Semantic Mathematics Operations in DIKWP 

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

In the DIKWP model proposed by Professor Yucong Duan, the Semantic Space (SemA) plays a crucial role in representing and processing meanings within cognitive systems, including artificial intelligence. Understanding the formation and evolution of SemA is essential for comprehending how meanings are constructed, interpreted, and transformed in the DIKWP framework.

This exploration will delve into:

1. Definition and Structure of the Semantic Space (SemA)

2. Formation of SemA

3. Evolution of SemA

4. Role of SemA in Meaning-Making

5. Mathematical Modeling of SemA Dynamics

6. Applications in Artificial Intelligence

7. Implications for Semantic Evolution and Cognitive Consistency

1. Definition and Structure of the Semantic Space (SemA)

The Semantic Space, denoted as SemA, is a conceptual framework representing the network of semantic associations between concepts within the cognitive system of an entity (human or artificial). It is formed through experiences, knowledge accumulation, and cognitive processing.

Mathematical Representation:

SemA=(VSemA,ESemA)\text{SemA} = (V_{\text{SemA}}, E_{\text{SemA}})SemA=(VSemA​,ESemA​)

• VSemAV_{\text{SemA}}VSemA​: Set of semantic units (nodes), such as words, concepts, or symbols.

• ESemAE_{\text{SemA}}ESemA​: Set of edges representing semantic relationships between nodes.

• Nodes (VSemAV_{\text{SemA}}VSemA​): Represent semantic units that carry meaning. These can be words, phrases, concepts, or any unit of meaning.

• Edges (ESemAE_{\text{SemA}}ESemA​): Represent semantic relationships, such as synonymy, antonymy, hypernymy, meronymy, causal relations, or associations.

Properties:

• Dynamic: SemA is not static; it evolves over time as new experiences and knowledge are acquired.

• Multidimensional: Relationships can have various types and strengths.

• Contextual: Meanings and associations can change depending on context.

2. Formation of SemA

SemA is initially formed through:

• Experiences: Direct interactions with the environment.

• Learning: Acquiring knowledge through education, observation, or instruction.

• Cognitive Processing: Internal processes that organize and interpret information.

• Definition: Linking new semantic units with existing ones based on perceived relationships.

• Process:

• Co-occurrence: Semantic units that frequently appear together become associated.

• Similarity: Units with similar features or contexts are linked.

• Definition: Grouping semantic units into categories based on shared properties.

• Process:

• Feature Extraction: Identifying attributes of units.

• Cluster Formation: Creating categories where units share common features.

• Definition: Combining multiple concepts to form new meanings.

• Process:

• Blending: Merging concepts to create novel ideas.

• Metaphor and Analogy: Understanding one concept in terms of another.

Association Strengths:

• Weighting Edges: Assigning weights to edges based on the strength of association.

  ESemA={(u,v,wuv)∣u,v∈VSemA,wuv∈R+}E_{\text{SemA}} = \{ (u, v, w_{uv}) \mid u, v \in V_{\text{SemA}}, w_{uv} \in \mathbb{R}^+ \}ESemA​={(u,v,wuv​)∣u,v∈VSemA​,wuv​∈R+}

  Where wuvw_{uv}wuv​ represents the weight (strength) of the association between nodes $u$ and $v$.

Formation Functions:

• Association Function (fAssocf_{\text{Assoc}}fAssoc​):

  fAssoc:VSemA×VSemA→ESemAf_{\text{Assoc}}: V_{\text{SemA}} \times V_{\text{SemA}} \rightarrow E_{\text{SemA}}fAssoc​:VSemA​×VSemA​→ESemA​

• Categorization Function (fCatf_{\text{Cat}}fCat​):

  fCat:VSemA→Cf_{\text{Cat}}: V_{\text{SemA}} \rightarrow CfCat​:VSemA​→C

  Where $C$ is a set of categories.

• Integration Function (fIntf_{\text{Int}}fInt​):

  fInt:VSemAn→VSemAf_{\text{Int}}: V_{\text{SemA}}^n \rightarrow V_{\text{SemA}}fInt​:VSemAn​→VSemA​

  Combines multiple nodes to form a new semantic unit.

