(t,y_{0}))$, $\forall t>0$. Since a factor of a product
lattice can be viewed as a sublattice, the factor lattices are both invariant
subspaces, such decomposition is obvious, and the conclusion follows. $\Box$

This can be viewed as an analogue of the results in [8]. The significance of
Theorem III.6 is that it simplifies the analysis of a system defined over a
product lattice by decomposing it into subsystems over factor sub-lattices.
This could not be done without the product lattice structure over the defining
set, and is different from conventional subspace decompositions since the
control system is nonlinear.

##  IV Recovering the Lattice Structures

After dealing with the general notions and expressions of networks over
lattices, we consider the verification of them. As aforementioned, networks
over lattices are special cases of multiple-valued logical networks; so a
natural question arises: how can one recognize them from the vast class of
networks over finite sets, and further, how can one construct or recover the
order relationships on the underlying sets from the algebraic state space
expressions of networks?

The aim of this section is twofold: first, to reconstruct the lattice
structure on ${\mathcal{D}}_{k}$ from the structural matrix of a network
defined over a lattice of $k$ elements; second, to verify if there exists a
partial order on ${\mathcal{D}}_{k}$ allowing a given network to be one
defined over a lattice, or at least be generated by some basic operators on
the lattice.

###  IV-A Recovering Order Through Lattice Functions

Consider a system over ${\mathcal{D}}_{k}$ with ASSR (7). The following
algorithm can decompose it into the component-wise expression, The proof of
which is straightforward.

###### Proposition IV.1

Consider a matrix $M\in{\mathcal{L}}_{k^{n}\times k^{n}}$, there exists a
unique decomposition $M=M_{1}*\cdots*M_{n}$, where
$M_{i}=R_{i}M\in{\mathcal{L}}_{k\times k^{n}}$, and $R_{i}={\bf
1}_{k^{i-1}}\otimes I_{k}\otimes{\bf 1}_{k^{n-i}}$, ${\bf 1}_{n}$ is the
$n$-dimensional row vector with all entries equal $1$.

In the following we will show that the information of comparability between
the elements in a lattice can be encoded in the structure matrix of any
lattice function over it. That is to say, if $M$ is the structure matrix of an
$n$-node network defined over a lattice, then any $M_{i}$ constructed as in
Proposition IV.1, which is the structure matrix of the dynamics of the $i$-th
node, $i=1,\cdots,n$, will be enough to rebuild the lattice structure.

The case of $2$-ary functions, which corresponds to a network of $2$ nodes, is
trivial, since by the so-called absorption property a binary lattice function
can only be $\vee$ or $\wedge$ (after some procedure of simplification).
Therefore it suffices to check the conditions in Proposition III.1 on its
structure matrix, and recover the partial order by $x\vee y=y\Leftrightarrow
x\leqslant y$.

The general $n$-ary case requires a different approach. Without loss of
generality, assume that functions discussed in the following do not have
“dumb” indices, i.e.

$\displaystyle\exists a\neq b\in{\mathcal{D}}_{k},\quad\text{s.t.}\quad$
$\displaystyle f(x_{1},\cdots,x_{i-1},a,x_{i+1},\cdots,x_{n})$
$\displaystyle\neq$ $\displaystyle
f(x_{1},\cdots,x_{i-1},b,x_{i+1},\cdots,x_{n}).$

In fact, one can verify if a variable in the expression of a multi-valued
logical function is dumb by using the method provided by [10], and further
remove them from the explicit expression of the function.

First we point out the fact that given two comparable elements $a$, $b$ in any
lattice $(L,\vee,\wedge)$, the operators $\vee$, $\wedge$ restricted to
$\\{a,b\\}$ behave in the same way as the disjunction and conjunction
operators over the classical Boolean algebra $\\{0,1\\}$. Hence we may try to
use the “restricted” structure matrix of a $k$-valued logical function to find
all comparable pairs on ${\mathcal{D}}_{k}$.

For an $n$-ary logical function $f$ over ${\mathcal{D}}_{k}$, consider its
restriction to a pair of elements $a,b\in{\mathcal{D}}_{k}$, that is to say,
the value $f(x_{1},\cdots,x_{n})$ takes when $x_{1},\cdots,x_{n}$ take values
in $\\{a,b\\}$ only. This transforms $f$ to another logical function
$f_{ab}:{\mathcal{D}}_{2}^{n}\rightarrow{\mathcal{D}}_{k}$.

