Mutual Expressing Capabilities and Limits Among DIKWP Components in a Mathematical Manner 

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)

Introduction

The DIKWP framework—comprising Data (D), Information (I), Knowledge (K), Wisdom (W), and Purpose (P)—provides a structured approach to understanding how raw data transforms into purposeful actions through cognitive processes. To fully grasp the interplay among these components, it's essential to investigate their mutual expressing capabilities and limits mathematically, while explicitly distinguishing between DIKWP Concepts and DIKWP Semantics.

1. Definitions and Distinctions

1.1 DIKWP Concepts

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

• Information (I): Processed data with identified patterns and relationships.

• Knowledge (K): Organized information structured into frameworks or models.

• Wisdom (W): Application of knowledge with ethical considerations to make judicious decisions.

• Purpose (P): Goals or objectives guiding actions and decisions.

1.2 DIKWP Semantics

• Data Semantics: The meaning assigned to data points, often based on context.

• Information Semantics: Interpretations and implications derived from information.

• Knowledge Semantics: Deeper understanding and insights gained from knowledge structures.

• Wisdom Semantics: Ethical and moral meanings influencing decisions.

• Purpose Semantics: The significance and value of goals guiding the system.

Distinction: While DIKWP Concepts refer to the structural elements and definitions within the framework, DIKWP Semantics pertain to the meanings, interpretations, and contextual associations attached to these concepts.

2. Mathematical Representations of DIKWP Components

2.1 Data (D)

• Conceptual Representation: D={d1,d2,...,dn}D = \{ d_1, d_2, ..., d_n \}D={d1​,d2​,...,dn​}

• Semantic Representation: Each data point did_idi​ has a meaning s(di)s(d_i)s(di​) assigned by a semantic function.

2.2 Information (I)

• Conceptual Representation: I=FD(D)I = F_D(D)I=FD​(D), where FDF_DFD​ is a function that processes data.

• Semantic Representation: s(I)={s(ij)∣ij∈I}s(I) = \{ s(i_j) \mid i_j \in I \}s(I)={s(ij​)∣ij​∈I}, meanings derived from patterns in data.

2.3 Knowledge (K)

• Conceptual Representation: $K=(N,E)$, a graph with nodes $N$ (concepts) and edges $E$ (relationships).

• Semantic Representation: $s(K)$ captures the insights and understandings from $K$.

2.4 Wisdom (W)

• Conceptual Representation: W=FK(K)W = F_K(K)W=FK​(K), applying knowledge to make decisions.

• Semantic Representation: $s(W)$ includes ethical and moral considerations.

2.5 Purpose (P)

• Conceptual Representation: P={p1,p2,...,pm}P = \{ p_1, p_2, ..., p_m \}P={p1​,p2​,...,pm​}, a set of goals.

• Semantic Representation: s(P)={s(pk)∣pk∈P}s(P) = \{ s(p_k) \mid p_k \in P \}s(P)={s(pk​)∣pk​∈P}, the significance of each goal.

3. Mutual Expressing Capabilities Among DIKWP Components

We will model how each component can express and be expressed by the others, using mathematical functions, while distinguishing between concepts and semantics.

3.1 From Data to Information

• Conceptual Transformation: I=FD(D)I = F_D(D)I=FD​(D)

• Semantic Transformation: s(I)=FsD(s(D))s(I) = F_{sD}(s(D))s(I)=FsD​(s(D)), where s(D)={s(di)}s(D) = \{ s(d_i) \}s(D)={s(di​)}

Explanation:

• DIKWP Concepts: Data is processed into information through function FDF_DFD​.

• DIKWP Semantics: Meanings of data are transformed into meanings of information via function FsDF_{sD}FsD​.

Limitations:

• Information Loss: Not all data semantics may be preserved in the information semantics.

• Ambiguity: Multiple data sets can produce similar information, leading to potential ambiguity.

3.2 From Information to Knowledge

• Conceptual Transformation: K=FI(I)K = F_I(I)K=FI​(I)

• Semantic Transformation: s(K)=FsI(s(I))s(K) = F_{sI}(s(I))s(K)=FsI​(s(I))

Explanation:

• DIKWP Concepts: Information is organized into knowledge structures via function FIF_IFI​.

• DIKWP Semantics: Meanings from information are integrated into knowledge semantics using FsIF_{sI}FsI​.

Limitations:

• Complexity: The transformation may not capture all nuances of information semantics.

