# A Hybrid Approach to Representing Shared Conceptualization in Decentralized AI Systems: Integrating Epistemology, Ontology, and Epistemic Logic

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Question and Key Contributions of the Paper

- Elucidation and definition of the core concepts pertinent to the research question, including ontology, epistemology, and epistemic logic within a decentralized environment.
- Identification and analysis of existing gaps in the literature regarding the representation of shared conceptualization in a decentralized context.
- Introduction of an innovative formal model for conceptualization in decentralized AI systems that amalgamates ontology, epistemology, and epistemic logic.
- Demonstration of the proposed model’s potential to enhance AI system performance in a decentralized setting, as evidenced by a healthcare sector scenario.
- Emphasis on the significance of comprehending the philosophical foundations of science and AI for the accurate and meaningful interpretation of research findings in interdisciplinary research.

#### 1.2. Research Methodology

## 2. Theoretical Foundations and Approaches

#### 2.1. Related Work

#### 2.2. Types of Environments

- Closed environments are characterized by strong agreement and coordination among observers. In these environments, observations are consistent with a shared understanding of the environment, enabling effective collaboration among participating entities. Closed environments are typically found in controlled settings, such as laboratory experiments and controlled studies, where variables are limited and conditions can be easily manipulated.
- Decentralized environments, on the other hand, are prevalent in autonomous and multi-agent systems where there is a lack of consensus and coordination among observers. While observers in decentralized environments may be autonomous and independent, they generally share a common understanding of the observed domain. In healthcare, for example, different healthcare providers may have different views on a patient’s condition, but they generally share a common understanding of medical terminology and best practices. Decentralized environments require the development of methods and models that enable efficient communication and collaboration of agents in the face of inconsistent information.
- Open environments represent complex and dynamic situations, such as natural disasters and intricate social systems, characterized by significant diversity and uncertainty among observers. In these environments, observers may have different or even conflicting concepts about the observed domain. For example, if the same individual is observed by different domains, such as agriculture, healthcare, and law, each domain may have a unique perspective on the individual, with different goals, values, and methods of evaluation. These environments are both decentralized and open, and they pose significant challenges due to conflicting or inconsistent observations, necessitating a shared understanding or shared conceptualization of the environment and its entities.

#### 2.3. Conceptualization

#### 2.4. Limitations of Ontology-Based Representation in Decentralized Environments

- Interpretation: Achieving a common understanding of concepts and relationships is crucial but difficult in decentralized environments due to the varying interpretations of concepts and relationships among different observer domains. While ontology offers an extensional representation by focusing on the extension of concepts and their relationships, the diverse interpretations necessitate an intensional representation that emphasizes the concepts themselves and the meanings attributed to them by different observer domains.
- Consequently, ontology-based representation methods, which rely on a shared understanding and a common ontology, are limited in these contexts [29].

- 3.
- Reasoning: Decentralized environments introduce unique challenges regarding the representation and reasoning of complex and heterogeneous knowledge [31]. Although ontology-based representation can categorize information, it is limited by the constraints of classical logics, such as first-order logic and description logic. These logics are ill-suited to represent and reason about multiple perspectives on a subject, and their incapacity to express modality and possibility hinders the representation and reasoning of relationships between various perspectives.

#### 2.5. Representing Conceptualization Structure: Challenges and Limitations of Classic Logics

- The individual interpretation of the conceptualization
- The intensional representation required to handle the diversity of these interpretations.

#### 2.6. The Importance of Understanding Extensions and Intensions in Representation

#### 2.7. Ontology and Epistemology in Conceptualization: A Modal Logic Approach

#### 2.8. Modal Logic for Representing and Reasoning in Decentralized Environments

- Extensional level (ontology): At the extensional level, the focus is on the relationships between specific objects, instances, or entities in the world. It deals with actual instances or examples of concepts and their relationships, aligning with ontology. By representing actual relationships among objects, the extensional level allows for more precise and accurate models of entities and their interactions in each domain.
- Intensional level (epistemology): At the intensional level, the focus is on abstract concepts, their properties, and the inherent relationships between them. This level deals with the general aspects of meaning, corresponding to epistemology. By capturing the essence of a concept or relationship, the intensional level provides a structured and meaningful representation of knowledge suitable for the development of ontologies and other knowledge representation systems.