Denote by $M\in{\mathcal{L}}_{k\times k^{n}}$ the structure matrix of $f$.
$\forall a,b\in{\mathcal{D}}_{k}$, if their vector forms are
$\delta_{k}^{i_{a}}$, $\delta_{k}^{i_{b}}$, then identify them as
$\delta_{2}^{1}$, $\delta_{2}^{2}$ respectively. By making no distinction
between a variable and its vector form ($a\sim\delta_{k}^{i_{a}}$,
$b\sim\delta_{k}^{i_{b}}$, $1\sim\delta_{2}^{1}$, $0\sim\delta_{2}^{2}$), we
may express a variable $x\in\\{a,b\\}$ through a Boolean one
$\tilde{x}\in\\{0,1\\}$ in vector form as

$\displaystyle x=\delta_{k}[i_{a},i_{b}]\tilde{x}$ (11)

hence by substitution, the restriction of $f$ on $\\{a,b\\}^{n}$, viewed as a
function ${f}_{ab}:{\mathcal{D}}_{2}^{n}\rightarrow{\mathcal{D}}_{k}$, has
vector form expression as

$\displaystyle
f(x_{1},\cdots,x_{n})\big{|}_{\\{a,b\\}^{n}}={f}_{ab}(\tilde{x}_{1},\cdots,\tilde{x}_{n})={M}_{ab}\tilde{x}_{1}\cdots\tilde{x}_{n},$
(12)

where

$\displaystyle{M}_{ab}=M\prod_{j=0}^{n-1}(I_{2^{j}}\otimes\delta_{k}[i_{a},i_{b}]),$
(13)

The structure matrix (13) is obtained from (11) and the commutative properties
in Proposition II.2.

###### Theorem IV.2

If there is an order $\leqslant$ on ${\mathcal{D}}_{k}$ making it a lattice,
and $f:{\mathcal{D}}_{k}^{n}\rightarrow{\mathcal{D}}_{k}$ is a lattice
function, then given two elements $a,b\in{\mathcal{D}}_{k}$ with vector
expressions $\delta_{k}^{i_{a}}$, $\delta_{k}^{i_{b}}$ respectively, $a$ and
$b$ are comparable ($a\leqslant b$ or $b\leqslant a$), if and only if,

$\displaystyle\operatorname{Col}(M_{ab})\subset\\{\delta_{k}^{i_{a}},\delta_{k}^{i_{b}}\\};$
(14a)
$\displaystyle\operatorname{Col}(M_{ab}^{i})\subset\\{\delta_{k}^{i_{a}}-\delta_{k}^{i_{b}},{\bf
0}\\},~{}i=1,\cdots,n,$ (14b) $\displaystyle
M_{ab}\delta_{2^{n}}^{1}=\delta_{k}^{i_{a}},~{}M_{ab}\delta_{2^{n}}^{2_{n}}=\delta_{k}^{i_{b}},$
(14c)

where ${\bf 0}=(0,\cdots,0)\in{\mathbb{R}}^{k}$, $M_{ab}$ is defined as in
(13), and

$\displaystyle M_{ab}^{i}:=M_{ab}W_{[2,2^{i-1}]}\begin{pmatrix}1\\\
-1\end{pmatrix}.$

Proof. (Necessity) When $a$, $b$ are comparable, since $\vee$, $\wedge$
restricted to $\\{a,b\\}$ are identical to disjunction and conjunction
operators on the classical Boolean algebra $\\{0,1\\}$ (denoted by $\vee_{2}$,
$\wedge_{2}$ respectively), $f|_{\\{a,b\\}^{n}}$ as a lattice function can be
viewed as belonging to the class generated by $\vee_{2}$, $\wedge_{2}$. Hence
its image are in the set of $\\{a,b\\}$, and by the property of these
operators, $f|_{\\{a,b\\}^{n}}$ is monotonic and reproducing, that is,
(adopting the notations in (12))
$\forall\tilde{x}_{1},\cdots,\tilde{x}_{n}\in\\{0,1\\}$,

