Mathematical Semantics of the 3-No Problems in the DIKWP Model's Semantic Space

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Abstract

Effective communication between stakeholders, whether human or artificial intelligence (AI) systems, is fundamental to achieving mutual understanding and collaboration. Within the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) model, communication processes can be impeded by three primary issues collectively termed the 3-No Problems: Incomplete Input/Output, Inconsistent Input/Output, and Imprecise Input/Output. This paper provides a comprehensive mathematical exposition of these problems within the DIKWP model's semantic space. By formally defining, quantifying, and analyzing these issues, we aim to facilitate their identification and remediation, thereby enhancing the reliability and effectiveness of DIKWP*DIKWP interactions between stakeholders.

Communication between stakeholders involves the exchange of cognitive components encapsulated within the DIKWP framework. However, this process is often impeded by deficiencies that disrupt the seamless transformation and integration of Data, Information, Knowledge, Wisdom, and Purpose. These deficiencies, known as the 3-No Problems, are:

1. Incomplete Input/Output (No-Incomplete)

2. Inconsistent Input/Output (No-Inconsistent)

3. Imprecise Input/Output (No-Imprecise)

Understanding these problems within the semantic space of the DIKWP model is crucial for enhancing mutual understanding and reducing the occurrence of hallucinations—erroneous perceptions or outputs—especially in human-AI collaborations.

The DIKWP model categorizes cognitive processes into five interconnected components:

1. Data (D): Raw, unprocessed facts or observations.

2. Information (I): Processed data highlighting patterns and relationships.

3. Knowledge (K): Organized and contextualized information forming a coherent understanding.

4. Wisdom (W): Judgments and decisions based on Knowledge, incorporating ethical and contextual considerations.

5. Purpose (P): The driving goals or intentions guiding cognitive processes.

Each component is represented as a vector in a high-dimensional semantic space, allowing for quantitative analysis and mathematical modeling.

Each stakeholder’s DIKWP profile is mathematically represented as:

Profilei={Di,Ii,Ki,Wi,Pi}\text{Profile}_i = \{\mathbf{D}_i, \mathbf{I}_i, \mathbf{K}_i, \mathbf{W}_i, \mathbf{P}_i\}Profilei​={Di​,Ii​,Ki​,Wi​,Pi​}

Where:

• Di∈Rn\mathbf{D}_i \in \mathbb{R}^nDi​∈Rn: Data vector

• Ii∈Rm\mathbf{I}_i \in \mathbb{R}^mIi​∈Rm: Information vector

• Ki∈Rp\mathbf{K}_i \in \mathbb{R}^pKi​∈Rp: Knowledge vector

• Wi∈Rq\mathbf{W}_i \in \mathbb{R}^qWi​∈Rq: Wisdom vector

• Pi∈Rr\mathbf{P}_i \in \mathbb{R}^rPi​∈Rr: Purpose vector

The semantic space enables the quantification of similarities and differences between these vectors, facilitating the analysis of communication effectiveness and the identification of the 3-No Problems.

Definition:

Incomplete Input/Output occurs when one stakeholder lacks sufficient data or information to fully comprehend the DIKWP components being communicated by the other stakeholder.

Mathematical Representation:

For each DIKWP component $X$, define the Completeness Score CXC_XCX​ as the ratio of the intersection to the union of the sender’s and receiver’s vectors:

CX=∣XA∩XB∣∣XA∪XB∣C_X = \frac{|\mathbf{X}_A \cap \mathbf{X}_B|}{|\mathbf{X}_A \cup \mathbf{X}_B|}CX​=∣XA​∪XB​∣∣XA​∩XB​∣​

Where:

• 0≤CX≤10 \leq C_X \leq 10≤CX​≤1

• CX=1C_X = 1CX​=1 indicates complete overlap (no incompleteness)

• CX<1C_X < 1CX​<1 indicates incompleteness

Gap Identification:

The Completeness Gap GXG_XGX​ quantifies the extent of incompleteness:

GX=1−CX=∣XA∪XB∣−∣XA∩XB∣∣XA∪XB∣G_X = 1 - C_X = \frac{|\mathbf{X}_A \cup \mathbf{X}_B| - |\mathbf{X}_A \cap \mathbf{X}_B|}{|\mathbf{X}_A \cup \mathbf{X}_B|}GX​=1−CX​=∣XA​∪XB​∣∣XA​∪XB​∣−∣XA​∩XB​∣​

Impact on Understanding:

Incomplete data or information leads to gaps in understanding, making it difficult for stakeholders to form a coherent Knowledge base, potentially resulting in misunderstandings or erroneous conclusions.