## 3. Formal Modeling of Conceptualization in Decentralized Environments

#### 3.1. Formal Modeling

#### 3.2. Epistemology: A Proposed Formulation

- $C$ denotes the conceptualization.
- $\mathfrak{I}$ stands for an intensional interpretation function, which maps entity elements of $D$ to constant symbols of $V$ and relation elements of $\mathfrak{R}$ to predicate symbols of $V$.$$\mathfrak{I}\text{}:\text{}V\text{}\to \text{}D\text{}\cup \mathfrak{R}$$

#### 3.3. Ontology: A Proposed Formulation

- ${S}_{w}C\in SC$.
- For all constant symbols $c\in V:I\left(c\right)=I\left(c\right).$
- There exists a world $w\in W$ such that for all predicate symbols $p\in V:I\left(p\right)=\rho and\rho \left(w\right)=I\left(p\right).$

_{Θ}(L) in accordance with $\mathsf{\Theta}$.

_{Θ}(L) within the conceptualization $C$.

- $\mathsf{\Theta}$ is deemed committed to $C$ when it is structured to characterize $C$ and effectively approximates reality $D$ through its extensions.
- A language $\text{}L$ exhibits commitment to $\mathsf{\Theta}$ if it adheres to the conceptualization $C$ in such a manner that $\mathsf{\Phi}$ aligns harmoniously with $C$.
- $L$ demonstrates commitment to $\mathsf{\Phi}$, given $\mathsf{\Theta}$, by ensuring that the models for $\Phi $ comprehensively encompass the intended extensions I
_{Θ}(L)

#### 3.4. Modal Logic Examples in the Healthcare Sector

- $Patien{t}_{\left(w\right)}^{1}\subseteq D$ represents the set of patients in world w.
- $Physicia{n}_{\left(w\right)}^{1}\subseteq D$ represents the set of physicians in world w.
- $Nurs{e}_{\left(w\right)}^{1}\subseteq D$ represents the set of nurses in world w.
- $Facilit{y}_{\left(w\right)}^{1}\subseteq D$ represents the set of medical facilities in world w.
- $Diagnosi{s}_{\left(w\right)}^{2}\subseteq D\times D$ represents the set of diagnoses made by physicians in world w.
- $Assis{t}_{\left(w\right)}^{2}\subseteq D\times D\mathrm{\u2008}$ represents the set of assistance provided by nurses in $world\text{}w$.

#### 3.4.1. Example 1: Representing Relations between Patients, Physicians, and Nurses

_{1}and w

_{2}. In world w

_{1}, there is a patient p

_{1}who is diagnosed with a disease d

_{1}by physician ph

_{1}. Nurse n

_{1}provides assistance to p

_{1}. The relations in this world can be represented using the function ρ as follows:

- $Patien{t}_{\left(w1\right)}^{1}={p}_{1}$
- $Physicia{n}_{\left(w1\right)}^{1}=p{h}_{1}$
- $Nurs{e}_{\left(w1\right)}^{1}={n}_{1}$
- $Diagnosi{s}_{\left(w1\right)}^{2}=\left({p}_{1},{d}_{1}\right)$
- $Assis{t}_{\left(w1\right)}^{2}=\left({p}_{1},{n}_{1}\right)$

_{2}, there is a different patient p

_{2}who is diagnosed with a different disease d

_{2}by physician ph

_{2}. Nurse n

_{2}provides assistance to p

_{2}. The relations in this world can be represented using the function ρ as follows:

- $Patien{t}_{\left(w2\right)}^{1}={p}_{2}$
- $Physicia{n}_{\left(w2\right)}^{1}=p{h}_{2}$
- $Nurs{e}_{\left(w2\right)}^{1}={n}_{2}$
- $Diagnosi{s}_{\left(w2\right)}^{2}=\left({p}_{2},{d}_{2}\right)$
- $Assis{t}_{\left(w2\right)}^{2}=\left({p}_{2},{n}_{2}\right)$

#### 3.4.2. Example 2: Representing and Reasoning about Different Perspectives in the Healthcare Sector

_{1}is diagnosed with disease d

_{1}and is being assisted by nurse n1. We can represent this belief using the epistemic modal operator K and the accessibility relation R as follows:

_{1}is diagnosed with disease d

_{1}and is being assisted by nurse n

_{1}in world w

_{1}, and also the agent believes that this is possible in world w

_{2}as well.

_{2}is diagnosed with disease d

_{2}and is being assisted by nurse n

_{2}. We can represent this belief using the epistemic modal operator K and the accessibility relation R as follows:

_{2}is diagnosed with disease d

_{2}and is being assisted by nurse n

_{2}in world w

_{2}, and also the agent believes that this is possible in world w

_{1}as well.