$\displaystyle f_{ab}(\tilde{x}_{1},\cdots,\tilde{x}_{n})\in\\{a,b\\},$
$\displaystyle
f_{ab}(\tilde{x},\cdots,1,\cdots,\tilde{x}_{n})-f_{ab}(\tilde{x},\cdots,0,\cdots,\tilde{x}_{n})\in\\{a-b,0\\},$
$\displaystyle f_{ab}(1,\cdots,1)=a,~{}f_{ab}(0,\cdots,0)=b,$

where in the second equation the variables only differ in the $i$-th entry,
$i=1,\cdots,n$, and it is equivalent to say that
$f(x_{1},\cdots,x_{n})\leqslant f(y_{1},\cdots,y_{n})$ if $x_{i}\leqslant
y_{i}$, $x_{i},y_{i}\in\\{a,b\\}$, $i=1,\cdots,n$. Translating the above
equations into the vector expression yields

$\displaystyle
M_{ab}\tilde{x}_{1}\cdots\tilde{x}_{n}\in\\{\delta_{k}^{i_{a}},\delta_{k}^{i_{b}}\\};$
$\displaystyle
M_{ab}\tilde{x}_{1}\cdots\delta_{2}^{1}\cdots\tilde{x}_{n}-M_{ab}\tilde{x}_{1}\cdots\delta_{2}^{2}\cdots\tilde{x}_{n}$
$\displaystyle=$ $\displaystyle
M_{ab}W_{[2,2^{i-1}]}(\delta_{2}^{1}-\delta_{2}^{2})\tilde{x}_{1}\cdots\tilde{x}_{i-1}\tilde{x}_{i+1}\cdots\tilde{x}_{n}\in\\{\delta_{k}^{i_{a}}-\delta_{k}^{i_{b}},{\bf
0}\\};$ $\displaystyle
M_{ab}\delta_{2}^{1}\cdots\delta_{2}^{1}=\delta_{k}^{i_{a}},~{}M_{ab}\delta_{2}^{2}\cdots\delta_{2}^{2}=\delta_{k}^{i_{b}},$
$\displaystyle\forall\tilde{x}_{1},\cdots,\tilde{x}_{n}\in\Delta_{2},~{}i=1,\cdots,n,$

which are (14a)(14b)(14c) respectively.

(Sufficiency) Conversely, if $f|_{\\{a,b\\}^{n}}$ satisfies the conditions
(14a)(14b)(14c), it can be viewed as generated by $\vee_{2}$, $\wedge_{2}$
over the lattice $\\{a,b\\}$, since according to the classical result by E.
Post [25], the class of reproducing and monotonic $n$-ary functions over
${\mathcal{D}}_{2}$ are exactly the one generated by $\vee_{2}$, $\wedge_{2}$.
Therefore $a$, $b$ has the order relation obtained from these operators.
$\Box$

Theorem IV.2 shows that given any lattice function $f$ over a $k$-element
lattice, we can find all the comparable pairs $a,b$ in ${\mathcal{D}}_{k}$ by
checking the conditions on corresponding $M_{ab}$, and further obtain an
undirected graph of $k$ nodes, called the comparability graph, where there is
an edge between two nodes if and only if they are comparable.

Then algorithms can be applied to the comparability graph, assigning to it an
orientation to recover the partial order on ${\mathcal{D}}_{k}$, which is,
however, not unique [6]. Details of these algorithms will be sketched later in
Section IV-B.

###### Remark IV.3

The significance of Theorem IV.2 is that, once the underlying lattice
structure of a network over a lattice is recovered, one may use it to simplify
the analysis of the control problems. For example, if we find out that the
network is defined over a product lattice, then the results in Section III-B
can be applied to decompose the network.

###  IV-B Realization of the Lattice Structure of a Network

Next we consider designing a lattice structure over ${\mathcal{D}}_{k}$ to
make a $k$-valued network be generated by some basic operators of the lattice.
The only data available is the structure matrix. The construction requires two
steps: first, derive a relation from $f$ according to Theorem IV.2; second,
verify if it is a partial order making ${\mathcal{D}}_{k}$ a lattice.

According to the results in [18], monotonic functions over a lattice are
generated by $\vee$, $\wedge$ and piecewise constant functions $m_{a,b}$,
defined as

$\displaystyle m_{a,b}(x)=\begin{cases}b,\quad a\leqslant x\\\ {\bf
0},\quad\text{otherwise}.\end{cases}$ (15)

We call $\vee$, $\wedge$, $\\{m_{a,b}\\}_{a,b\in{\mathcal{D}}_{k}}$ basic
operators on a lattice.

Here is the algorithm to construct a lattice structure on ${\mathcal{D}}_{k}$
to make the network generated by basic operators on this lattice.

