Differences Among Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space in the DIKWP 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)

Introduction

The DIKWP framework—comprising Data (D), Information (I), Knowledge (K), Wisdom (W), and Purpose (P)—provides a structured approach to understanding cognitive processes and semantic transformations. To model the differences among Conceptual Space (ConC), Semantic Space (SemA), Cognitive Space (ConN), and Conscious Space, we can map each space to specific components and processes within the DIKWP framework. This mapping will highlight the clear distinctions among the DIKWP concepts and their semantics as they manifest in each space.

Overview of DIKWP Components and Their Mathematical Representations

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

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

• Semantics: Recognized by shared semantic attributes.

2. Information (I): Processed data with identified patterns and differences.

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

• Semantics: Emerges by identifying differences and contextual relationships.

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

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

• Semantics: Represents "complete" understanding through relationships and rules.

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

• Representation: Decision function W:{D,I,K,W,P}→D∗W: \{D, I, K, W, P\} \rightarrow D^*W:{D,I,K,W,P}→D∗

• Semantics: Guides decision-making by integrating all DIKWP components.

5. Purpose (P): Goals or objectives guiding the system's actions.

• Representation: $P=(\text{Input},\text{Output})$

• Semantics: Aligns transformations towards desired outcomes.

Mapping Spaces to DIKWP Components

We will model each space in terms of the DIKWP components and highlight the differences among DIKWP concepts and semantics within each space.

1. Conceptual Space (ConC)

Role in DIKWP:

• Focus: Data (D) and Knowledge (K)

• DIKWP Concepts: Definitions and structures of data and knowledge.

• DIKWP Semantics: Static representation of concepts and their relationships.

Mathematical Modeling:

• Data Concepts: Semantic Attribute Set S={f1,f2,...,fn}S = \{f_1, f_2, ..., f_n\}S={f1​,f2​,...,fn​}

• Data Concept Set: $D=\{d\mid d\text{ shares }S\}$

• Knowledge Graph: $K=(N,E)$

• Nodes $N$: Concepts defined in ConC.

• Edges $E$: Relationships among concepts.

Differences in DIKWP Concepts and Semantics:

• DIKWP Concepts: Emphasizes the definitions of data and knowledge elements.

• DIKWP Semantics: Concerned with the static, symbolic representation of concepts without contextual meanings.

Boundary Characteristics:

• Limits: Restricted to predefined concepts and their structural relationships.

• No Processing: Does not involve transformations or cognitive functions.

2. Semantic Space (SemA)

Role in DIKWP:

• Focus: Information (I)

• DIKWP Concepts: Meanings and associations of information derived from data.

• DIKWP Semantics: Dynamic representation of meanings and contextual relationships.

Mathematical Modeling:

• Semantic Mapping Function: ϕ:D→Rm\phi: D \rightarrow \mathbb{R}^mϕ:D→Rm

• Maps data elements to semantic vectors.

• Information Extraction Function: FI:D→IF_I: D \rightarrow IFI​:D→I

• Processes data to generate information with meanings.

Differences in DIKWP Concepts and Semantics:

• DIKWP Concepts: Focuses on differences and contextualization of data to form information.

• DIKWP Semantics: Deals with the meanings and associations of information elements, enriching data with context.

Boundary Characteristics:

• Limits: Meanings are constrained by the semantic mappings and context provided.

• Interaction with ConC: Extends concepts by adding semantic richness.

3. Cognitive Space (ConN)

Role in DIKWP:

• Focus: Transformation Processes among D, I, K, W

• DIKWP Concepts: Cognitive functions that process and transform DIKWP components.

• DIKWP Semantics: Dynamic operations that produce understanding and actions.

Mathematical Modeling:

• Cognitive Functions: FConN={fi∣fi:Xi→Yi}F_{\text{ConN}} = \{ f_i \mid f_i: X_i \rightarrow Y_i \}FConN​={fi​∣fi​:Xi​→Yi​}

• Data to Information: fD→I:D→If_{\text{D→I}}: D \rightarrow IfD→I​:D→I

• Information to Knowledge: fI→K:I→Kf_{\text{I→K}}: I \rightarrow KfI→K​:I→K

• Knowledge to Wisdom: fK→W:K→Wf_{\text{K→W}}: K \rightarrow WfK→W​:K→W

• Function Composition: fCognition=fK→W∘fI→K∘fD→If_{\text{Cognition}} = f_{\text{K→W}} \circ f_{\text{I→K}} \circ f_{\text{D→I}}fCognition​=fK→W​∘fI→K​∘fD→I​

Differences in DIKWP Concepts and Semantics:

• DIKWP Concepts: Emphasizes processes and transformations among components.

• DIKWP Semantics: Involves the dynamic interpretation and manipulation of meanings to generate new understanding.

Boundary Characteristics:

• Limits: Bound by the capabilities of the cognitive functions defined.

• Interaction with SemA and ConC: Utilizes concepts (ConC) and their meanings (SemA) in processing.

4. Conscious Space

Role in DIKWP:

• Focus: Wisdom (W) and Purpose (P)

• DIKWP Concepts: Awareness and integration of DIKWP components at a higher level.

• DIKWP Semantics: Emergent properties involving self-awareness and ethical considerations.

Mathematical Modeling:

• Meta-Cognitive Function: $\Phi :W\to \mathcal{C}$

• Maps wisdom to states of consciousness or self-awareness.

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

• Aligns actions with overarching goals.

Differences in DIKWP Concepts and Semantics:

• DIKWP Concepts: Involves higher-order reasoning, self-reflection, and goal alignment.

• DIKWP Semantics: Concerned with the subjective experience, ethical values, and purposeful actions.