• Overgeneralization: Simplification in knowledge structures may lead to loss of specific meanings.

3.3 From Knowledge to Wisdom

• Conceptual Transformation: W=FK(K)W = F_K(K)W=FK​(K)

• Semantic Transformation: s(W)=FsK(s(K))s(W) = F_{sK}(s(K))s(W)=FsK​(s(K))

Explanation:

• DIKWP Concepts: Knowledge is applied to make wise decisions through FKF_KFK​.

• DIKWP Semantics: Ethical and moral meanings are derived from knowledge semantics via FsKF_{sK}FsK​.

Limitations:

• Subjectivity: Wisdom semantics are influenced by values, which can vary.

• Incompleteness: Not all knowledge may contribute to wisdom; some aspects might be overlooked.

3.4 From Wisdom to Purpose

• Conceptual Transformation: P=FW(W)P = F_W(W)P=FW​(W)

• Semantic Transformation: s(P)=FsW(s(W))s(P) = F_{sW}(s(W))s(P)=FsW​(s(W))

Explanation:

• DIKWP Concepts: Wisdom informs the setting of goals through FWF_WFW​.

• DIKWP Semantics: The significance of wisdom guides the purpose semantics via FsWF_{sW}FsW​.

Limitations:

• Alignment Issues: The purpose derived may not fully encompass all wisdom semantics.

• Dynamic Changes: As wisdom evolves, purpose may need to be reassessed.

3.5 Feedback Loop: From Purpose to Data

• Conceptual Transformation: D′=FP(P)D' = F_P(P)D′=FP​(P)

• Semantic Transformation: s(D′)=FsP(s(P))s(D') = F_{sP}(s(P))s(D′)=FsP​(s(P))

Explanation:

• DIKWP Concepts: Purpose influences the collection or generation of new data via FPF_PFP​.

• DIKWP Semantics: The meanings of purpose guide what new data is considered relevant.

Limitations:

• Bias: Purpose may bias data collection, leading to incomplete or skewed data.

• Relevance: Not all data collected may be pertinent to the purpose semantics.

4. Mathematical Modeling of Mutual Expressions and Limits

We will formalize the mutual expressing capabilities among DIKWP components, highlighting the explicit distinction between concepts and semantics.

4.1 Mathematical Functions and Their Domains

• FD:D→IF_D: D \rightarrow IFD​:D→I

• FI:I→KF_I: I \rightarrow KFI​:I→K

• FK:K→WF_K: K \rightarrow WFK​:K→W

• FW:W→PF_W: W \rightarrow PFW​:W→P

• FP:P→D′F_P: P \rightarrow D'FP​:P→D′

Semantic Functions:

• FsD:s(D)→s(I)F_{sD}: s(D) \rightarrow s(I)FsD​:s(D)→s(I)

• FsI:s(I)→s(K)F_{sI}: s(I) \rightarrow s(K)FsI​:s(I)→s(K)

• FsK:s(K)→s(W)F_{sK}: s(K) \rightarrow s(W)FsK​:s(K)→s(W)

• FsW:s(W)→s(P)F_{sW}: s(W) \rightarrow s(P)FsW​:s(W)→s(P)

• FsP:s(P)→s(D′)F_{sP}: s(P) \rightarrow s(D')FsP​:s(P)→s(D′)

4.2 Mutual Expressing Capabilities

A. Data and Information

• Expressing Data in Terms of Information:

• Concept Level: $D$ is mapped to $I$ via FDF_DFD​.

• Semantic Level: $s(D)$ is expressed in $s(I)$ via FsDF_{sD}FsD​.

• Limitations:

• Expressiveness: Not all data semantics can be fully captured in information semantics.

• Loss of Granularity: Detailed data may be abstracted in information.

B. Information and Knowledge

• Expressing Information in Terms of Knowledge:

• Concept Level: $I$ is structured into $K$ via FIF_IFI​.

• Semantic Level: $s(I)$ contributes to $s(K)$ via FsIF_{sI}FsI​.

• Limitations:

• Structural Constraints: Knowledge structures may limit the expression of all information semantics.

• Context Dependence: Knowledge semantics may depend heavily on context not fully captured from information.

C. Knowledge and Wisdom

• Expressing Knowledge in Terms of Wisdom:

• Concept Level: $K$ is applied to generate $W$ via FKF_KFK​.