###### Algorithm IV.4

Consider a network over ${\mathcal{D}}_{k}$ with $n$ nodes and its ASSR as
$x(t+1)=Mx(t)$, where $x(t)=x(t)=\ltimes_{j=1}^{n}x_{j}(t)$,
$M\in{\mathcal{L}}_{k^{n}\times k^{n}}$.

  * •

Step 1: Decompose $M$ into component-wise form $M_{1},\cdots,M_{n}$, where
$M_{i}$ is the structure matrix of the $i$-th node
$x_{i}(t+1)=f_{i}(x_{1}(t),\cdots,x_{n}(t))$ with ASSR $x_{i}(t+1)=M_{i}x(t)$,
$i=1,\cdots,n$. Remove the dumb indices in each function $M_{i}$ (see the
algorithm provided in [10]).

  * •

Step 2: For the $i$-th function $f_{i}$ with structure matrix $M_{i}$, draw
the corresponding graph $G_{i}$ of $n$ nodes, where there is an edge between
nodes $a,b\in{\mathcal{D}}_{k}$ if and only if $(a,b)$ is a pair on which
$f_{i}$ satisfies the conditions (14a)(14b). Take the conjunction of the
graphs $G_{i}$, $i=1,\cdots,n$, denote it by $G$.

  * •

Step 3: Give an orientation to the resulting graph $G$, verify whether it is
transitive to derive a partial order on ${\mathcal{D}}_{k}$, and then see if
it is a lattice.

###### Remark IV.5

The algorithm in Step 3 for transitive orientation have been well established
since 1999 [6], and we only give a sketch of it due to the limitation of
space. Let $G$ be a prime undirected graph with vertex set $V$, start with a
partition $\\{\\{v\\},V\backslash\\{v\\}\\}$ and define it as an ordered list.
Then choose a pivot vertex $x\in V$, and split each partition class $Y$ into
two parts: the vertices adjacent to $x$ (denoted by $Y_{a}$) and those non-
adjacent to $x$ (denoted by $Y_{n}$) and place them in consecutive positions
in the ordered list. Let $Y_{n}$ occupy the earlier in the two new positions
if $x$ proceeds $Y$, and the latter if not. Then move on to another pivot
vertex. Refining the partition through the procedure of choosing pivots, one
will obtain an ordered partition where each class only contains a single
element, which is the topological sorting of the orientation. One may refer to
[21, 22] for proofs and details.

Applying this algorithm to any network over ${\mathcal{D}}_{k}$, we have the
following proposition, which ensures a network be generated from three classes
of operators when a lattice structure is obtained.

###### Proposition IV.6

If the partial order derived from a network over ${\mathcal{D}}_{k}$ by
Algorithm IV.4 makes a lattice, then the network is generated from the class
of functions $\\{\vee,\wedge,m_{a,b}\\}$, where $\vee$, $\wedge$ are the
$\sup$, $\inf$ operators of the lattice respectively, and $m_{a,b}$ are
piecewise constant functions defined as in (15), $a,b\in{\mathcal{D}}_{k}$.

Proof. The pairs on which the function with structure matrix $M_{i}$ is
monotonic (i.e. satisfies (14a)(14b), by the proof of Theorem IV.2) are
candidates of comparable pairs to allow this function to be generated from
basic operators of a lattice [18]. After checking the updating function of the
$i$-th node on each pair $\\{a,b\\}\subset{\mathcal{D}}_{k}$, we obtain the
comparability graph of the partial order for the dynamics of the $i$-th node.
Since only when two elements are comparable in the order relation derived from
each node can they be labelled as comparable in the partial order
corresponding to the whole system, one need to take conjunction of the $n$
graphs (that is, there is an edge between $\\{a,b\\}$ in graph $G$ if and only
if such edge exists for each graph $G_{i}$, $i=1,\cdots,n$), and check the
conditions for being a lattice on the corresponding order relation after
orienting the graph. $\Box$

The above proposition gives a criterion for whether a network can be realized
through basic operators of some lattice when only its structure matrix is
known.

In the end, we claim that there exist canonical expressions for monotonic
functions over a lattice.