Remediation Mechanism:

To address incompleteness, the sender must provide the missing components:

ΔXA=XA−XB\Delta \mathbf{X}_A = \mathbf{X}_A - \mathbf{X}_BΔXA​=XA​−XB​

The receiver updates their profile:

XB′=XB∪ΔXA\mathbf{X}_B' = \mathbf{X}_B \cup \Delta \mathbf{X}_AXB′​=XB​∪ΔXA​

Where:

• ΔXA\Delta \mathbf{X}_AΔXA​ represents the missing elements from the sender’s DIKWP component $X$

• XB′\mathbf{X}_B'XB′​ is the updated receiver’s DIKWP component after integration

Example:

In a medical diagnosis scenario, if the AI lacks specific patient data that the human possesses, the human must provide this missing data to the AI to enhance the Completeness Score for the Data component.

Definition:

Inconsistent Input/Output arises when there are conflicting data or information between the DIKWP components of the two stakeholders, leading to confusion and misunderstanding.

Mathematical Representation:

For each DIKWP component $X$, define the Consistency Score SXS_XSX​ using a similarity measure (e.g., cosine similarity):

SX=sim(XA,XB)=XA⋅XB∥XA∥∥XB∥S_X = \text{sim}(\mathbf{X}_A, \mathbf{X}_B) = \frac{\mathbf{X}_A \cdot \mathbf{X}_B}{\|\mathbf{X}_A\| \|\mathbf{X}_B\|}SX​=sim(XA​,XB​)=∥XA​∥∥XB​∥XA​⋅XB​​

Where:

• 0≤SX≤10 \leq S_X \leq 10≤SX​≤1

• SX=1S_X = 1SX​=1 indicates perfect consistency

• SX<1S_X < 1SX​<1 indicates inconsistency

Inconsistency Quantification:

The Inconsistency Measure IXI_XIX​ quantifies the level of inconsistency:

IX=1−SXI_X = 1 - S_XIX​=1−SX​

Impact on Understanding:

Inconsistencies disrupt the formation of a coherent Knowledge base, leading to confusion and potentially incorrect Wisdom application, which can undermine the Purpose of the interaction.

Conflict Resolution Mechanism:

To resolve inconsistencies, stakeholders engage in dialogue to reconcile conflicting components. A simple mathematical approach involves averaging the conflicting vectors:

XA′=XA+XB2\mathbf{X}_A' = \frac{\mathbf{X}_A + \mathbf{X}_B}{2}XA′​=2XA​+XB​​XB′=XA+XB2\mathbf{X}_B' = \frac{\mathbf{X}_A + \mathbf{X}_B}{2}XB′​=2XA​+XB​​

Alternatively, more sophisticated reconciliation algorithms can be employed based on context and stakeholder priorities.

Example:

If the AI suggests a diagnosis that conflicts with the human’s medical expertise, averaging the Knowledge vectors or engaging in further dialogue can reconcile the inconsistency.

Definition:

Imprecise Input/Output occurs when the data or information exchanged is vague or ambiguous, leading to misunderstandings in the DIKWP components.

Mathematical Representation:

For each DIKWP component $X$, define the Precision Score PXP_XPX​ as the inverse of the entropy HXH_XHX​:

PX=1−HXHmaxP_X = 1 - \frac{H_X}{H_{\text{max}}}PX​=1−Hmax​HX​​

Where:

• HXH_XHX​ is the entropy of component $X$, measuring uncertainty or ambiguity

• HmaxH_{\text{max}}Hmax​ is the maximum possible entropy for component $X$

Entropy HXH_XHX​ is calculated using Shannon’s entropy formula for discrete variables:

HX=−∑i=1Np(xi)log⁡p(xi)H_X = -\sum_{i=1}^{N} p(x_i) \log p(x_i)HX​=−i=1∑N​p(xi​)logp(xi​)

Where:

• p(xi)p(x_i)p(xi​) is the probability of occurrence of the ithi^{th}ith element in $X$

Imprecision Quantification:

The Imprecision Measure MXM_XMX​ quantifies the level of imprecision:

MX=1−PX=HXHmaxM_X = 1 - P_X = \frac{H_X}{H_{\text{max}}}MX​=1−PX​=Hmax​HX​​

Impact on Understanding:

Imprecise data or information introduces ambiguity, making it challenging to accurately interpret and integrate Knowledge, thereby impeding the formation of coherent Wisdom and alignment with Purpose.

