Expression of Semantics in DIKWP 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

In the DIKWP Semantic Mathematics framework proposed by Prof. Yucong Duan, it is essential to explicitly express the specific semantics of Sameness, Difference, and Completeness as explicit targets, rather than referencing them implicitly. This document revisits the iterative derivation of these semantics, ensuring that each step explicitly identifies and utilizes the targeted semantics. By doing so, we provide a clear and transparent exposition of how complex semantic structures are built upon existing identified closures of the three fundamental semantics, strictly adhering to the principles of the DIKWP model.

The DIKWP Semantic Mathematics aims to model semantics in Artificial Intelligence by exclusively using the three fundamental semantics derived from the DIKWP model:

• Sameness (Data)

• Difference (Information)

• Completeness (Knowledge)

The key principle is that each step in the semantic derivation process should explicitly target and express these specific semantics. This ensures clarity and aligns with the objective of objectifying subjective semantics without introducing additional concepts.

When evaluating the Sameness semantics between two concepts (Concept A and Concept B), we begin by explicitly identifying all existing Sameness semantics associated with each concept.

• Concept A has Sameness Semantics Set SA={sA1,sA2,...,sAn}S_A = \{ s_{A1}, s_{A2}, ..., s_{An} \}SA​={sA1​,sA2​,...,sAn​}

• Concept B has Sameness Semantics Set SB={sB1,sB2,...,sBm}S_B = \{ s_{B1}, s_{B2}, ..., s_{Bm} \}SB​={sB1​,sB2​,...,sBm​}

We explicitly target the Sameness semantics that are common to both Concept A and Concept B:

• Shared Sameness Semantics SAB=SA∩SBS_{AB} = S_A \cap S_BSAB​=SA​∩SB​

This set SABS_{AB}SAB​ contains all Sameness semantics that both concepts share.

By explicitly using SABS_{AB}SAB​ as our target, we derive new Sameness semantics between Concept A and Concept B:

• New Sameness Semantics snews_{new}snew​ is explicitly defined as:

  snew=Sameness(SAB)s_{new} = \text{Sameness}(S_{AB})snew​=Sameness(SAB​)

This means that Concept A and Concept B are considered semantically the same concerning the specific semantics in SABS_{AB}SAB​.

We can further iterate by explicitly identifying additional Sameness semantics through their relationships with other concepts.

Example:

• Concept C shares Sameness semantics sC1s_{C1}sC1​ with Concept A.

• Concept D shares Sameness semantics sD1s_{D1}sD1​ with Concept B.

We explicitly check if sC1s_{C1}sC1​ and sD1s_{D1}sD1​ themselves share Sameness semantics, thus potentially deriving new Sameness semantics between Concept A and Concept B.

For Difference semantics between Concept A and Concept B, we explicitly list their known Difference semantics.

• Concept A has Difference Semantics Set DA={dA1,dA2,...,dAn}D_A = \{ d_{A1}, d_{A2}, ..., d_{An} \}DA​={dA1​,dA2​,...,dAn​}

• Concept B has Difference Semantics Set DB={dB1,dB2,...,dBm}D_B = \{ d_{B1}, d_{B2}, ..., d_{Bm} \}DB​={dB1​,dB2​,...,dBm​}

We explicitly target the Difference semantics that are unique to each concept:

• Unique Difference Semantics of A DA−unique=DA−DBD_{A-unique} = D_A - D_BDA−unique​=DA​−DB​

• Unique Difference Semantics of B DB−unique=DB−DAD_{B-unique} = D_B - D_ADB−unique​=DB​−DA​

By explicitly examining DA−uniqueD_{A-unique}DA−unique​ and DB−uniqueD_{B-unique}DB−unique​, we derive new Difference semantics that distinguish Concept A from Concept B.

• New Difference Semantics dnewd_{new}dnew​ is explicitly defined as:

  dnew=Difference(DA−unique,DB−unique)d_{new} = \text{Difference}(D_{A-unique}, D_{B-unique})dnew​=Difference(DA−unique​,DB−unique​)

This explicitly identifies how Concept A and Concept B differ based on their unique Difference semantics.

To further refine the Difference semantics, we can explicitly target and compare their Difference semantics with those of other concepts.

Example:

• Concept A differs from Concept E by Difference semantics dAEd_{AE}dAE​.

• Concept B differs from Concept E by Difference semantics dBEd_{BE}dBE​.

We explicitly analyze dAEd_{AE}dAE​ and dBEd_{BE}dBE​ to derive new Difference semantics between Concept A and Concept B.

Each concept has an existing Completeness semantics that encapsulates its entirety.

• Concept A has Completeness Semantics CAC_ACA​

• Concept B has Completeness Semantics CBC_BCB​

We explicitly assess the Completeness semantics of Concept A and Concept B to determine their scope and boundaries.

By explicitly integrating the Completeness semantics, we derive new Completeness semantics:

• Combined Completeness Semantics CABC_{AB}CAB​ is explicitly defined as:

  CAB=CA∪CBC_{AB} = C_A \cup C_BCAB​=CA​∪CB​

This represents a new Completeness semantics that encompasses both Concept A and Concept B.

We can iteratively refine the Completeness semantics by explicitly incorporating additional Sameness and Difference semantics identified in previous steps.

Example:

• From the Sameness semantics SABS_{AB}SAB​, we explicitly include shared attributes into CABC_{AB}CAB​.