######  Proposition IV.7 ([18])

If $f:{\mathcal{D}}_{k}^{n}\rightarrow{\mathcal{D}}_{k}$ is monotonic over a
lattice $({\mathcal{D}}_{k},\leqslant)$. Then
$\forall\bm{x}=(x_{1},\cdots,x_{n})\in{\mathcal{D}}_{k}^{n}$,

$\displaystyle
f(\bm{x})=\bigvee_{\bm{a}=(a_{1},\cdots,a_{n})\in{\mathcal{D}}_{k}^{n}}\Big{(}\bigwedge_{i=1}^{n}m_{a_{i},f(\bm{a})}(x_{i})\Big{)},$
(16)

where $m_{a_{i},f(\bm{a})}$ is defined as in (15).

One can see that additional constraints are needed to make sure that a
function is a lattice function (that is, generated solely by $\vee$,
$\wedge$); however, finding such conditions would be difficult due to (16),
since any lattice function is monotonic and hence will always allow an
expression that contains $m_{ab}$.

##  V Numerical Examples

This section provides numerical examples to illustrate the results in sections
III and IV.

We first consider the observability problem of a control network defined over
a product lattice, and see how it can be solved through decomposing the system
into subsystems over factor lattices.

###### Example V.1

Consider a network over the lattice
${\mathcal{D}}_{2}\times{\mathcal{D}}_{3}$, where
${\mathcal{D}}_{2}=\\{0,1\\}$, ${\mathcal{D}}_{3}=\\{0,1,2\\}$ are ordered as
chains canonically. The Hasse diagram of the product lattice is shown in
Figure 2.

${\bf 0}=(0,0)$$(1,0)$$(0,1)$$(1,1)$$(0,2)$${\bf 1}=(1,2)$ Figure 2: Hasse
diagram of ${\mathcal{D}}_{2}\times{\mathcal{D}}_{3}$

The dynamics of the network is

$\displaystyle\begin{cases}x_{1}(t+1)=x_{2}(t)\wedge u(t)\\\
x_{2}(t+1)=x_{1}(t)\vee x_{2}(t)\\\ \end{cases},\quad y(t)=x_{2}(t).$ (17)

Converting (17) to ASSR, we obtain the structure matrices of the dynamics and
the output respectively as

$\displaystyle L$
$\displaystyle=\delta_{36}[1,3,3,5,5,\cdots,32,33,34,35,36]\in{\mathcal{L}}_{36\times
216},$ $\displaystyle E$
$\displaystyle=\delta_{6}[1,2,3,4,5,6,\cdots,1,2,3,4,5,6]\in{\mathcal{L}}_{6\times
36}.$

We will consider the observability of the system (17), showing how existing
results can be applied to the ASSR, and how it could be solved through its
subnetworks over sublattices. Suppose $x_{1}(t)=(x^{1}_{1}(t),x^{1}_{2}(t))$,
$x_{2}(t)=(x^{2}_{1}(t),x^{2}_{2}(t))$,
$y(t)=(y_{1}(t),y_{2}(t))\in\Delta_{2}\times\Delta_{3}$, then one can
calculate the ASSR of the corresponding subnetworks over the factor rings
respectively as

$\displaystyle\begin{array}[]{rl}\begin{cases}z_{1}(t+1)=\tilde{L}_{1}z_{1}(t),\\\
y_{1}(t)=E_{1}z_{1}(t);\end{cases}\begin{cases}z_{2}(t+1)=\tilde{L}_{2}z_{2}(t),\\\
y_{2}(t)=E_{2}z_{2}(t),\end{cases}\end{array}$

where $z_{1}(t)=x^{1}_{1}(t)x^{1}_{2}(t)$,
$z_{2}(t)=x^{1}_{2}(t)x^{2}_{2}(t)$, and

$\displaystyle\begin{array}[]{rcl}\tilde{L}&=\delta_{4}&[1,2,2,2,1,2,4,4],\quad
E_{1}=\delta_{2}[1,2,1,2],\\\
\tilde{L}_{2}&=\delta_{9}&[1,2,3,2,2,3,3,3,3,1,2,3,5,5,\\\
{}\hfil&{}\hfil&6,6,6,6,1,2,3,5,5,6,9,9,9],\\\
E_{2}&=\delta_{3}&[1,2,3,1,2,3,1,2,3].\end{array}$

First consider the network over ${\mathcal{D}}_{2}$. Following [9], construct
an auxiliary network as

$\displaystyle\begin{cases}z_{1}(t+1)=\tilde{L}_{1}u(t)z_{1}(t)\\\
z^{*}_{1}(t+1)=\tilde{L}_{1}u(t)z^{*}_{1}(t),\end{cases}$

set $w(t)=z(t)z^{*}(t)$, then the ASSR is

$w(t+1)=Gu(t)w(t),$

where $G=\tilde{L}_{1}(I_{8}\otimes\tilde{L}_{1})(I_{2}\otimes
W_{[2,4]})\mathrm{diag}(\delta_{2}^{1},\delta_{2}^{2})$.