Clarification Mechanism:

To mitigate imprecision, stakeholders seek clarification:

Clarified Component XA′=XA+ΔXA\text{Clarified Component } \mathbf{X}_A' = \mathbf{X}_A + \Delta \mathbf{X}_AClarified Component XA′​=XA​+ΔXA​Clarified Component XB′=XB+ΔXB\text{Clarified Component } \mathbf{X}_B' = \mathbf{X}_B + \Delta \mathbf{X}_BClarified Component XB′​=XB​+ΔXB​

Where:

• ΔXA\Delta \mathbf{X}_AΔXA​ and ΔXB\Delta \mathbf{X}_BΔXB​ represent the clarified and disambiguated additions to the respective DIKWP components

Example:

In an educational AI tutor, if the AI provides vague feedback, specifying precise information and contextual examples can reduce imprecision, enhancing the student’s Understanding.

The interaction between two stakeholders, considering the 3-No Problems, is modeled as:

Interaction=EntityA×EntityB={DAB,IAB,KAB,WAB,PAB}\text{Interaction} = \text{Entity}_A \times \text{Entity}_B = \{\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB}\}Interaction=EntityA​×EntityB​={DAB​,IAB​,KAB​,WAB​,PAB​}

Where each component XAB\mathbf{X}_{AB}XAB​ is adjusted based on the presence of Incompleteness, Inconsistency, and Imprecision:

XAB′=fX(XA,XB)−Remediation Terms\mathbf{X}_{AB}' = f_X(\mathbf{X}_A, \mathbf{X}_B) - \text{Remediation Terms}XAB′​=fX​(XA​,XB​)−Remediation Terms

Remediation terms are functions that address GXG_XGX​, IXI_XIX​, and MXM_XMX​ for each $X\in \{D,I,K,W,P\}$.

To encapsulate all three No Problems, define a Deficiency Vector DefABX\mathbf{Def}_{AB}^XDefABX​ for each component $X$:

DefABX=[GXIXMX]\mathbf{Def}_{AB}^X = \begin{bmatrix} G_X \\ I_X \\ M_X \end{bmatrix}DefABX​=​GX​IX​MX​​​

Where:

• GXG_XGX​: Completeness Gap

• IXI_XIX​: Inconsistency Measure

• MXM_XMX​: Imprecision Measure

Define the Overall Deficiency Measure DAB\mathcal{D}_{AB}DAB​ as the aggregation of deficiencies across all DIKWP components:

DAB=∑X∈{D,I,K,W,P}∥DefABX∥2\mathcal{D}_{AB} = \sum_{X \in \{D, I, K, W, P\}} \|\mathbf{Def}_{AB}^X\|_2DAB​=X∈{D,I,K,W,P}∑​∥DefABX​∥2​

Where ∥⋅∥2\|\cdot\|_2∥⋅∥2​ denotes the Euclidean norm, providing a scalar value representing the total deficiency in the interaction.

The objective is to minimize DAB\mathcal{D}_{AB}DAB​ while maximizing mutual Understanding UAB\mathcal{U}_{AB}UAB​:

max⁡UAB−λ⋅DAB\max \quad \mathcal{U}_{AB} - \lambda \cdot \mathcal{D}_{AB}maxUAB​−λ⋅DAB​

Where $\lambda$ is a weighting factor balancing the importance of minimizing deficiencies against maximizing Understanding.

Objective: Maximize the Completeness Score CXC_XCX​ to ensure robust connectivity across DIKWP components.

Mechanism:

1. Identify Missing Components:

   ΔXA=XA−XB\Delta \mathbf{X}_A = \mathbf{X}_A - \mathbf{X}_BΔXA​=XA​−XB​

2. Integrate Missing Data/Information:

   XB′=XB∪ΔXA\mathbf{X}_B' = \mathbf{X}_B \cup \Delta \mathbf{X}_AXB′​=XB​∪ΔXA​

Example:

In a medical diagnosis scenario, if the AI lacks specific patient data that the human possesses, the human must provide this missing data to the AI to enhance the Completeness Score for the Data component.