- $Studen{t}_{\left(w\right)}^{1}\subseteq D$, which denotes the set of students in world $w$.
- $Teache{r}_{\left(w\right)}^{1}\subseteq D$, which denotes the set of teachers in world $w$.
- $Subjec{t}_{\left(w\right)}^{1}\subseteq D$, which denotes the set of subjects in world $w$.
- $Institutio{n}_{\left(w\right)}^{1}\subseteq D$, which denotes the set of educational institutions in world $w.$
- $Teac{h}_{\left(w\right)}^{2}\subseteq D\times D$, which denotes the set of teaching relations between teachers and subjects in world $w$.
- $Lear{n}_{\left(w\right)}^{2}\subseteq D\times D$, which denotes the set of learning relations between students and subjects in world $w$.

#### 3.4.3. Example 3: Formal Representation of Relations within the Education Sector

_{1}, a student ${s}_{1}$ is learning a subject $su{b}_{1}$ from a teacher ${t}_{1}$. The relations within this world can be represented using the function $\rho $ as follows:

- $Studen{t}_{\left(w1\right)}^{1}={s}_{1}$
- $Teache{r}_{\left(w1\right)}^{1}={t}_{1}$
- $Subjec{t}_{\left(w1\right)}^{1}=su{b}_{1}$
- $Teac{h}_{\left(w1\right)}^{2}=\left({t}_{1},su{b}_{1}\right)$
- $Lear{n}_{\left(w1\right)}^{2}=\left({s}_{1},su{b}_{1}\right)$

_{2}is learning a different subject sub

_{2}from a teacher t

_{2}. The relations within this world can be represented using the function $\rho $ as follows:

- $Studen{t}_{\left(w2\right)}^{1}={s}_{2}$
- $Teache{r}_{\left(w2\right)}^{1}={t}_{2}$
- $Subjec{t}_{\left(w2\right)}^{1}=su{b}_{2}$
- $Teac{h}_{\left(w2\right)}^{2}=\left({t}_{2},su{b}_{2}\right)$
- $Lear{n}_{\left(w2\right)}^{2}=\left({s}_{2},su{b}_{2}\right)$

#### 3.4.4. Example 4: Formal Representation and Reasoning about Different Perspectives within the Education Sector

_{1}is learning subject sub

_{1}from teacher t

_{1}. This belief can be formally represented using the epistemic modal operator K and the accessibility relation R as follows:

_{1}is learning subject sub

_{1}from teacher t

_{1}in world w

_{1}, and the agent also believes that this situation is plausible in world w

_{2}.

_{2}from teacher t

_{2}. This belief can be formally represented using the epistemic modal operator K and the accessibility relation R as follows:

#### 3.5. Applying the Hybrid Model to a Multi-Specialist Healthcare Scenario

- ${w}_{1}\left(endocrinologist\mathrm{\u2019}sperspective\right)$
- ${w}_{2}\left(nephrologist\mathrm{\u2019}sperspective\right)$

- $Patient\left(x\right)$
- $PrimaryCarePhysician\left(x\right)$
- $Endocrinologist\left(x\right)$
- $Nephrologist\left(x\right)$
- $TreatmentPlan\left(x\right)$
- $InsulinType\left(x\right)$
- $Dosage\left(x\right)$
- $HasDiabetes\left(x\right)$
- $ResponsibleFor(x,y)$
- $ProvidesInput(x,y,z)$
- $TreatmentPlanFor(x,y)$
- $TreatmentPlanIncludes(x,y,z)$

- $V:p,d,e,n,t,i,dos$
- $\mathfrak{I}:V\to D\cup \mathfrak{R}$
- $\mathfrak{I}\left(p\right)=patient$
- $\mathfrak{I}\left(d\right)=primaryCarePhysician\mathfrak{I}\left(e\right)=endocrinologist\mathfrak{I}\left(n\right)=nephrologist$
- $\mathfrak{I}\left(t\right)=treatmentPlan$
- $\mathfrak{I}\left(i\right)=insulinType$
- $\mathfrak{I}\left(dos\right)=dosage$

- ${S}_{w},{C}_{w}$ = $\u27e8D,{R}_{w},{C}_{w1}\u27e9$

- ${S}_{w},{C}_{w}$ = $\u27e8D,{R}_{w},{C}_{w1}\u27e9$

_{1}and w

_{2}, respectively) of the elements of R. These sets of extensions will differ based on the perspectives of the two specialists, capturing their unique viewpoints and understanding of the domain.

