Empirical Mechanisms of the DIKWP Semantic Mathematics 

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

This document provides a detailed examination of the concrete mechanisms underlying the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework proposed by Prof. Yucong Duan. The DIKWP framework represents a revolutionary approach to mathematics, integrating semantics, purpose, dynamism, and human consciousness into mathematical practice. This exploration delves into the specific mathematical structures, processes, and methodologies that constitute the framework, illustrating how each component operates and interacts with the others. By detailing these mechanisms, the document aims to provide a clear understanding of how DIKWP Semantic Mathematics functions in practice and how it can be applied to advance mathematical modeling, artificial intelligence (AI), and human cognition.

1. Introduction

• 1.1. Overview of DIKWP Semantic Mathematics

• 1.2. Objectives of the Document

2. Foundational Concepts and Notations

• 2.1. Mathematical Representation of Semantics

• 2.2. Notational Conventions

3. Mechanisms of the DIKWP Components

• 3.5.1. Purpose-Driven Mathematical Processes

• 3.5.2. Goal Alignment and Optimization

• 3.4.1. From Knowledge to Wisdom

• 3.4.2. Decision-Making Mechanisms

• 3.3.1. Integrating Information into Knowledge

• 3.3.2. Formalization of Completeness

• 3.2.1. Deriving Information from Data

• 3.2.2. Quantifying Difference

• 3.1.1. Mathematical Modeling of Data

• 3.1.2. Representation of Sameness

• 3.1. Data (Sameness)

• 3.2. Information (Difference)

• 3.3. Knowledge (Completeness)

• 3.4. Wisdom

• 3.5. Purpose

4. Semantic Integration in Mathematical Operations

• 4.1. Formal Bundling of Concepts with Semantics

• 4.2. Semantic Abstraction and Reification

5. Dynamic Evolution of Mathematical Models

• 5.1. Evolutionary Algorithms and Processes

• 5.2. Handling Inconsistencies: The "BUG" Theory

6. Human Cognition Modeling

• 6.1. Cognitive Structures in Mathematics

• 6.2. Subjectivity and First-Person Perspective

7. Application in Artificial Intelligence

• 7.1. Constructing Artificial Consciousness

• 7.2. Semantic AI Systems

8. Examples and Case Studies

• 8.1. Example: Semantic Data Processing

• 8.2. Example: Purpose-Driven Optimization

9. Conclusion

• 9.1. Summary of Mechanisms

• 9.2. Implications and Future Work

10. References

The DIKWP Semantic Mathematics framework is an innovative approach that redefines mathematical practice by integrating semantics, human cognition, dynamism, and purpose into mathematical constructs. It extends the traditional Data-Information-Knowledge-Wisdom (DIKW) hierarchy by adding Purpose as a critical component, forming a comprehensive system that mirrors human cognitive development and decision-making processes.

The primary objective is to detail the concrete mechanisms of the DIKWP Semantic Mathematics framework, explaining how each component operates mathematically and how they interconnect to form a cohesive system. This includes:

• Defining mathematical representations for each component.

• Describing processes and operations that transform data through the DIKWP hierarchy.

• Illustrating how semantics are integrated into mathematical models.

• Demonstrating the application of these mechanisms in AI and problem-solving.

In DIKWP Semantic Mathematics, semantics are explicitly represented within mathematical constructs. This involves associating mathematical entities with semantic metadata or meanings, ensuring that operations on these entities preserve and reflect their underlying semantics.

• Semantic Tuple: A basic unit combining a mathematical value with its semantic context, represented as $(v,s)$, where $v$ is the value, and $s$ is the semantic annotation.

• Sets and Elements: Uppercase letters denote sets (e.g., $D$ for Data), and lowercase letters denote elements (e.g., $d\in D$).

• Functions and Mappings: Functions are represented as $f:X\to Y$, mapping elements from set $X$ to set $Y$.

• Operators: Mathematical operators are defined with consideration of semantic contexts.

Data represents raw observations or facts, characterized by their inherent properties.

• Data Set $D$: A collection of data points did_idi​, each associated with semantic annotations sis_isi​.

  D={(di,si)∣i∈I}D = \{ (d_i, s_i) \mid i \in I \}D={(di​,si​)∣i∈I}

• Sameness Relation RsR_sRs​: A relation defining the sameness between data points based on shared attributes.

