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

The phenomenon of hallucination in both human cognition and artificial intelligence (AI) systems presents significant challenges in understanding and mitigating erroneous perceptions and outputs. Building upon Prof. Yucong Duan's Data-Information-Knowledge-Wisdom-Purpose (DIKWP) model and his Theory of Relativity of Consciousness, this paper introduces the Theory of Relativity of Hallucination. This theory employs a mathematical framework to model interactions between two stakeholders using their respective DIKWP profiles. By formalizing DIKWP*DIKWP interactions, we aim to quantify and address issues related to connectivity, inconsistency, uncertainty, and complexity within these interactions, thereby reducing the incidence of hallucinations. The proposed framework offers a systematic approach to enhancing mutual understanding and alignment in human-AI collaborations.

Hallucinations, defined as perceptions or outputs not grounded in external reality, are prevalent in both human cognition and AI systems. In humans, hallucinations are often linked to neurological or psychological conditions, while in AI, particularly in Large Language Models (LLMs) like GPT-4, hallucinations manifest as factually incorrect or nonsensical outputs. Understanding and mitigating hallucinations are critical for improving the reliability and effectiveness of AI systems and enhancing human cognitive health.

Prof. Yucong Duan's DIKWP model and his Theory of Relativity of Consciousness provide a robust framework for analyzing cognitive processes. Extending these concepts, the Theory of Relativity of Hallucination introduces a mathematical approach to model and measure interactions between two cognitive entities (stakeholders) using their DIKWP profiles. This paper delineates this theory, focusing on mathematical formulations to remediate connectivity issues, eliminate inconsistencies and uncertainties, and reduce interaction complexities within DIKWP*DIKWP compositions.

Each cognitive entity is characterized by a DIKWP profile, encapsulating its Data, Information, Knowledge, Wisdom, and Purpose. Mathematically, we represent each component as vectors within a multidimensional space:

Entityi={Di,Ii,Ki,Wi,Pi}\text{Entity}_i = \{\mathbf{D}_i, \mathbf{I}_i, \mathbf{K}_i, \mathbf{W}_i, \mathbf{P}_i\}Entityi​={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 interaction between two entities, EntityA\text{Entity}_AEntityA​ and EntityB\text{Entity}_BEntityB​, is modeled as a composition of their DIKWP profiles:

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 of the interaction is a function of the corresponding components of the individual entities:

DAB=fD(DA,DB)\mathbf{D}_{AB} = f_D(\mathbf{D}_A, \mathbf{D}_B)DAB​=fD​(DA​,DB​)IAB=fI(IA,IB)\mathbf{I}_{AB} = f_I(\mathbf{I}_A, \mathbf{I}_B)IAB​=fI​(IA​,IB​)KAB=fK(KA,KB)\mathbf{K}_{AB} = f_K(\mathbf{K}_A, \mathbf{K}_B)KAB​=fK​(KA​,KB​)WAB=fW(WA,WB)\mathbf{W}_{AB} = f_W(\mathbf{W}_A, \mathbf{W}_B)WAB​=fW​(WA​,WB​)PAB=fP(PA,PB)\mathbf{P}_{AB} = f_P(\mathbf{P}_A, \mathbf{P}_B)PAB​=fP​(PA​,PB​)

Functions fD,fI,fK,fW,fPf_D, f_I, f_K, f_W, f_PfD​,fI​,fK​,fW​,fP​ define how each DIKWP component interacts and transforms during the interaction process.

Understanding arises from the coherent integration of DIKWP components during interactions. We define Understanding ($\mathcal{U}$) between two entities as a scalar measure derived from their DIKWP*DIKWP interaction:

UAB=g(DAB,IAB,KAB,WAB,PAB)\mathcal{U}_{AB} = g(\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB})UAB​=g(DAB​,IAB​,KAB​,WAB​,PAB​)

Where $g$ is a function that aggregates the interaction components to quantify understanding.