• Semantic Level: $s(K)$ informs $s(W)$ via FsKF_{sK}FsK​.

• Limitations:

• Subjective Interpretation: Different interpretations of knowledge semantics can lead to varying wisdom semantics.

• Ethical Divergence: Wisdom semantics are influenced by ethical frameworks that may not be inherent in knowledge semantics.

D. Wisdom and Purpose

• Expressing Wisdom in Terms of Purpose:

• Concept Level: $W$ shapes $P$ via FWF_WFW​.

• Semantic Level: $s(W)$ guides $s(P)$ via FsWF_{sW}FsW​.

• Limitations:

• Goal Alignment: The purpose derived may not fully reflect all aspects of wisdom semantics.

• Complexity of Translation: Capturing the full depth of wisdom semantics in purpose semantics can be challenging.

E. Purpose and Data

• Expressing Purpose in Terms of Data Collection:

• Concept Level: $P$ influences new data D′D'D′ via FPF_PFP​.

• Semantic Level: $s(P)$ determines s(D′)s(D')s(D′) via FsPF_{sP}FsP​.

• Limitations:

• Feedback Delays: The effect of purpose on data may not be immediate.

• Circular Dependencies: Purpose guided by wisdom derived from data can create loops with potential for inconsistency.

5. Explicit Distinction Between DIKWP Concepts and Semantics

5.1 Examples Illustrating Distinctions

Example 1: Weather Forecasting System

• Data (D):

• Concepts: Temperature readings, humidity levels.

• Semantics: What does a specific temperature indicate about the weather?

• Information (I):

• Concepts: Weather patterns identified from data.

• Semantics: The meaning of these patterns in predicting weather changes.

• Knowledge (K):

• Concepts: Models that explain weather behavior.

• Semantics: Understanding why certain patterns lead to specific weather events.

• Wisdom (W):

• Concepts: Decision-making guidelines for issuing weather warnings.

• Semantics: Ethical considerations in balancing public safety and preventing panic.

• Purpose (P):

• Concepts: Goals such as "Ensure public safety during extreme weather."

• Semantics: The importance and value of this goal.

Mutual Expressing Capabilities and Limits:

• Data to Information: Raw temperature data is processed into patterns (concept), but some data nuances may be lost (semantic limit).

• Knowledge to Wisdom: Models inform decision guidelines (concept), but ethical considerations may vary (semantic limit).

6. Tables Summarizing Mutual Expressions and Limits

Table 1: Mutual Expressing Capabilities Among DIKWP Components

7. Analytical Discussion of Limits

7.1 Expressiveness Limits

• Data to Information: The transformation may abstract away critical data semantics, leading to information that lacks depth.

• Information to Knowledge: Overgeneralization in knowledge structures may omit important information semantics.

7.2 Semantic Loss

• Knowledge to Wisdom: Ethical nuances in knowledge semantics might not fully translate into wisdom, causing incomplete ethical considerations.

• Wisdom to Purpose: The broad nature of wisdom semantics can be challenging to distill into specific, actionable purposes.

7.3 Mutual Dependence and Circularity

• The components are interdependent, and limitations in one can affect the others.

• For instance, if the purpose is not well-defined, it may lead to irrelevant data collection, impacting the entire DIKWP chain.

8. Conclusion

By investigating the mutual expressing capabilities and limits among the DIKWP components in a mathematical manner, and explicitly distinguishing between DIKWP Concepts and DIKWP Semantics, we gain a deeper understanding of how data transforms into purposeful actions through cognitive processes.

• DIKWP Concepts provide the structural framework for transformation.

• DIKWP Semantics add depth and meaning, influencing how transformations are interpreted and applied.

Key Insights:

• Mathematical functions model the transformations but have limitations due to abstraction and potential loss of semantics.

• Explicit distinctions between concepts and semantics highlight where meanings may be lost or altered.

• Understanding these limitations is crucial for designing systems that effectively integrate all DIKWP components.

Further Study

• Refinement of Functions: Developing more sophisticated functions that preserve semantics.

• Contextual Modeling: Incorporating context to enhance semantic transformations.

• Feedback Mechanisms: Implementing iterative processes to adjust for limitations.

References

• Duan, Y. (Year). Relevant publications on DIKWP Semantic Mathematics.

• Cognitive Science Literature: Exploring semantic processing in cognition.

• Mathematical Modeling Texts: For advanced function and transformation modeling.

References for Further Reading

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. ".