• From the Difference semantics DA−uniqueD_{A-unique}DA−unique​ and DB−uniqueD_{B-unique}DB−unique​, we explicitly define the boundaries of CABC_{AB}CAB​.

At the Data level, we explicitly identify and target Sameness semantics.

• Explicit Targeting: For each data element, we explicitly list its Sameness semantics.

• Example: Data elements D1 and D2 share explicit Sameness semantics sD1D2s_{D1D2}sD1D2​.

At the Information level, we explicitly identify and target Difference semantics.

• Explicit Targeting: For each data element, we explicitly list its Difference semantics.

• Example: Data elements D1 and D2 have explicit Difference semantics dD1D2d_{D1D2}dD1D2​.

At the Knowledge level, we explicitly integrate Sameness and Difference semantics to form Completeness semantics.

• Explicit Targeting: We explicitly define the Completeness semantics for a concept.

• Example: Concept K has Completeness semantics CK=SK∪DKC_K = S_K \cup D_KCK​=SK​∪DK​, where SKS_KSK​ and DKD_KDK​ are the explicitly targeted Sameness and Difference semantics, respectively.

• Targeted Sameness Semantics: Explicitly identify initial Sameness semantics.

• Targeted Difference Semantics: Explicitly identify initial Difference semantics.

• Targeted Completeness Semantics: Explicitly form initial Completeness semantics.

• Explicitly Utilize Previously Derived Semantics: Use the explicitly identified semantics from previous iterations as the basis for new derivations.

• Maintain Explicit Expression: At each step, ensure that the specific semantics are explicitly stated and targeted.

• Sameness Semantics Set for Concept A:

  SA={sA1,sA2,...,sAn}S_A = \{ s_{A1}, s_{A2}, ..., s_{An} \}SA​={sA1​,sA2​,...,sAn​}

• Explicit Derivation of Shared Sameness Semantics:

  SAB=SA∩SBS_{AB} = S_A \cap S_BSAB​=SA​∩SB​

• New Sameness Semantics:

  sAB=Sameness(SAB)s_{AB} = \text{Sameness}(S_{AB})sAB​=Sameness(SAB​)

• Difference Semantics Set for Concept A:

  DA={dA1,dA2,...,dAn}D_A = \{ d_{A1}, d_{A2}, ..., d_{An} \}DA​={dA1​,dA2​,...,dAn​}

• Explicit Derivation of Unique Difference Semantics:

  DA−unique=DA−DBD_{A-unique} = D_A - D_BDA−unique​=DA​−DB​DB−unique=DB−DAD_{B-unique} = D_B - D_ADB−unique​=DB​−DA​

• New Difference Semantics:

  dAB=Difference(DA−unique,DB−unique)d_{AB} = \text{Difference}(D_{A-unique}, D_{B-unique})dAB​=Difference(DA−unique​,DB−unique​)

• Completeness Semantics for Concept A:

  CA=Completeness(SA,DA)C_A = \text{Completeness}(S_A, D_A)CA​=Completeness(SA​,DA​)

• Explicit Derivation of Combined Completeness Semantics:

  CAB=Completeness(SAB,DAB)C_{AB} = \text{Completeness}(S_{AB}, D_{AB})CAB​=Completeness(SAB​,DAB​)

• At each iteration $n$, we explicitly derive:

• Sameness Semantics S(n)S^{(n)}S(n)

• Difference Semantics D(n)D^{(n)}D(n)

• Completeness Semantics C(n)C^{(n)}C(n)

• Using the explicit targets from $n-1$:

  S(n)=Sameness(S(n−1))S^{(n)} = \text{Sameness}(S^{(n-1)})S(n)=Sameness(S(n−1))D(n)=Difference(D(n−1))D^{(n)} = \text{Difference}(D^{(n-1)})D(n)=Difference(D(n−1))C(n)=Completeness(S(n),D(n))C^{(n)} = \text{Completeness}(S^{(n)}, D^{(n)})C(n)=Completeness(S(n),D(n))

• Explicit Semantic Networks: Build networks where nodes and edges explicitly represent Sameness, Difference, and Completeness semantics.

• Transparent Learning Process: At each learning step, the AI system explicitly identifies and utilizes the targeted semantics.

• Explicit Semantic Parsing: Parse language by explicitly mapping words and phrases to the targeted semantics.

By explicitly expressing and targeting the specific semantics of Sameness, Difference, and Completeness at every step, the DIKWP Semantic Mathematics ensures clarity and precision in modeling semantics. This explicit approach aligns with Prof. Yucong Duan's vision and provides a robust foundation for AI systems to develop complex semantic understanding without introducing ambiguity or relying on implicit references.

1. 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 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. ".

2. Russell, S., & Norvig, P. (2021). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

3. Sowa, J. F. (2000). Knowledge Representation: Logical, Philosophical, and Computational Foundations. Brooks/Cole.

4. Smith, B. (2004). Beyond Concepts: Ontology as Reality Representation. In Formal Ontology in Information Systems.

I extend sincere gratitude to Prof. Yucong Duan for his guidance on ensuring the explicit expression of semantics in the DIKWP Semantic Mathematics framework. Appreciation is also given to colleagues in artificial intelligence and semantic modeling for their valuable feedback.

For further discussion on the explicit expression of semantics in DIKWP Semantic Mathematics, please contact [Author's Name] at [Contact Information].

Keywords: DIKWP Model, Semantic Mathematics, Explicit Semantics, Sameness, Difference, Completeness, Prof. Yucong Duan, Artificial Intelligence, Semantic Modeling, Knowledge Representation