_{w}

_{1}and SwC

_{w}

_{2}for ${\mathsf{\Phi}}_{1}$ (endocrinologist’s perspective) and ${\mathsf{\Phi}}_{2}$ (nephrologist’s perspective), respectively, along with their sets of extensions ${R}_{w}{C}_{w1}$ and ${R}_{w}{C}_{w2}$:

- ${S}_{w}{C}_{w1}$ = $\u27e8D,{R}_{w}{C}_{w1}\u27e9$ (the intended extensional structure for w
_{1}, the endocrinologist’s perspective) - ${S}_{w}{C}_{w2}$ = $\u27e8D,{R}_{w}{C}_{w2}\u27e9$ (the intended extensional structure for w
_{2}, the nephrologist’s perspective)

_{1}(endocrinologist’s perspective on the ontological view

_{2}), we can define the set of extensions RwC

_{w}

_{1}as follows:

- $Patient\left(p\right)$
- $PrimaryCarePhysician\left(d\right)$
- $Endocrinologist\left(e\right)$
- $TreatmentPlan\left(t\right)$
- $InsulinType(t,i)$
- $Dosage(t,dos)$
- $HasDiabetes\left(p\right)$
- $ResponsibleFor(d,p)$
- $ProvidesInput(e,d,t)$
- $TreatmentPlanFor(t,p)$
- $TreatmentPlanIncludes(t,i,dos)$

_{2}(nephrologist’s perspective), we can define the set of extensions RwC

_{w}

_{2}as follows:

- $Patient\left(p\right)$
- $PrimaryCarePhysician\left(d\right)$
- $Nephrologist\left(n\right)$
- $TreatmentPlan\left(t\right)$
- $InsulinType(t,i)$
- $Dosage(t,dos)$
- $HasDiabetes\left(p\right)$
- $ResponsibleFor(d,p)$
- $ProvidesInput(n,d,t)$
- $TreatmentPlanFor(t,p)$
- $TreatmentPlanIncludes(t,i,dos)$

_{w}

_{1}and RwC

_{w}

_{2}, we capture the different perspectives of the two specialists, the endocrinologist (e) and nephrologist (n). The main difference between RwC

_{w}

_{1}and RwC

_{w}

_{2}is the type of specialist providing input to the primary care physician (d) regarding the treatment plan (t) for the patient (p).

_{w}

_{1}, the endocrinologist (e) provides input, while in RwC

_{w}

_{2}, the nephrologist (n) provides input. These unique perspectives and inputs will contribute to the medical team’s overall understanding of the patient’s condition and inform their decision-making process for the treatment plan.

- ${K}_{e}:$ Represents the knowledge of the endocrinologist (in ${w}_{1}$)
- ${K}_{n}:$ Represents the knowledge of the nephrologist (in ${w}_{2}$)
- R: Represents the accessibility relation between ${w}_{1}$ and ${w}_{2}$

_{w}

_{1}and RwC

_{w}

_{2}:

_{1}(endocrinologist’s perspective), we have:

- ${K}_{e},Patient\left(p\right)$
- ${K}_{e},PrimaryCarePhysician\left(d\right)$
- ${K}_{e},Endocrinologist\left(e\right)$
- ${K}_{e},TreatmentPlan\left(t\right)$
- ${K}_{e},InsulinType\left(t,i\right)$
- ${K}_{e},Dosage\left(t,dos\right)$
- ${K}_{e},HasDiabetes\left(p\right)$
- ${K}_{e},ResponsibleFor\left(d,p\right)$
- ${K}_{e},ProvidesInput\left(e,d,t\right)$
- ${K}_{e},TreatmentPlanFor\left(t,p\right)$
- ${K}_{e},TreatmentPlanIncludes\left(t,i,dos\right)$

_{2}(nephrologist’s perspective), we have:

- ${K}_{n},Patient\left(p\right)$
- ${K}_{n},PrimaryCarePhysician\left(d\right)$
- ${K}_{n},Nephrologist\left(n\right)$
- ${K}_{n},TreatmentPlan\left(t\right)$
- ${K}_{n},InsulinType\left(t,i\right)$
- ${K}_{n},Dosage\left(t,dos\right)$
- ${K}_{n},HasDiabetes\left(p\right)$
- ${K}_{n},ResponsibleFor\left(d,p\right)$
- ${K}_{n},ProvidesInput\left(n,d,t\right)$
- ${K}_{n},TreatmentPlanFor\left(t,p\right)$
- ${K}_{n},TreatmentPlanIncludes\left(t,i,dos\right)$

_{1}(endocrinologist’s perspective) is accessible from w

_{2}(nephrologist’s perspective), and vice versa.

_{2}(nephrologist’s perspective) is accessible from w

_{1}(endocrinologist’s perspective).

