Expressing Capabilities Among 4 Spaces with DIKWP Mathematical Model

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

This exploration aims to model the mutual expressing capabilities among the Conceptual Space (ConC), Semantic Space (SemA), Cognitive Space (ConN), and Conscious Space, using the DIKWP (Data, Information, Knowledge, Wisdom, Purpose) mathematical model. We will explicitly distinguish between the DIKWP concepts and the DIKWP semantics to clarify their roles and interactions within each space.

1. Overview of the DIKWP Model

The DIKWP model represents the transformation of content through five interconnected components:

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

• Information (I): Data processed to reveal patterns and relationships.

• Knowledge (K): Structured information organized into coherent frameworks.

• Wisdom (W): Application of knowledge with ethical considerations.

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

Mathematical Representations:

• Data (D): D={d1,d2,...,dn}D = \{ d_1, d_2, ..., d_n \}D={d1​,d2​,...,dn​}

• Information (I): I={i1,i2,...,im}I = \{ i_1, i_2, ..., i_m \}I={i1​,i2​,...,im​}

• Knowledge (K): $K=(N,E)$, where $N$ is a set of nodes (concepts) and $E$ is a set of edges (relationships).

• Wisdom (W): Decision function W:{D,I,K,W,P}→D∗W: \{ D, I, K, W, P \} \rightarrow D^*W:{D,I,K,W,P}→D∗, where D∗D^*D∗ is the optimal decision.

• Purpose (P): $P=(\text{Input},\text{Output})$, aligning actions towards goals.

2. Overview of the Spaces

• Conceptual Space (ConC): The cognitive representation where concepts (DIKWP concepts) are defined, organized, and related through language and symbols.

• Semantic Space (SemA): The network of semantic associations (DIKWP semantics) between concepts, capturing meanings and relationships.

• Cognitive Space (ConN): The dynamic processing environment where DIKWP components are transformed into understanding and actions through cognitive functions.

• Conscious Space: The emergent layer representing awareness and subjective experience arising from cognitive and semantic interactions.

3. Distinction Between DIKWP Concepts and DIKWP Semantics

• DIKWP Concepts: The structural elements (Data, Information, Knowledge, etc.) and their relationships, as defined and organized within the model.

• DIKWP Semantics: The meanings, interpretations, and contextual associations attached to the DIKWP concepts.

Example:

• DIKWP Concept: "Knowledge" represented as a node in a knowledge graph.

• DIKWP Semantics: The meaning and significance of that knowledge in a particular context.

4. Modeling Mutual Expressing Capabilities Among Spaces Using the DIKWP Model

We will model how each space can express and be expressed by the others, using the DIKWP concepts and semantics, with explicit mathematical representations.

A. Conceptual Space (ConC) and Semantic Space (SemA)

From ConC to SemA: Expressing Concepts as Semantics

• DIKWP Concepts in ConC: C={c1,c2,...,cn}C = \{ c_1, c_2, ..., c_n \}C={c1​,c2​,...,cn​}

• Mapping Function (Semantic Assignment): $\varphi :C\to S$, where $S$ is the set of semantic representations.

Mathematical Representation:

• For each concept ci∈Cc_i \in Cci​∈C, assign semantics si=ϕ(ci)s_i = \phi(c_i)si​=ϕ(ci​) in SemA.

Distinction:

• DIKWP Concept: The structural definition of cic_ici​ in ConC.

• DIKWP Semantics: The meaning sis_isi​ associated with cic_ici​ in SemA.

From SemA to ConC: Extracting Concepts from Semantics

• Extraction Function: $\psi :S\to C$

Limitations:

• Ambiguity: Multiple concepts may share similar semantics, leading to potential confusion.

• Information Loss: Not all semantic nuances can be captured in the concept definitions.

B. Semantic Space (SemA) and Cognitive Space (ConN)

From SemA to ConN: Processing Semantics through Cognitive Functions

• DIKWP Semantics in SemA: S={s1,s2,...,sn}S = \{ s_1, s_2, ..., s_n \}S={s1​,s2​,...,sn​}

• Cognitive Functions: FConN={fi∣fi:S→I or K}F_{\text{ConN}} = \{ f_i \mid f_i: S \rightarrow I \text{ or } K \}FConN​={fi​∣fi​:S→I or K}

Mathematical Representation:

• Processing Semantics: ij=fi(sj)i_j = f_i(s_j)ij​=fi​(sj​), where ij∈Ii_j \in Iij​∈I (Information) or kj∈Kk_j \in Kkj​∈K (Knowledge).

Distinction:

• DIKWP Semantics: The meanings sjs_jsj​ being processed.

• DIKWP Concepts: The resulting information iji_jij​ or knowledge kjk_jkj​ produced by cognitive functions.

