Semantic Completeness and Mathematical Properties of DIKWP Mathematical System

Yucong Duan

Benefactor: Shiming Gong

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

DIKWP-AC Artificial Consciousness Standardization Committee

World Conference on Artificial Consciousness

World Artificial Consciousness Association

(Email：duanyucong@hotmail.com)

Catalog

Introduction

1. Mathematical Characteristics of the DIKWP Model

1.1 Data

1.2 Information

1.3 Knowledge

2. Mathematical Logic of Semantic Space

2.1 Conceptual Space

2.2 Semantic Space

2.3 Cognitive Space

3. Semantic Completeness and Mathematical Characteristics

3.1 Semantic Completeness

3.2 Semantic Consistency

3.3 Dynamic Update

4. Construction of a Semantic Processing Mathematical System

4.1 Concept Representation and Processing

4.2 Semantic Matching and Association

4.3 Cognitive Process and Dynamic Update

5. Comparative Analysis of DIKWP Mathematical System with Other Similar Mathematical Systems

5.1 Comparison of DIKWP Mathematical System and DIKW Model

5.2 Comparison of DIKWP Mathematical System and SECI Model

5.3 Comparison of DIKWP Mathematical System and Polanyi's Tacit Knowledge Theory

5.4 Comparison of DIKWP Mathematical System and Cynefin Framework

Conclusion

Introduction

Professor Yucong Duan's DIKWP model, by distinctly differentiating data, information, and knowledge and further dividing conceptual space, semantic space, and cognitive space, provides a high-level, dynamic, and structured cognitive framework for cognitive entities (such as humans or AI systems). As an expert in mathematical logic, this report will delve into the semantic completeness and mathematical characteristics of this model and, based on this, construct a semantic processing mathematical system.

1. Mathematical Characteristics of the DIKWP Model

1.1 Data

Definition: Original facts or observational records confirmed by the cognitive subject, classified and organized through the conceptual space to form preliminary cognitive objects. Mathematical Representation: Data can be represented as a set D, where each data item d∈D is an atomic fact.

1.2 Information

Definition: Through the intention of the cognitive subject, data is semantically associated with existing cognitive objects, identifying differences and forming new cognitive content. Mathematical Representation: Information can be represented as a mapping f:D→I from the data set D to the information set I, where each information item i∈I is generated through semantic association.

1.3 Knowledge

Definition: Through higher-order cognitive activities and hypotheses, data and information are systematically understood and interpreted, forming a deep understanding and explanation of the world. Mathematical Representation: Knowledge can be represented as a mapping g:I→K from the information set I to the knowledge set K, where each knowledge item k∈K is generated through higher-order cognitive activities.

2. Mathematical Logic of Semantic Space

2.1 Conceptual Space

Definition: The conceptual space is where cognitive subjects communicate and understand through natural language, symbols, etc. Data, information, and knowledge exist as specific concepts in this space and are expressed through semantic networks and conceptual graphs. Mathematical Representation: The conceptual space can be represented as a directed graph G=(V,E), where nodes V represent concepts, and edges E represent relationships between concepts. Each node v∈V in the semantic network has a unique identifier and is connected to other nodes through semantic relationships.

2.2 Semantic Space

Definition: The semantic space is where the cognitive subject understands and processes the intrinsic semantic connections of concepts. Data, information, and knowledge in this space are understood and new knowledge is generated through semantic matching, association, and transformation. Mathematical Representation: The semantic space can be represented as a semantic matching function h:V×V→[0,1], where h(v_i,v_j) indicates the degree of semantic matching between two concept nodes. The matching function h in the semantic space satisfies the following properties:

Symmetry: h(v_i,v_j)=h(v_j,v_i)

Reflexivity: h(v_i,v_i)=1

Non-negativity: 0≤h(v_i,v_j)≤1

2.3 Cognitive Space

Definition: The cognitive space is the internal psychological space where cognitive subjects think, learn, and understand. Data, information, and knowledge in this space form a deep understanding and interpretation of the world through cognitive activities such as observation, hypothesis, abstraction, and verification. Mathematical Representation: The cognitive space can be represented as a dynamic system S=(X,F,Y), where:

X represents the cognitive state space, containing all possible cognitive states x∈X;

F represents the transition function f:X×U→X of the cognitive process, where U is external input;

Y represents the output space, containing all possible cognitive outputs y∈Y.