_{1}and w

_{2}, we allow the sharing of knowledge between the two specialists’ perspectives. This, in turn, helps the medical team to collaborate and make informed decisions about the patient’s treatment plan. By combining the knowledge from both perspectives, the primary care physician can consider the input from both specialists and adjust the treatment plan accordingly.

#### Summary of Hybrid Model for Formal Modeling of Conceptualization in AI Systems

## 4. Implications, Limitations, and Future Research

#### 4.1. Implications for Future Research and Practice

- Improved AI system performance: By providing a consistent and accurate representation of knowledge, beliefs, and relationships within a domain, the proposed model may contribute to enhanced decision-making and problem-solving capabilities in AI systems.
- Interdisciplinary collaboration: The integration of ontology, epistemology, and modal logic in the proposed model highlights the importance of interdisciplinary collaboration in addressing complex problems in AI systems. This approach can inspire future research to draw upon diverse fields to create innovative solutions.
- Scalability: The proposed model’s flexibility and adaptability can support the growth and expansion of AI systems in various domains, allowing for the seamless integration of new entities and relationships.
- Standardization: The formal modeling approach can contribute to the development of standard protocols and guidelines for representing knowledge and beliefs in AI systems, facilitating interoperability and compatibility among diverse systems.

#### 4.2. Limitations

- Complexity: The integration of ontology, epistemology, and modal logic in the proposed model may increase its complexity, which could present challenges in implementation and maintenance.
- Applicability: Although the patient treatment journey scenario demonstrated the effectiveness of the proposed model, its applicability to other sectors and use cases remains to be further explored.
- Evolution of knowledge: The proposed model may not fully account for the dynamic nature of knowledge and its continuous evolution in various domains, potentially requiring periodic updates to maintain accuracy and relevance.

#### 4.3. Future Research Directions

- Development of tools and frameworks: To facilitate the implementation and maintenance of the proposed model, future research could focus on developing tools, frameworks, and techniques that streamline the process. For example, in the healthcare domain using a graph database like Neo4j, researchers could define axioms in modal logic, design a graph schema capturing modal logic relationships, assign properties to nodes and relationships, import healthcare data adhering to the schema, and query the graph using Cypher to analyze the healthcare domain while incorporating modal logic. By developing tools, frameworks, and techniques that support these steps, future research can streamline the implementation and maintenance of the proposed model, making it more accessible and practical across various domains.
- Evaluation in diverse contexts: Additional case studies in various sectors and use cases could be conducted to assess the generalizability and robustness of the proposed model across different contexts.
- Adaptive modeling: Incorporating mechanisms for handling the dynamic nature of knowledge and its evolution in various domains could enhance the proposed model’s effectiveness and applicability.
- Integration with other AI techniques: Exploring the synergies between the proposed model and other AI techniques, such as machine learning and natural language processing, could provide additional insights and further improve the representation and reasoning capabilities in AI systems.

- Enhanced understanding: By representing the underlying structure and meaning of information from diverse sources, the model can facilitate a more comprehensive understanding of the data, leading to better decision making and problem solving.
- Conflict resolution: The model’s ability to represent and reason about different perspectives and beliefs can help identify and resolve inconsistencies or contradictions that may arise during the integration process.
- Improved interoperability: By providing a standardized and formal approach to representing knowledge and beliefs, the model can enhance the compatibility and interoperability of information from different systems, facilitating seamless integration.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Visualization of the Intensional and Extensional Semantic Levels in Ontology and Epistemology.

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**MDPI and ACS Style**

Adhnouss, F.M.A.; El-Asfour, H.M.A.; McIsaac, K.; El-Feghi, I.
A Hybrid Approach to Representing Shared Conceptualization in Decentralized AI Systems: Integrating Epistemology, Ontology, and Epistemic Logic. *AppliedMath* **2023**, *3*, 601-624.
https://doi.org/10.3390/appliedmath3030032

**AMA Style**

Adhnouss FMA, El-Asfour HMA, McIsaac K, El-Feghi I.
A Hybrid Approach to Representing Shared Conceptualization in Decentralized AI Systems: Integrating Epistemology, Ontology, and Epistemic Logic. *AppliedMath*. 2023; 3(3):601-624.
https://doi.org/10.3390/appliedmath3030032

**Chicago/Turabian Style**

Adhnouss, Fateh Mohamed Ali, Husam M. Ali El-Asfour, Kenneth McIsaac, and Idris El-Feghi.
2023. "A Hybrid Approach to Representing Shared Conceptualization in Decentralized AI Systems: Integrating Epistemology, Ontology, and Epistemic Logic" *AppliedMath* 3, no. 3: 601-624.
https://doi.org/10.3390/appliedmath3030032