From ConN to SemA: Generating New Semantics from Cognitive Processing

• Generation Function: g:I or K→S′g: I \text{ or } K \rightarrow S'g:I or K→S′, where S′S'S′ is a new set of semantics.

Limitations:

• Complex Transformations: Cognitive processes may alter semantics in non-linear ways.

• Representational Differences: The semantics generated may not directly map back to initial concepts.

C. Cognitive Space (ConN) and Conscious Space

From ConN to Conscious Space: Emergence of Conscious States

• Cognitive Processes: FConNF_{\text{ConN}}FConN​

• Consciousness Function: Φ:FConN→W\Phi: F_{\text{ConN}} \rightarrow WΦ:FConN​→W, where $W$ represents wisdom in the context of conscious awareness.

Mathematical Representation:

• Emergent Wisdom: w=Φ(fi)w = \Phi(f_i)w=Φ(fi​), integrating ethical considerations and self-awareness.

Distinction:

• DIKWP Concepts: Cognitive functions fif_ifi​ operating on data, information, and knowledge.

• DIKWP Semantics: The conscious understanding and ethical implications $w$ derived from fif_ifi​.

From Conscious Space to ConN: Conscious Influence on Cognitive Processing

• Modulation Function: Θ:W→FConN′\Theta: W \rightarrow F_{\text{ConN}}'Θ:W→FConN′​, where FConN′F_{\text{ConN}}'FConN′​ represents adjusted cognitive functions.

Limitations:

• Subjectivity: Conscious experiences are subjective and may not be fully captured mathematically.

• Non-Linearity: The influence of consciousness on cognition can be complex and non-linear.

D. Conceptual Space (ConC) and Cognitive Space (ConN)

From ConC to ConN: Concepts Guiding Cognitive Functions

• Function Definition: Cognitive functions are defined based on concepts from ConC.

Mathematical Representation:

• Cognitive Function Formation: fi:D→If_i: D \rightarrow Ifi​:D→I, where fif_ifi​ is designed using concepts cj∈Cc_j \in Ccj​∈C.

Distinction:

• DIKWP Concepts: The definitions and structures cjc_jcj​ in ConC.

• DIKWP Semantics: The meanings and applications of these concepts within cognitive processing.

From ConN to ConC: Abstraction of Cognitive Processes into Concepts

• Abstraction Function: κ:FConN→C′\kappa: F_{\text{ConN}} \rightarrow C'κ:FConN​→C′, where C′C'C′ is an expanded set of concepts.

Limitations:

• Abstraction Loss: Some details of cognitive processes may be lost when abstracted into concepts.

• Static Representation: ConC may not fully capture the dynamic nature of cognition.

5. Reasoning with the DIKWP Model and Explicit Distinctions

We will now reason through an example to illustrate the mutual expressing capabilities, explicitly distinguishing between DIKWP concepts and semantics.

Example: AI-Powered Decision Support System

• Purpose (P): Assist in making ethical business decisions.

• Data (D): Market data, customer feedback, financial reports.

• Information (I): Processed insights, trends, risk assessments.

• Knowledge (K): Business rules, ethical guidelines, best practices.

• Wisdom (W): Recommendations that consider ethical implications and long-term goals.

Spaces and Interactions:

A. ConC and SemA

• ConC (DIKWP Concepts): Definitions of "Market Data," "Ethical Guidelines," "Risk Assessment."

• SemA (DIKWP Semantics): Meanings of market trends, ethical considerations in different cultural contexts.

Mathematical Mapping:

• ϕ("Market Data")=sMarket Data\phi(\text{"Market Data"}) = s_{\text{Market Data}}ϕ("Market Data")=sMarket Data​, assigning semantics to the concept.

• Distinction:

• Concept: "Market Data" as a defined term.

• Semantics: Understanding what market data indicates about consumer behavior.

B. SemA and ConN

• SemA (DIKWP Semantics): Meanings of identified trends and ethical implications.

• ConN (DIKWP Concepts and Semantics):

• Cognitive Function: fAnalysis:sMarket Trends→iInsightsf_{\text{Analysis}}: s_{\text{Market Trends}} \rightarrow i_{\text{Insights}}fAnalysis​:sMarket Trends​→iInsights​

• Processing: Analyzing meanings to generate actionable information.

Distinction:

• Concepts: The function fAnalysisf_{\text{Analysis}}fAnalysis​.

• Semantics: The insights iInsightsi_{\text{Insights}}iInsights​ derived from processing meanings.

C. ConN and Conscious Space

• ConN (DIKWP Concepts and Semantics):

• Knowledge Application: Using business rules to interpret information.