Boundary Characteristics:

• Limits: Emergent properties that may not be fully captured by standard mathematical functions.

• Interaction with ConN: Builds upon cognitive processes to achieve conscious awareness.

Comparative Analysis of DIKWP Concepts and Semantics Across Spaces

Space	DIKWP Focus	DIKWP Concepts	DIKWP Semantics	Mathematical Boundaries
Conceptual Space (ConC)	Data (D), Knowledge (K)	Definitions, structures of data and knowledge	Static representation of concepts and relationships	Limited to predefined concepts and structural relationships
Semantic Space (SemA)	Information (I)	Meanings and associations of information	Dynamic representation of meanings and context	Constrained by semantic mappings and contextual meanings
Cognitive Space (ConN)	Transformations among D, I, K, W	Cognitive functions processing DIKWP components	Dynamic operations producing understanding and actions	Bound by defined cognitive functions and processing capabilities
Conscious Space	Wisdom (W), Purpose (P)	Awareness, ethical reasoning, goal alignment	Emergent properties involving self-awareness	Emergent; may require higher-order or non-standard mathematical models

Modeling Differences Using the DIKWP Mathematical Model

A. Between Conceptual Space and Semantic Space

• Transition Function: $\varphi :D\to I$

• From Data Concepts (ConC) to Information Meanings (SemA)

• DIKWP Concepts Difference:

• ConC: Deals with data definitions without inherent meanings.

• SemA: Adds meanings and context to data, forming information.

• DIKWP Semantics Difference:

• ConC Semantics: Static and symbolic.

• SemA Semantics: Dynamic and meaningful.

Mathematical Boundary:

• Mapping Function $\varphi$ defines the boundary.

• Limit: Data not mapped cannot have associated meanings in SemA.

B. Between Semantic Space and Cognitive Space

• Cognitive Functions: FConNF_{\text{ConN}}FConN​ operate on information from SemA.

• DIKWP Concepts Difference:

• SemA: Focuses on meanings of information.

• ConN: Processes information to generate knowledge and wisdom.

• DIKWP Semantics Difference:

• SemA Semantics: Meanings and associations.

• ConN Semantics: Interpretation and transformation of meanings.

Mathematical Boundary:

• Function Domains: Dom(fI→K)=I\text{Dom}(f_{\text{I→K}}) = IDom(fI→K​)=I

• Limit: Cognitive functions cannot process information without semantic meanings.

C. Between Cognitive Space and Conscious Space

• Meta-Cognitive Function: $\Phi :W\to \mathcal{C}$

• DIKWP Concepts Difference:

• ConN: Executes cognitive processes without self-awareness.

• Conscious Space: Involves self-awareness and reflection on cognitive processes.

• DIKWP Semantics Difference:

• ConN Semantics: Operational and procedural.

• Conscious Space Semantics: Subjective and experiential.

Mathematical Boundary:

• Meta-Cognitive Mapping $\Phi$ defines the boundary.

• Limit: Conscious properties emerge only when cognitive processes are subject to self-reflection.

D. Between Conceptual Space and Cognitive Space

• Processing Function: fD→If_{\text{D→I}}fD→I​ operates on data concepts from ConC.

• DIKWP Concepts Difference:

• ConC: Static concepts of data.

• ConN: Active processing of data into information.

• DIKWP Semantics Difference:

• ConC Semantics: Definitions without processing.

• ConN Semantics: Dynamic transformation leading to new semantics.

Mathematical Boundary:

• Function Input Domain: Dom(fD→I)=D\text{Dom}(f_{\text{D→I}}) = DDom(fD→I​)=D

• Limit: Without data concepts, cognitive processing cannot begin.

Visualization of Transitions and Boundaries

mathematicaCopy code[Conceptual Space (ConC)]        |        | Mapping Function φ (D → I)        ↓[Semantic Space (SemA)]        |        | Cognitive Functions F_ConN (I → K → W)        ↓[Cognitive Space (ConN)]        |        | Meta-Cognitive Function Φ (W → Conscious States)        ↓[Conscious Space]

• Each mapping or function represents a boundary where DIKWP concepts and semantics differ.

Implications for Modeling and System Design

1. Awareness of Boundaries:

• Understanding where DIKWP concepts and semantics change is crucial for accurate modeling.

2. Defining Mappings and Functions:

• Precise mathematical definitions are necessary to ensure proper transitions between spaces.

3. Handling Limits:

• Recognize that each space has limits based on its DIKWP focus and semantics.

4. Designing for Emergence:

• Conscious properties may require complex interactions and cannot be directly programmed.

Conclusion

By modeling the differences among the spaces in terms of the DIKWP mathematical model, we have highlighted how each space focuses on different DIKWP components, concepts, and semantics. The distinctions among Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space are defined by the roles they play in processing and transforming DIKWP components, as well as the mathematical boundaries established by mappings and functions.

Understanding these differences is essential for:

• Developing Advanced AI Systems: Ensuring that each component and space is accurately modeled.

• Modeling Cognitive Processes: Capturing the nuances of how data becomes information, knowledge, and wisdom.

• Exploring Consciousness in AI: Recognizing the emergent nature of consciousness and its dependence on underlying processes.

References

• DIKWP Framework Literature: For detailed explanations of the components and their mathematical representations.

• Cognitive Science and AI Research: Studies on modeling cognitive processes and consciousness.

• Mathematical Modeling Resources: Guides on set theory, graph theory, function mappings, and higher-order functions.

Note: This modeling is an abstract representation intended to illustrate the differences among the spaces within the DIKWP framework. Specific implementations may require additional considerations based on the context and objectives of the system being designed.

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. ".

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