Objective: Minimize the Inconsistency Measure IXI_XIX​ to ensure alignment and coherence in DIKWP components.

Mechanism:

1. Calculate Consistency Scores SXS_XSX​:

   SX=sim(XA,XB)=XA⋅XB∥XA∥∥XB∥S_X = \text{sim}(\mathbf{X}_A, \mathbf{X}_B) = \frac{\mathbf{X}_A \cdot \mathbf{X}_B}{\|\mathbf{X}_A\| \|\mathbf{X}_B\|}SX​=sim(XA​,XB​)=∥XA​∥∥XB​∥XA​⋅XB​​

2. Identify and Quantify Inconsistencies IX=1−SXI_X = 1 - S_XIX​=1−SX​:

3. Resolve Conflicts through Reconciliation:

   A simple approach involves averaging the conflicting vectors:

   XA′=XA+XB2\mathbf{X}_A' = \frac{\mathbf{X}_A + \mathbf{X}_B}{2}XA′​=2XA​+XB​​XB′=XA+XB2\mathbf{X}_B' = \frac{\mathbf{X}_A + \mathbf{X}_B}{2}XB′​=2XA​+XB​​

   Alternatively, implement more sophisticated reconciliation algorithms based on context and stakeholder priorities.

Example:

If the AI suggests a diagnosis that conflicts with the human’s medical expertise, averaging the Knowledge vectors or engaging in further dialogue can reconcile the inconsistency.

Objective: Minimize the Imprecision Measure MXM_XMX​ to ensure clarity and precision in DIKWP components.

Mechanism:

1. Calculate Precision Scores PX=1−MXP_X = 1 - M_XPX​=1−MX​:

   PX=1−HXHmaxP_X = 1 - \frac{H_X}{H_{\text{max}}}PX​=1−Hmax​HX​​

2. Identify Imprecise Components:

3. Clarify and Disambiguate Content:

   XA′=XA+ΔXA\mathbf{X}_A' = \mathbf{X}_A + \Delta \mathbf{X}_AXA′​=XA​+ΔXA​XB′=XB+ΔXB\mathbf{X}_B' = \mathbf{X}_B + \Delta \mathbf{X}_BXB′​=XB​+ΔXB​

   Where ΔXA\Delta \mathbf{X}_AΔXA​ and ΔXB\Delta \mathbf{X}_BΔXB​ represent the clarified and disambiguated additions to the respective DIKWP components.

Example:

In an educational AI tutor, if the AI provides vague feedback, specifying precise information and contextual examples can reduce imprecision, enhancing the student’s Understanding.

Formulate an optimization problem to maximize Understanding while minimizing Deficiency:

max⁡UAB−λ⋅DAB\max \quad \mathcal{U}_{AB} - \lambda \cdot \mathcal{D}_{AB}maxUAB​−λ⋅DAB​

Where:

• UAB\mathcal{U}_{AB}UAB​: Mutual Understanding measure

• DAB\mathcal{D}_{AB}DAB​: Overall Deficiency measure

• $\lambda$: Weighting factor balancing the trade-off

Ensure that deficiencies do not exceed acceptable thresholds:

DAB≤θ\mathcal{D}_{AB} \leq \thetaDAB​≤θ

Where:

• $\theta$: Predefined threshold for maximum allowable deficiency

Incorporate penalties for exceeding thresholds:

Penalty=η⋅max⁡(0,DAB−θ)\text{Penalty} = \eta \cdot \max(0, \mathcal{D}_{AB} - \theta)Penalty=η⋅max(0,DAB​−θ)

Where:

• $\eta$: Penalty coefficient

Integrate the penalty into the objective function:

max⁡(UAB−λ⋅DAB−Penalty)\max \left( \mathcal{U}_{AB} - \lambda \cdot \mathcal{D}_{AB} - \text{Penalty} \right)max(UAB​−λ⋅DAB​−Penalty)

This formulation ensures that the optimization prioritizes enhancing Understanding while adhering to deficiency constraints.