  Rs={((di,si),(dj,sj))∣si=sj}R_s = \{ ((d_i, s_i), (d_j, s_j)) \mid s_i = s_j \}Rs​={((di​,si​),(dj​,sj​))∣si​=sj​}

Sameness is identified through equivalence classes formed by the sameness relation.

• Equivalence Classes:

  [di]={dj∈D∣si=sj}[d_i] = \{ d_j \in D \mid s_i = s_j \}[di​]={dj​∈D∣si​=sj​}

• Mathematical Operations: Functions can be defined over these classes, respecting the semantics.

Information is obtained by identifying differences and patterns within data.

• Difference Function $\delta$:

  $$\delta :D\times D\to I$$

  where $I$ is the set of information units.

• Information Unit IkI_kIk​:

  Ik=δ((di,si),(dj,sj))I_k = \delta((d_i, s_i), (d_j, s_j))Ik​=δ((di​,si​),(dj​,sj​))

Differences can be quantified using distance metrics or similarity measures that consider semantics.

• Semantic Distance Δs\Delta_sΔs​:

  Δs((di,si),(dj,sj))=fsem(si,sj)\Delta_s((d_i, s_i), (d_j, s_j)) = f_{\text{sem}}(s_i, s_j)Δs​((di​,si​),(dj​,sj​))=fsem​(si​,sj​)

• Information Content: The amount of information is proportional to the semantic distance.

Knowledge is formed by integrating information units into a coherent structure.

• Knowledge Structure $K$:

  K=⋃kIkK = \bigcup_{k} I_kK=k⋃​Ik​

• Integration Function $\varphi$:

  $$\varphi :\mathcal{P}(I)\to K$$

  where $\mathcal{P}(I)$ is the power set of information units.

Completeness refers to the extent to which knowledge encompasses all relevant information.

• Completeness Metric $C(K)$:

  C(K)=∣K∣∣Itotal∣C(K) = \frac{|K|}{|I_{\text{total}}|}C(K)=∣Itotal​∣∣K∣​

  where ∣Itotal∣|I_{\text{total}}|∣Itotal​∣ is the total number of relevant information units.

Wisdom involves applying knowledge with judgment, considering context and ethical implications.

• Wisdom Function $\Psi$:

  $$\Psi :K\times \Theta \to W$$

  where $\Theta$ represents contextual and ethical parameters, and $W$ is the set of wisdom units.

• Decision Function $\Delta$:

  $$\Delta :W\to A$$

  where $A$ is the set of actions or decisions.

• Optimization Criteria: Decisions are optimized based on purpose and ethical considerations.

Purpose guides the selection and application of mathematical processes.

• Purpose Function $P$:

  $$P:\{\text{Processes}\}\to \{\text{Outcomes}\}$$

• Goal Alignment: Mathematical models are adjusted to align with specified purposes.

• Objective Function $O$:

  $$O:A\to \mathbb{R}$$

  where $O$ evaluates actions based on their alignment with the purpose.

• Optimization Problem:

  $$\text{Maximize or Minimize }O(A)\text{ subject to constraints}$$

Mathematical entities are bundled with their semantic annotations, ensuring operations respect meanings.

• Semantic Operation $\otimes$:

  For two semantic tuples (v1,s1)(v_1, s_1)(v1​,s1​) and (v2,s2)(v_2, s_2)(v2​,s2​):

  (v1,s1)⊗(v2,s2)=(vresult,scombined)(v_1, s_1) \otimes (v_2, s_2) = (v_{\text{result}}, s_{\text{combined}})(v1​,s1​)⊗(v2​,s2​)=(vresult​,scombined​)

  where scombineds_{\text{combined}}scombined​ reflects the combined semantics.

• Semantic Abstraction: Extracting general principles from specific semantic contexts.

  Abstract(S)=Sgeneral\text{Abstract}(S) = S_{\text{general}}Abstract(S)=Sgeneral​

• Reification: Applying abstract concepts back to specific semantics.

  Reify(Sgeneral,scontext)=Sspecific\text{Reify}(S_{\text{general}}, s_{\text{context}}) = S_{\text{specific}}Reify(Sgeneral​,scontext​)=Sspecific​

Mathematical models evolve over time, adapting to new data and contexts.

• Evolutionary Function $E$:

  E:Mt×Dnew→Mt+1E: M_t \times D_{\text{new}} \rightarrow M_{t+1}E:Mt​×Dnew​→Mt+1​

  where MtM_tMt​ is the model at time $t$, and DnewD_{\text{new}}Dnew​ is new data.