Solving the output distinguishable pairs, we derive the set

$W=\\{\delta_{16}^{2},\delta_{16}^{4},\delta_{16}^{5},\delta_{16}^{7},\delta_{16}^{10},\delta_{16}^{12},\delta_{16}^{13},\delta_{16}^{15}\\},$

Then the controllability matrix from $\Delta_{16}$ to $W$ is

$\displaystyle C_{W}^{1}$
$\displaystyle=I_{W}[\sum_{i=1}^{16}(G\delta_{2}^{1}+G\delta_{2}^{2})^{(i)}]$
$\displaystyle=[0,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0],$

where $I_{W}$ is the index matrix of $W$.

Therefore the distinguishable pairs of subnetwork is
$S_{1}=\\{(\delta_{4}^{1},\delta_{4}^{2}),(\delta_{4}^{1},\delta_{4}^{3}),(\delta_{4}^{1},\delta_{4}^{4})\\}$,
the system is not observable.

A similar argument can be applied to the subsystem over ${\mathcal{D}}_{3}$,
showing that

$\displaystyle\begin{array}[]{ccl}C_{W}^{2}&=&[0,1,1,1,1,1,1,1,1,1,0,1,0,0,1,1,1,1,1,1,\\\
&{}\hfil&0,1,1,0,0,0,0,1,0,1,0,0,1,1,1,1,1,0,1,0,\\\
&{}\hfil&0,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,\\\
&{}\hfil&0,0,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0],\\\ \end{array}$

and the distinguishable pairs of the subnetwork is

$\displaystyle\begin{array}[]{rcl}S_{2}&=&\\{(\delta_{9}^{1},\delta_{9}^{2}),(\delta_{9}^{1},\delta_{9}^{3}),(\delta_{9}^{1},\delta_{9}^{4}),(\delta_{9}^{1},\delta_{9}^{5}),(\delta_{9}^{1},\delta_{9}^{6}),(\delta_{9}^{1},\delta_{9}^{7}),\\\
&{}\hfil&(\delta_{9}^{1},\delta_{9}^{8}),(\delta_{9}^{1},\delta_{9}^{9}),(\delta_{9}^{2},\delta_{9}^{3}),(\delta_{9}^{2},\delta_{9}^{6}),(\delta_{9}^{2},\delta_{9}^{7}),(\delta_{9}^{2},\delta_{9}^{8}),\\\
&{}\hfil&(\delta_{9}^{2},\delta_{9}^{9}),(\delta_{9}^{3},\delta_{9}^{4}),(\delta_{9}^{3},\delta_{9}^{5}),(\delta_{9}^{4},\delta_{9}^{6}),(\delta_{9}^{4},\delta_{9}^{7}),(\delta_{9}^{4},\delta_{9}^{8}),\\\
&{}\hfil&(\delta_{9}^{4},\delta_{9}^{9}),(\delta_{9}^{5},\delta_{9}^{6}),(\delta_{9}^{5},\delta_{9}^{7}),(\delta_{9}^{5},\delta_{9}^{8}),(\delta_{9}^{5},\delta_{9}^{9})\\}\end{array}$

Therefore we conclude that the whole system is not observable. To be precise,
$(\delta_{2}^{i_{1}},\delta_{2}^{j_{1}}),(\delta_{2}^{i_{2}},\delta_{2}^{j_{2}})\in({\Delta}_{2}\times{\Delta}_{3})^{2}$
is distinguishable if and only if, $(\delta_{2}^{i_{1}},\delta_{2}^{j_{1}})\in
S_{1}$, $(\delta_{2}^{i_{2}},\delta_{3}^{j_{2}})\in S_{2}$.

Next, we give an example of how Algorithm IV.4 works to construct a lattice
structure on a finite set from the structure matrix of a network, so that this
network can be expressed by basic operators on the lattice.