A possible formulation for $g$ could involve weighted sums and similarity measures:

UAB=α⋅sim(DAB,Dshared)+β⋅sim(IAB,Ishared)+γ⋅sim(KAB,Kshared)+δ⋅sim(WAB,Wshared)+ϵ⋅sim(PAB,Pshared)\mathcal{U}_{AB} = \alpha \cdot \text{sim}(\mathbf{D}_{AB}, \mathbf{D}_{shared}) + \beta \cdot \text{sim}(\mathbf{I}_{AB}, \mathbf{I}_{shared}) + \gamma \cdot \text{sim}(\mathbf{K}_{AB}, \mathbf{K}_{shared}) + \delta \cdot \text{sim}(\mathbf{W}_{AB}, \mathbf{W}_{shared}) + \epsilon \cdot \text{sim}(\mathbf{P}_{AB}, \mathbf{P}_{shared})UAB​=α⋅sim(DAB​,Dshared​)+β⋅sim(IAB​,Ishared​)+γ⋅sim(KAB​,Kshared​)+δ⋅sim(WAB​,Wshared​)+ϵ⋅sim(PAB​,Pshared​)

Where:

• $\alpha ,\beta ,\gamma ,\delta ,ϵ\ge 0$: Weights assigned to each DIKWP component based on their importance.

• Xshared\mathbf{X}_{shared}Xshared​: Shared or ideal vectors representing consensus in each DIKWP component.

• $\text{sim}(\cdot ,\cdot )$: Similarity function (e.g., cosine similarity) measuring alignment between interaction components and shared components.

To enhance Understanding and mitigate hallucinations, we must remediate connectivity issues, eliminate inconsistencies and uncertainties, and reduce interaction complexities.

Connectivity ($\mathcal{C}$) refers to the strength and coherence of the paths connecting DIKWP components during interactions.

CAB=h(DAB,IAB,KAB,WAB,PAB)\mathcal{C}_{AB} = h(\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB})CAB​=h(DAB​,IAB​,KAB​,WAB​,PAB​)

Where $h$ measures the integrity and strength of connections. For instance:

CAB=∑x∈{D,I,K,W,P}λx⋅conn(XAB)\mathcal{C}_{AB} = \sum_{x \in \{D, I, K, W, P\}} \lambda_x \cdot \text{conn}(\mathbf{X}_{AB})CAB​=x∈{D,I,K,W,P}∑​λx​⋅conn(XAB​)

• λx≥0\lambda_x \geq 0λx​≥0: Weights for each DIKWP component.

• $\text{conn}(\cdot )$: Connectivity function assessing the coherence within each DIKWP component.

Objective: Maximize CAB\mathcal{C}_{AB}CAB​ to ensure robust connectivity, enhancing the pathways that facilitate Understanding.

Inconsistency ($\mathcal{I}$) and Uncertainty ($\mathcal{U}$) disrupt the transformation paths within DIKWP*DIKWP interactions.

IAB=k(DAB,IAB,KAB,WAB,PAB)\mathcal{I}_{AB} = k(\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB})IAB​=k(DAB​,IAB​,KAB​,WAB​,PAB​)UAB=l(DAB,IAB,KAB,WAB,PAB)\mathcal{U}_{AB} = l(\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB})UAB​=l(DAB​,IAB​,KAB​,WAB​,PAB​)

Where:

• $k$ and $l$ are functions quantifying inconsistency and uncertainty, respectively. These could involve variance measures or entropy calculations.

Objective: Minimize IAB\mathcal{I}_{AB}IAB​ and UAB\mathcal{U}_{AB}UAB​ to ensure consistency and clarity in interactions, thereby reducing the likelihood of hallucinations.

Complexity ($\mathcal{X}$) within DIKWP*DIKWP interactions can obscure Understanding and foster hallucinations.

XAB=m(DAB,IAB,KAB,WAB,PAB)\mathcal{X}_{AB} = m(\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB})XAB​=m(DAB​,IAB​,KAB​,WAB​,PAB​)

Where $m$ quantifies the complexity of the interaction, possibly through measures like computational complexity or network density.

Objective: Minimize XAB\mathcal{X}_{AB}XAB​ by simplifying interaction pathways, reducing redundancies, and streamlining communication processes.

To achieve optimal Understanding (UAB\mathcal{U}_{AB}UAB​), we formulate an optimization problem balancing connectivity, consistency, and complexity.

max⁡DAB,IAB,KAB,WAB,PABUAB\max_{\mathbf{D}_{AB}, \mathbf{I}_{AB}, \mathbf{K}_{AB}, \mathbf{W}_{AB}, \mathbf{P}_{AB}} \quad \mathcal{U}_{AB}DAB​,IAB​,KAB​,WAB​,PAB​max​UAB​

Subject to:

CAB≥θ1\mathcal{C}_{AB} \geq \theta_1CAB​≥θ1​IAB≤θ2\mathcal{I}_{AB} \leq \theta_2IAB​≤θ2​XAB≤θ3\mathcal{X}_{AB} \leq \theta_3XAB​≤θ3​

Where θ1,θ2,θ3\theta_1, \theta_2, \theta_3θ1​,θ2​,θ3​ are predefined thresholds ensuring adequate connectivity, minimal inconsistency, and manageable complexity.

