The DIKWP Model Based on 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 report, based on Professor Duan Yucong's "Semantic Mathematics" concept and the existing DIKWP (Data, Information, Knowledge, Wisdom, Purpose) framework, attempts to generate the semantics of DIKWP directly within a mathematical structure without requiring an initial transformation from natural language to mathematical abstraction. The report first reviews the fundamental ideas of semantic mathematics and the semantic descriptions of DIKWP components, then proposes a preliminary mathematical framework, aiming to refine and improve it through subsequent interactions and iterations.

1. Background and Motivation

1. Semantic Mathematics Concept:

Semantic Mathematics is a concept proposed by Professor Duan Yucong, aiming to integrate the rigor of mathematics with semantic understanding. This approach seeks to bridge the gap between the abstract, formal world of mathematics and the often ambiguous and context-dependent world of natural language semantics.

1. DIKWP Framework:

DIKWP is a hierarchical/network model commonly used to describe cognitive, information processing, and decision-making processes. It comprises five elements:

• Data: Emphasizes the semantics of "sameness" from a cognitive perspective.

• Information: Emphasizes the extraction of "difference" semantics from data.

• Knowledge: Emphasizes the comprehensive semantic understanding through the synthesis of information.

• Wisdom: Integrates ethical, social moral, and humanistic semantic dimensions into decision-making.

• Purpose: Sets goals and directions to guide the transformations and decisions within the DIKWP system.

1. Research Problem:

How can we utilize the principles of Semantic Mathematics to directly construct the semantics of DIKWP within a mathematical framework, thereby eliminating the reliance on natural language-based subjective definitions? The goal is to develop an operational and verifiable mathematical framework where the semantics of each DIKWP component emerge naturally from internal relationships within the framework, without prior dependence on natural language abstraction.

2. Semantic Foundation and Mathematical Intuition of DIKWP Components

Based on Professor Duan Yucong's semantic descriptions (source: link), the semantic features of each DIKWP component are summarized as follows:

1. Data (D):

• Define a semantic feature space F, where any f ∈ F represents a fundamental semantic feature (e.g., a distinguishable semantic attribute).

• Define data D as a subset D⊆F or as a tuple D=(f1, f2, ..., fn), where each fi is a feature selected from F.

• "Sameness" can be defined via an equivalence relation ~ on F, where f_i ~ f_j indicates that two features are considered equivalent or similar under certain semantic criteria. This allows data semantics to appear as equivalence classes.

• Core Semantics: Data represents the manifestation of "sameness" in cognition. It is not merely raw facts but requires classification and recognition within a semantic or conceptual space.

• Preliminary Mathematical Representation:

2. Information (I):

• Information generation can be viewed as a mapping I: D→I_space, where I_space is the space of information constructs.

• Define I_space as the structure formed by semantic mappings based on data features, e.g., I = { φ(D) | φ: D→Σ } Here Σ is a set of semantic symbols or semantic values, and φ is a function that maps data features to distinguishable semantic symbols.

• "Difference" can be embodied by defining a distance or metric d: F×F→R on F. Information arises when d(f_i, f_j) exceeds a threshold or generates a new distinction. Information processing involves classification, differentiation, and reorganization of data features.

• Core Semantics: Information corresponds to the "difference" semantics in cognition, involving the extraction of new semantic units driven by specific intents.

• Preliminary Mathematical Representation:

3. Knowledge (K):

• Define knowledge as a graph structure K = (N, E), where N is a set of semantic nodes (concepts abstracted from information), and E represents the relationships between these concepts.

• The "completeness" of knowledge can be represented by graph properties such as connectivity, coverage, or redundancy measures. For example, a completeness function C(K) can measure the semantic completeness of the knowledge network, with higher C(K) indicating more comprehensive knowledge.

• The formation of knowledge can be seen as the mapping K = Ψ(I), where Ψ transforms information into a stable network of semantic concepts and meaningful relational edges based on the inherent differences and associations in the information.

• Core Semantics: Knowledge is the condensed result of structured and comprehensive synthesis of information, forming a robust semantic network.

• Preliminary Mathematical Representation:

4. Wisdom (W):

• Define wisdom as a decision function W: K×P→A, where A is a set of possible actions or decisions, and P is purpose (see below).

• Wisdom is represented by an optimization function on the knowledge graph, choosing actions that maximize a value function E(K, a).

• The function E incorporates social values and ethical constraints, which are embedded into the mathematical framework via a set of axioms or rules reflecting ethical principles.

• Core Semantics: Wisdom integrates ethics, social morals, and humanistic semantics into decision-making.

• Preliminary Mathematical Representation:

5. Purpose (P):

• Define purpose as a tuple P=(X_in, X_out, Goal), where X_in and X_out ∈ {D, I, K, W, P} represent the input and output spaces of the transformation, and Goal is a conditional or objective function guiding the transformation.

• Purpose directs the processing of Data, Information, Knowledge, Wisdom by setting target objectives and aligning transformations with these goals.

• Core Semantics: Purpose corresponds to a tuple (input, output), where both input and output are semantic contents of Data, Information, Knowledge, Wisdom, or Purpose.