• Adaptation Mechanism: Models are updated to incorporate new information and correct inconsistencies.

Inconsistencies ("bugs") are identified and resolved to improve models.

• Bug Detection Function $B$:

  $$B:M\to \{\text{Bugs}\}$$

• Correction Function $C$:

  C:M×{Bugs}→MupdatedC: M \times \{ \text{Bugs} \} \rightarrow M_{\text{updated}}C:M×{Bugs}→Mupdated​

• Iterative Improvement: Models evolve through cycles of bug detection and correction.

Mathematical models incorporate structures that mirror human cognitive processes.

• Cognitive Functions $\kappa$:

  κ:Sinput→Soutput\kappa: S_{\text{input}} \rightarrow S_{\text{output}}κ:Sinput​→Soutput​

  where $S$ represents semantic structures.

• Memory Representation: Models include mechanisms for storing and retrieving information akin to human memory.

• Subjective Parameters: Models include parameters representing individual perspectives.

  $$\text{Subjective State }\sigma =\{\text{beliefs},\text{preferences},\text{intentions}\}$$

• First-Person Modeling: Mathematical constructs account for subjective experiences and interpretations.

• Consciousness Model $C$:

  $$C:(D,I,K,W,P)\to \text{Conscious State}$$

• Self-Awareness Mechanisms: AI systems include models that reflect on their own states and processes.

• Semantic Processing: AI systems process information with semantic understanding.

  $$\text{Semantic Function }\Sigma :\text{Input}\to \text{Meaningful Output}$$

• Contextual Adaptation: AI adjusts its operations based on the semantic context.

• Scenario: Processing sensor data in an autonomous vehicle.

• Data: Raw sensor readings with semantic annotations (e.g., (di,"distance to obstacle")(d_i, \text{"distance to obstacle"})(di​,"distance to obstacle")).

• Information Extraction:

  Ik=δ((di,"distance"),(dj,"velocity"))I_k = \delta((d_i, \text{"distance"}), (d_j, \text{"velocity"}))Ik​=δ((di​,"distance"),(dj​,"velocity"))

• Knowledge Formation:

  K=ϕ({Iobstacle detection,Ispeed analysis})K = \phi(\{ I_{\text{obstacle detection}}, I_{\text{speed analysis}} \})K=ϕ({Iobstacle detection​,Ispeed analysis​})

• Wisdom Application:

  W=Ψ(K,Θtraffic laws)W = \Psi(K, \Theta_{\text{traffic laws}})W=Ψ(K,Θtraffic laws​)

• Purposeful Action:

  $$A=\Delta (W)\text{ to safely navigate}$$

• Problem: Optimizing energy consumption in a smart grid.

• Purpose: Minimize energy usage while maintaining service quality.

• Objective Function:

  $$O(A)=\alpha \times \text{Energy Savings}-\beta \times \text{Service Degradation}$$

• Optimization:

  $$\text{Minimize }O(A)\text{ subject to }\text{regulatory constraints}$$

The DIKWP Semantic Mathematics framework operates by:

• Integrating Semantics: Associating mathematical entities with semantic meanings.

• Evolving Dynamically: Updating models through evolutionary processes and handling inconsistencies via the "BUG" theory.

• Aligning with Purpose: Guiding mathematical processes and decisions based on specified purposes.

• Modeling Cognition: Reflecting human cognitive structures and subjectivity in mathematical constructs.

The concrete mechanisms of DIKWP Semantic Mathematics provide a foundation for:

• Advanced AI Systems: Developing AI with semantic understanding and consciousness-like properties.

• Enhanced Problem-Solving: Applying mathematics to complex, real-world problems with greater effectiveness.

• Interdisciplinary Research: Fostering collaboration across fields to explore new applications and methodologies.

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".

3. Russell, S., & Norvig, P. (2021). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

4. Floridi, L. (2011). The Philosophy of Information. Oxford University Press.

5. Hofstadter, D. R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.

6. Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.

Keywords: DIKWP Semantic Mathematics, Concrete Mechanisms, Prof. Yucong Duan, Semantics Integration, Mathematical Modeling, Data-Information-Knowledge-Wisdom-Purpose, Artificial Intelligence, Human Cognition, Dynamic Mathematics, Purposeful Mathematics.

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