###### Example V.2

Consider the following network over ${\mathcal{D}}_{5}$ with $3$ nodes and
ASSR as

$\displaystyle x(t+1)=Mx(t),$ (18)

where $x(t)=x_{1}(t)x_{2}(t)x_{3}(t)$, and

$\displaystyle\begin{array}[]{ccl}M=&\delta_{125}&[1,31,61,91,121,26,31,61,91,121,51,56,61,\\\
{}\hfil&{}\hfil&91,121,76,81,86,91,121,101,106,111,116,121,\\\
{}\hfil&{}\hfil&1,32,61,92,122,32,32,67,92,117,51,57,61,92,\\\
{}\hfil&{}\hfil&122,82,82,92,92,117,102,107,112,117,122,1,\\\
{}\hfil&{}\hfil&31,63,93,123,26,31,63,93,123,63,68,63,93,\\\
{}\hfil&{}\hfil&118,88,93,88,93,118,103,108,113,118,123,1,\\\
{}\hfil&{}\hfil&32,63,94,124,32,32,69,94,119,63,69,63,94,\\\
{}\hfil&{}\hfil&119,94,94,94,94,119,124,119,119,119,124,1,\\\
{}\hfil&{}\hfil&32,63,94,125,27,32,64,94,125,53,59,63,94,\\\
{}\hfil&{}\hfil&125,99,94,94,94,125,125,120,120,120,125]\\\ \end{array}$

We aim to check if there is an underlying partial order on ${\mathcal{D}}_{5}$
to make this system be one defined over a lattice, or to realize the
transition law by some lattice operators.

Decomposition of $M$ according to Proposition IV.1, we derive the structure
matrix of the dynamics of each node as

$M_{i}={\bf 1}_{5^{i-1}}\otimes I_{5}\otimes{\bf
1}_{5^{3-i}}\in{\mathcal{L}}_{5\times 5^{3}},~{}i=1,2,3.$

Next, we try to derive comparability graphs from them. Checking the conditions
(14a)(14b) on $M_{1}$, $M_{2}$ and $M_{3}$ for each pair of
$a,b\in{\mathcal{D}}_{5}$, one can obtain their comparability graphs, as shown
in Figure 3(a), 3(b) and 3(c) respectively.

$2$$1$$4$$0$$3$ (a)

$2$$1$$4$$0$$3$ (b)

$2$$1$$4$$0$$3$ (c)

Figure 3: Comparability graphs corresponding to $M_{1}$, $M_{2}$, $M_{3}$

Then take their conjunction to get the comparability graph of the system
(which is, naturally, the one in Figure 3(b)) and orient it according to the
vertex-partitioning algorithm in [21]. First, choose $1$ as a source and the
initial partition is $\\{\\{1\\},\\{2,3,4,0\\}\\}$. After a pivot on $2$,
since $\\{3,0\\}$ is not adjacent to $2$, it goes before $\\{2,4\\}$; then
after a pivot on $3$, we obtain the topological sorting of the graph as

$\displaystyle 1~{}|~{}0~{}|~{}3~{}|~{}2~{}|~{}4$ (19)

where an element is less than the ones following it in the sequence. Applying
(19) to Figure 4(a) gives the conjunction of comparability graphs a transitive
orientation, as shown in Figure 4(b). Hence we obtain a partial order on
${\mathcal{D}}_{5}$, denoted by $R$.

$2$$1$$4$$0$$3$ (a)

$1$$2$$3$$0$$4$ (b)

Figure 4: Transitive orientation of the conjunction of comparability graphs
and the Hasse diagram of the lattice

Finally, after checking the conditions on this orientation for being a lattice
over ${\mathcal{D}}_{5}$, we may draw the Hasse diagram of this order
relation, shown as in Figure 4.

Therefore, $({\mathcal{D}}_{5},R)$ is the lattice structure that makes the
network (18) become one generated by basic operators on the lattice, and one
can write it explicitly in these operators according to (16).

##  VI Conclusions

In this paper control networks over finite lattices are firstly considered.
Using STP, the ASSR of such networks is obtained, making control problems
easily solvable. Algorithms are proposed to recover the partial order from a
lattice function, and construct lattice structures for an arbitrary network to
allow it to be generated by basic operators over the lattice.

Although due to the canonical expression (16), the difficulty of finding
restrictions on the monotonic function class over a lattice to exclude the
functions $m_{a,b}$ seems intrinsic, a criteria for a multiple-valued logical
function to be generated solely by the operators $\vee$ and $\wedge$ is still
a problem worth trying.

Meanwhile, using the framework provided in this paper, problems concerning
switched, delayed and probabilistic networks over finite lattices can be
further investigated.

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