• Preliminary Mathematical Representation:

3. Preliminary Mathematical Framework of DIKWP Based on Semantic Mathematics

1. Overall Structure:

• Define the feature space F and semantic operations to represent data D with inherent semantic "sameness".

• Define transformation operators Φ to map Data to Information, emphasizing "difference" semantics.

• Define knowledge construction operators Ψ to organize Information into Knowledge networks, embodying "completeness".

• Define wisdom decision functions Ω to utilize Knowledge and Purpose in optimizing actions, incorporating social and ethical values.

• Define Purpose as guiding parameters, P=(X_in, X_out, Goal), steering the transformations toward specific semantic objectives.

• To achieve direct semantic generation without translation, we model DIKWP components as a set of interrelated mathematical structures and transformation rules. These rules themselves encapsulate the semantics, ensuring that each transformation step retains semantic integrity.

• Framework Highlights:

2. Data to Information Transformation (D→I):

• Mapping: Φ: D→I_space represents the transformation from Data to Information, embodying the introduction of "difference" semantics.

• Semantic Representation: Φ is designed to capture and encode distinguishable patterns within Data, ensuring Information reflects new semantic distinctions.

3. Information to Knowledge Transformation (I→K):

• Mapping: Ψ: I_space→K_space transforms Information into Knowledge, structured as semantic networks.

• Semantic Representation: Ψ constructs Knowledge networks K=(N, E) from Information, ensuring completeness and robustness by incorporating relational semantics.

4. Knowledge to Wisdom Transformation (K×P→W):

• Mapping: Ω: K×P→A leverages Knowledge and Purpose to make decisions, optimizing actions within ethical and social constraints.

• Semantic Representation: Ω applies an evaluation function E(K, a) to select actions a that maximize E, aligning with Purpose.

5. Purpose (P):

• Mapping: P=(X_in, X_out, Goal) defines the transformation targets and objectives within the DIKWP framework.

• Semantic Representation: Purpose directs transformations by setting goals that guide data processing, information extraction, knowledge synthesis, and wisdom application toward desired outcomes.

4. Expandability and Iterative Optimization

1. Refining Features:

• Further define and specify the feature space F, distance metric d, transformation functions Φ, knowledge construction Ψ, decision value functions E, and Purpose constraints. These require domain-specific knowledge and empirical data.

2. Context and Case Studies:

• Apply the framework to specific cognitive task scenarios, such as differentiating "security" from "safety", to instantiate and validate the framework's applicability and effectiveness.

3. Axioms and Rules:

• Incorporate defined axioms, rules, or theorems to constrain DIKWP's behavior within the semantic space, ensuring transformations adhere to formalized semantic integrity.

4. Verification and Testing:

• Use simulation or reasoning tests to validate the framework, ensuring that semantic integrity is preserved according to expectations. Adjust mathematical definitions as necessary based on discrepancies.

5. Conclusion and Outlook

This report attempts to leverage the principles of Semantic Mathematics to provide a preliminary mathematical model for the DIKWP framework, allowing the semantics of Data, Information, Knowledge, Wisdom, and Purpose to be generated directly within mathematical structures without relying on natural language abstraction. The initial draft presents foundational ideas and framework structures, yet remains in need of precise definition and refinement.

Future Work:

• Collaboration with Domain Experts: Refine definitions of feature spaces, information metrics, knowledge networks, wisdom decision functions, and Purpose objectives.

• Selection of Specific Scenarios: Empirically test and validate the framework using scenarios like differentiating "security" from "safety" in decision-making contexts.

• Introduction of Axiom Systems and Logical Rules: Tighten the mathematical definitions with semantic constraints to ensure semantic integrity and losslessness.

• Continuous Verification and Refinement: Iteratively test and adjust the framework based on feedback and empirical results to develop a semantic mathematics-based framework that accurately and consistently represents DIKWP concepts.

Through ongoing iteration and improvement, it is expected that a semantic mathematical framework for DIKWP can be developed, maintaining semantic integrity and minimizing loss during the transformation processes. This would provide a solid and unified mathematical foundation for addressing complex cognitive and decision-making problems.

Summary:

This report provides an initial exploration of constructing a semantic mathematical framework for the DIKWP model, aiming to preserve the semantics of Data, Information, Knowledge, Wisdom, and Purpose within a mathematical system without the intermediary of natural language abstraction. The approach focuses on defining mathematical structures and transformation rules that inherently encapsulate the intended semantics, thereby maintaining semantic integrity and avoiding loss or distortion typically associated with language-to-math translations.

Conclusion:

By focusing on abstract interactions and transformations within the DIKWP framework, we reduce reliance on subjective natural language terms. Instead, we use a series of systematic transformations and constructs that differentiate "security" and "safety" based purely on their mathematical or logical properties:

• Security: Concerned with minimizing deviations in transformations, focusing on maintaining stable, non-volatile states by preserving system integrity.

• Safety: Ensures transformations remain within predefined safe boundaries, preventing any harmful deviations and maintaining operational safety.

This approach emphasizes that security focuses on maintaining a stable, controlled environment, while safety emphasizes preventing harmful conditions. This table-based abstract method allows us to understand the distinction between security and safety from the perspective of system interactions and transformations within the DIKWP framework.