To incorporate the 3-No Problem into the optimization, we introduce penalty functions:

Penalty=η1⋅max⁡(0,θ1−CAB)+η2⋅max⁡(0,IAB−θ2)+η3⋅max⁡(0,XAB−θ3)\text{Penalty} = \eta_1 \cdot \max(0, \theta_1 - \mathcal{C}_{AB}) + \eta_2 \cdot \max(0, \mathcal{I}_{AB} - \theta_2) + \eta_3 \cdot \max(0, \mathcal{X}_{AB} - \theta_3)Penalty=η1​⋅max(0,θ1​−CAB​)+η2​⋅max(0,IAB​−θ2​)+η3​⋅max(0,XAB​−θ3​)

Where η1,η2,η3≥0\eta_1, \eta_2, \eta_3 \geq 0η1​,η2​,η3​≥0 are penalty coefficients.

The modified objective becomes:

max⁡(UAB−Penalty)\max \left( \mathcal{U}_{AB} - \text{Penalty} \right)max(UAB​−Penalty)

Employing optimization techniques such as Lagrangian multipliers or gradient-based methods can solve this constrained optimization problem, adjusting the DIKWP*DIKWP interaction components to maximize Understanding while adhering to the constraints imposed by connectivity, inconsistency, and complexity.

Understanding evolves over time as interactions between entities continue. We model this as a dynamic system where DIKWP profiles are functions of time:

Entityi(t)={Di(t),Ii(t),Ki(t),Wi(t),Pi(t)}\text{Entity}_i(t) = \{\mathbf{D}_i(t), \mathbf{I}_i(t), \mathbf{K}_i(t), \mathbf{W}_i(t), \mathbf{P}_i(t)\}Entityi​(t)={Di​(t),Ii​(t),Ki​(t),Wi​(t),Pi​(t)}

The interaction at time $t$ is:

Interaction(t)=EntityA(t)×EntityB(t)\text{Interaction}(t) = \text{Entity}_A(t) \times \text{Entity}_B(t)Interaction(t)=EntityA​(t)×EntityB​(t)

We define the rate of change of each DIKWP component based on interactions:

dDAB(t)dt=fD(DA(t),DB(t))\frac{d\mathbf{D}_{AB}(t)}{dt} = f_D(\mathbf{D}_A(t), \mathbf{D}_B(t))dtdDAB​(t)​=fD​(DA​(t),DB​(t))dIAB(t)dt=fI(IA(t),IB(t))\frac{d\mathbf{I}_{AB}(t)}{dt} = f_I(\mathbf{I}_A(t), \mathbf{I}_B(t))dtdIAB​(t)​=fI​(IA​(t),IB​(t))dKAB(t)dt=fK(KA(t),KB(t))\frac{d\mathbf{K}_{AB}(t)}{dt} = f_K(\mathbf{K}_A(t), \mathbf{K}_B(t))dtdKAB​(t)​=fK​(KA​(t),KB​(t))dWAB(t)dt=fW(WA(t),WB(t))\frac{d\mathbf{W}_{AB}(t)}{dt} = f_W(\mathbf{W}_A(t), \mathbf{W}_B(t))dtdWAB​(t)​=fW​(WA​(t),WB​(t))dPAB(t)dt=fP(PA(t),PB(t))\frac{d\mathbf{P}_{AB}(t)}{dt} = f_P(\mathbf{P}_A(t), \mathbf{P}_B(t))dtdPAB​(t)​=fP​(PA​(t),PB​(t))

These equations model the temporal evolution of the interaction components, capturing how Understanding is dynamically shaped by ongoing interactions.

Consider a human user (EntityH\text{Entity}_HEntityH​) interacting with an AI system (EntityA\text{Entity}_AEntityA​). Their DIKWP profiles interact to facilitate communication and Understanding.

1. Assess Connectivity (CHA\mathcal{C}_{HA}CHA​):

• Evaluate the strength of connections across DIKWP components.

• Identify weak or missing connections that may hinder Understanding.

2. Identify Inconsistencies and Uncertainties (IHA,UHA\mathcal{I}_{HA}, \mathcal{U}_{HA}IHA​,UHA​):

• Detect conflicting information or knowledge gaps.