A medical practitioner (Human) collaborates with an AI diagnostic system (AI) to diagnose a patient's condition. Their interaction involves sharing and integrating DIKWP components to achieve an accurate diagnosis.

• Human (ProfileH\text{Profile}_HProfileH​):

• Data (DH\mathbf{D}_HDH​): Patient symptoms, medical history.

• Information (IH\mathbf{I}_HIH​): Identified potential diagnoses.

• Knowledge (KH\mathbf{K}_HKH​): Medical expertise and clinical guidelines.

• Wisdom (WH\mathbf{W}_HWH​): Ethical considerations in patient care.

• Purpose (PH\mathbf{P}_HPH​): Accurate and timely diagnosis.

• AI (ProfileA\text{Profile}_AProfileA​):

• Data (DA\mathbf{D}_ADA​): Extensive medical datasets, research articles.

• Information (IA\mathbf{I}_AIA​): Pattern recognition in patient data.

• Knowledge (KA\mathbf{K}_AKA​): Compiled medical knowledge from training data.

• Wisdom (WA\mathbf{W}_AWA​): Predefined ethical guidelines.

• Purpose (PA\mathbf{P}_APA​): Assist in diagnosis and treatment planning.

1. Data Exchange:

• Human to AI: Sends patient data (DH\mathbf{D}_HDH​) to AI.

• Gap Identification: AI identifies any missing Data components relevant for diagnosis.

2. Information Sharing:

• AI Analysis: Processes DH\mathbf{D}_HDH​ to generate Information (IA\mathbf{I}_AIA​).

• Shared Information: AI communicates patterns and potential diagnoses back to the Human.

3. Knowledge Integration:

• Human Review: Evaluates AI's Information (IA\mathbf{I}_AIA​) against personal Knowledge (KH\mathbf{K}_HKH​).

• Knowledge Update: Human incorporates relevant Knowledge from AI into KH\mathbf{K}_HKH​.

4. Wisdom Application:

• Ethical Judgment: Both stakeholders apply Wisdom (WH\mathbf{W}_HWH​ and WA\mathbf{W}_AWA​) to ensure ethical considerations in diagnosis.

5. Purpose Alignment:

• Goal Confirmation: Both entities confirm their Purpose (PH\mathbf{P}_HPH​ and PA\mathbf{P}_APA​) to ensure alignment towards accurate diagnosis.

• Mathematical Optimization: Adjust DIKWP components to minimize DHA\mathcal{D}_{HA}DHA​ and maximize UHA\mathcal{U}_{HA}UHA​.

max⁡UHA−λ⋅DHA\max \quad \mathcal{U}_{HA} - \lambda \cdot \mathcal{D}_{HA}maxUHA​−λ⋅DHA​

• Result: Enhanced Understanding leads to a more accurate and reliable diagnosis, reducing the likelihood of AI-generated hallucinations (erroneous diagnoses).

The 3-No Problems—Incomplete, Inconsistent, and Imprecise Input/Output—are critical challenges within the DIKWP*DIKWP interaction framework that impede effective communication and mutual Understanding between stakeholders. By establishing precise mathematical semantics for these problems, stakeholders can systematically identify and remediate deficiencies, thereby enhancing the reliability and coherence of their interactions. This structured approach not only mitigates the risk of hallucinations in AI systems but also fosters more effective human-AI collaboration, ensuring that interactions are aligned with shared goals and cognitive frameworks.

Future research should focus on refining these mathematical models, integrating advanced optimization techniques, and conducting empirical studies to validate the framework in diverse real-world scenarios. Such advancements will significantly contribute to the fields of cognitive science and artificial intelligence, promoting more reliable and harmonious human-AI interactions.

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The author extends gratitude to Prof. Yucong Duan for his pioneering work on the DIKWP model and the Theory of Relativity of Consciousness, which have significantly influenced the conceptual framework of this analysis. Appreciation is also given to colleagues in cognitive science and artificial intelligence for their invaluable feedback and insights.

Correspondence and requests for materials should be addressed to [Author's Name and Contact Information].

Keywords: DIKWP Model, 3-No Problems, Relativity of Consciousness, Human-AI Interaction, Cognitive Enclosure, Mathematical Framework, Data-Information-Knowledge-Wisdom-Purpose, Hallucination Mitigation, Understanding Enhancement