3. Semantic Completeness and Mathematical Characteristics

3.1 Semantic Completeness

Definition: Semantic completeness refers to the ability to fully express and compute all possible semantic connections and matching relationships in the semantic space. Implementation: Through the definition and properties of the semantic matching function h, semantic completeness can be ensured. The semantic matching function h can be trained and optimized through machine learning algorithms and semantic search engines to achieve semantic completeness.

3.2 Semantic Consistency

Definition: Semantic consistency refers to the consistency and stability of concepts and semantic relationships in different cognitive and semantic spaces. Implementation: By the symmetry and reflexivity of the semantic matching function h, semantic consistency can be ensured. The structured representation of semantic networks and conceptual graphs also helps maintain semantic consistency.

3.3 Dynamic Update

Definition: Dynamic update refers to the ability to adjust and update knowledge and semantic relationships dynamically with the input of new data and information during the cognitive process. Implementation: Through the dynamic system S and transition function F in the cognitive space, dynamic updating of knowledge and semantic relationships can be achieved. Deep learning and reinforcement learning techniques can be used to train and optimize cognitive processes.

4. Construction of a Semantic Processing Mathematical System

4.1 Concept Representation and Processing

Semantic Network: Construct a semantic network G=(V,E) to represent concepts and their relationships. Ontology: Use ontology to define concepts and their hierarchical structure, ensuring consistency and completeness of concept representation.

4.2 Semantic Matching and Association

Semantic Matching Function: Define a semantic matching function h:V×V→[0,1] to calculate the semantic matching degree between concepts. Semantic Reasoning Engine: Construct a semantic reasoning engine based on rules and logic reasoning to achieve semantic association and reasoning.

4.3 Cognitive Process and Dynamic Update

Dynamic System Model: Construct a dynamic system S=(X,F,Y) in the cognitive space to simulate cognitive processes. Deep Learning and Reinforcement Learning: Use deep learning and reinforcement learning techniques to train and optimize cognitive processes, achieving dynamic updating of knowledge and semantic relationships.

5. Comparative Analysis of DIKWP Mathematical System with Other Similar Mathematical Systems

To deeply understand the unique advantages and application potential of Professor Yucong Duan's DIKWP mathematical system, we will compare it with several other well-known mathematical systems. These systems include the DIKW model, the SECI model, Polanyi's tacit knowledge theory, and the Cynefin framework.

5.1 Comparison of DIKWP Mathematical System and DIKW Model

5.2 Comparison of DIKWP Mathematical System and SECI Model

5.3 Comparison of DIKWP Mathematical System and Polanyi's Tacit Knowledge Theory

5.4 Comparison of DIKWP Mathematical System and Cynefin Framework

Conclusion

Professor Yucong Duan's DIKWP mathematical system, by distinctly differentiating data, information, and knowledge and further dividing conceptual space, semantic space, and cognitive space, provides a more efficient and accurate method for knowledge generation and understanding. Compared with other knowledge models and frameworks, the DIKWP mathematical system has the following unique advantages:

Semantic Completeness: Achieves semantic completeness through the semantic matching function h, ensuring the accuracy and consistency of semantic association and matching.

Dynamic Update: Achieves dynamic updating of knowledge and semantic relationships through the dynamic system S in the cognitive space, ensuring the timeliness and adaptability of knowledge.

Higher-order Cognitive Activities: Emphasizes knowledge generation and verification through higher-order cognitive activities, ensuring the systematicity and deep understanding of knowledge.

Mathematical Representation: Uses sets, mappings, dynamic systems, and other mathematical structures to provide systematic and formal mathematical representation methods, making knowledge representation and processing more structured and precise.

These advantages make the DIKWP mathematical system particularly outstanding in handling complex systems and abstract concepts and show broad application potential in natural language processing, knowledge graphs, and intelligent decision support systems. Future research can further explore and optimize the application of this model in more fields, promoting the development and progress of artificial intelligence technology.