• Conscious Space (Wisdom):

• Conscious Decision Function: Φ(fDecision)→wRecommendation\Phi(f_{\text{Decision}}) \rightarrow w_{\text{Recommendation}}Φ(fDecision​)→wRecommendation​

Distinction:

• Concepts: The decision function fDecisionf_{\text{Decision}}fDecision​.

• Semantics: The ethical and purposeful recommendations wRecommendationw_{\text{Recommendation}}wRecommendation​.

D. ConC and ConN

• ConC (DIKWP Concepts): Business rules, ethical guidelines.

• ConN (DIKWP Concepts and Semantics):

• Function Definition: Cognitive functions are designed based on these concepts.

• Processing: Applying rules to data and information.

Distinction:

• Concepts: The rules and guidelines as defined entities.

• Semantics: The interpretations and applications of these rules within cognitive processes.

6. Mathematical Modeling of Interactions with Explicit Distinctions

A. Mapping Concepts to Semantics

• Function: $\varphi :C\to S$

• Example: ϕ("Ethical Guideline 1")=sEthical Guideline 1\phi(\text{"Ethical Guideline 1"}) = s_{\text{Ethical Guideline 1}}ϕ("Ethical Guideline 1")=sEthical Guideline 1​

DIKWP Concepts vs. Semantics:

• Concept: "Ethical Guideline 1" as a defined rule.

• Semantics: The meaning and implications of following this guideline.

B. Cognitive Processing of Semantics

• Function: fEthical Analysis:S→If_{\text{Ethical Analysis}}: S \rightarrow IfEthical Analysis​:S→I

• Example: fEthical Analysis(sMarket Trends,sEthical Guidelines)=iEthical Insightsf_{\text{Ethical Analysis}}(s_{\text{Market Trends}}, s_{\text{Ethical Guidelines}}) = i_{\text{Ethical Insights}}fEthical Analysis​(sMarket Trends​,sEthical Guidelines​)=iEthical Insights​

DIKWP Concepts vs. Semantics:

• Concepts: The function fEthical Analysisf_{\text{Ethical Analysis}}fEthical Analysis​ and the resulting information iEthical Insightsi_{\text{Ethical Insights}}iEthical Insights​.

• Semantics: The meanings processed and the ethical considerations derived.

C. Emergence of Wisdom

• Function: Φ:FConN→W\Phi: F_{\text{ConN}} \rightarrow WΦ:FConN​→W

• Example: wRecommendation=Φ(fEthical Analysis)w_{\text{Recommendation}} = \Phi(f_{\text{Ethical Analysis}})wRecommendation​=Φ(fEthical Analysis​)

DIKWP Concepts vs. Semantics:

• Concept: The process of transforming cognitive functions into wisdom.

• Semantics: The ethical and purposeful meaning of the recommendation.

D. Purpose Alignment

• Function: TW→P:W→PT_{\text{W→P}}: W \rightarrow PTW→P​:W→P

• Example: pAction Plan=TW→P(wRecommendation)p_{\text{Action Plan}} = T_{\text{W→P}}(w_{\text{Recommendation}})pAction Plan​=TW→P​(wRecommendation​)

DIKWP Concepts vs. Semantics:

• Concept: The transformation function aligning wisdom with purpose.

• Semantics: The intended outcomes and goals encapsulated in the action plan.

7. Conclusion

By reasoning with the DIKWP mathematical model and explicitly distinguishing between DIKWP concepts and DIKWP semantics, we have modeled the mutual expressing capabilities among the Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space. Each space interacts through defined functions and mappings, transforming concepts and semantics to achieve purposeful actions guided by wisdom.

This explicit distinction clarifies:

• DIKWP Concepts: The structural elements and processes within each space.

• DIKWP Semantics: The meanings and interpretations that provide context and significance.

Understanding these distinctions enhances our ability to model complex cognitive processes and design AI systems that effectively integrate data, information, knowledge, wisdom, and purpose.

Key Takeaways:

• Conceptual Space (ConC) provides the structural definitions (DIKWP concepts) that are assigned meanings (DIKWP semantics) in Semantic Space (SemA).

• Semantic Space (SemA) contains the meanings that are processed by Cognitive Space (ConN) through cognitive functions, transforming semantics into higher-level DIKWP concepts (information and knowledge).

• Cognitive Space (ConN) utilizes both DIKWP concepts and semantics to perform cognitive processing, leading to the emergence of wisdom in the Conscious Space.

• Conscious Space integrates DIKWP concepts and semantics at the highest level, where wisdom and purpose guide actions and decisions.

By maintaining clear distinctions between concepts and semantics, we can better understand and model the intricate interactions within the DIKWP framework across different cognitive and semantic spaces.

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. ".