• Measure uncertainties in the interaction trajectories.

3. Simplify Interaction Paths (XHA\mathcal{X}_{HA}XHA​):

• Reduce complexity by streamlining communication protocols.

• Eliminate redundant or convoluted pathways in DIKWP transformations.

4. Optimize Understanding (UHA\mathcal{U}_{HA}UHA​):

• Adjust DIKWP components to enhance similarity with shared or ideal DIKWP vectors.

• Implement feedback mechanisms to iteratively refine Understanding.

Utilize the optimization framework to adjust the AI's DIKWP components, ensuring that outputs align closely with human expectations, thereby minimizing hallucinations.

• Human (EntityH\text{Entity}_HEntityH​):

• 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 guidelines.

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

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

• AI (EntityA\text{Entity}_AEntityA​):

• Data (DA\mathbf{D}_ADA​): Training data from medical literature.

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

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

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

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

InteractionHA=EntityH×EntityA={DHA,IHA,KHA,WHA,PHA}\text{Interaction}_{HA} = \text{Entity}_H \times \text{Entity}_A = \{\mathbf{D}_{HA}, \mathbf{I}_{HA}, \mathbf{K}_{HA}, \mathbf{W}_{HA}, \mathbf{P}_{HA}\}InteractionHA​=EntityH​×EntityA​={DHA​,IHA​,KHA​,WHA​,PHA​}

Suppose the AI generates a diagnosis not supported by the patient's data:

Hallucination⇒Incorrect Knowledge Integration (KHA)\text{Hallucination} \Rightarrow \text{Incorrect Knowledge Integration } (\mathbf{K}_{HA})Hallucination⇒Incorrect Knowledge Integration (KHA​)

1. Evaluate Similarity:

sim(KHA,Kshared)=cos⁡(KHA,Kshared)\text{sim}(\mathbf{K}_{HA}, \mathbf{K}_{shared}) = \cos(\mathbf{K}_{HA}, \mathbf{K}_{shared})sim(KHA​,Kshared​)=cos(KHA​,Kshared​)

1. Apply Optimization:

Maximize:

UHA=α⋅sim(KHA,Kshared)\mathcal{U}_{HA} = \alpha \cdot \text{sim}(\mathbf{K}_{HA}, \mathbf{K}_{shared})UHA​=α⋅sim(KHA​,Kshared​)

Subject to:

CHA≥θ1\mathcal{C}_{HA} \geq \theta_1CHA​≥θ1​IHA≤θ2\mathcal{I}_{HA} \leq \theta_2IHA​≤θ2​XHA≤θ3\mathcal{X}_{HA} \leq \theta_3XHA​≤θ3​

1. Adjust AI's Knowledge Vector:

KA′=KA+ΔKA\mathbf{K}_A' = \mathbf{K}_A + \Delta \mathbf{K}_AKA′​=KA​+ΔKA​

Where ΔKA\Delta \mathbf{K}_AΔKA​ is determined through gradient ascent to maximize UHA\mathcal{U}_{HA}UHA​ while satisfying constraints.

1. Result:

Enhanced similarity between AI's Knowledge and shared medical guidelines reduces the likelihood of incorrect diagnoses, mitigating hallucinations.

The current framework models interactions between two stakeholders. Future work should extend this to multi-agent systems, considering interactions among multiple entities and the emergent properties therein.

Integrating dynamic learning algorithms can allow the DIKWP profiles to evolve over time, adapting to feedback and enhancing Understanding iteratively.

Conduct empirical studies to validate the mathematical formulations and optimization strategies, assessing their effectiveness in real-world human-AI interactions.

Explore the ethical implications of manipulating DIKWP interactions to enhance Understanding, ensuring that such interventions respect autonomy and promote fairness.

The Theory of Relativity of Hallucination provides a mathematical framework to model and mitigate hallucinations in DIKWP*DIKWP interactions between cognitive entities. By formalizing the interactions and addressing connectivity, consistency, and complexity, this theory offers a systematic approach to enhancing Understanding and reducing erroneous outputs in both human cognition and AI systems. Future advancements in this framework hold the potential to significantly improve human-AI collaboration, fostering more reliable and coherent 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: Relativity of Hallucination, DIKWP Model, Theory of Relativity of Consciousness, Human-AI Interaction, Cognitive Enclosure, Mathematical Framework, Artificial Intelligence, Cognitive Science