3. Evolution of SemA

• New Experiences: Introduction of new semantic units and relationships.

• Learning and Adaptation: Updating associations based on feedback.

• Contextual Changes: Shifts in context that alter meanings.

• Addition: Incorporating new nodes and edges.

• Deletion: Removing outdated or irrelevant nodes and edges.

• Weight Updates: Adjusting edge weights based on new information.

• Relationship Changes: Changing the type of relationship between nodes.

• Re-Categorization: Changing the grouping of semantic units.

• Structural Changes: Altering the topology of the semantic network.

Temporal Dynamics:

• Time-Dependent SemA:

  SemA(t)=(VSemA(t),ESemA(t))\text{SemA}(t) = (V_{\text{SemA}}(t), E_{\text{SemA}}(t))SemA(t)=(VSemA​(t),ESemA​(t))

• Evolution Functions:

• Addition Function (fAddf_{\text{Add}}fAdd​):

  fAdd:SemA(t)→SemA(t+Δt)f_{\text{Add}}: \text{SemA}(t) \rightarrow \text{SemA}(t + \Delta t)fAdd​:SemA(t)→SemA(t+Δt)

  Adds new nodes and edges.

• Deletion Function (fDelf_{\text{Del}}fDel​):

  fDel:SemA(t)→SemA(t+Δt)f_{\text{Del}}: \text{SemA}(t) \rightarrow \text{SemA}(t + \Delta t)fDel​:SemA(t)→SemA(t+Δt)

  Removes nodes and edges.

• Update Function (fUpdf_{\text{Upd}}fUpd​):

  fUpd:ESemA(t)→ESemA(t+Δt)f_{\text{Upd}}: E_{\text{SemA}}(t) \rightarrow E_{\text{SemA}}(t + \Delta t)fUpd​:ESemA​(t)→ESemA​(t+Δt)

  Updates weights or types of edges.

4. Role of SemA in Meaning-Making

SemA is foundational in constructing meanings by:

• Contextualizing Semantic Units: Meanings arise from the position of a unit within the network.

• Facilitating Inference: Relationships allow for the inference of new meanings.

• Supporting Disambiguation: Network structure helps resolve ambiguities.

• Activation Spreading: Activation flows through the network to retrieve related meanings.

• Path Traversal: Navigation through edges to explore associations.

• Combining Meanings: Building complex meanings from simpler units.

• Syntax and Semantics: Structures in SemA correspond to syntactic constructions.

• Dynamic Contextualization: Meanings can change based on context, which is represented as subgraphs or activation patterns within SemA.

• Polysemy and Homonymy: Multiple meanings are managed through network structures.

5. Mathematical Modeling of SemA Dynamics

Applying graph theory and network science to analyze SemA.

• Degree Centrality: Importance of a node based on connections.

  Degree(v)=∑u∈VSemAδ(u,v)\text{Degree}(v) = \sum_{u \in V_{\text{SemA}}} \delta(u, v)Degree(v)=u∈VSemA​∑​δ(u,v)

  Where $\delta (u,v)=1$ if there is an edge between $u$ and $v$, else 0.

• Betweenness Centrality: Nodes that act as bridges.

• Clustering Algorithms: Identifying modules or clusters within SemA.

• Implications: Communities may represent semantic fields or topics.

• Path-Based Measures: Semantic similarity based on shortest paths.

• Edge Weight Consideration: Weights influence the calculation of similarity.

• Modeling the change of activation levels over time.

  dAv(t)dt=−λAv(t)+∑u∈VSemAwuvAu(t)\frac{dA_v(t)}{dt} = -\lambda A_v(t) + \sum_{u \in V_{\text{SemA}}} w_{uv} A_u(t)dtdAv​(t)​=−λAv​(t)+u∈VSemA​∑​wuv​Au​(t)

  Where Av(t)A_v(t)Av​(t) is the activation level of node $v$ at time $t$, $\lambda$ is a decay constant, and wuvw_{uv}wuv​ are edge weights.

• Random Walks: Modeling the probabilistic traversal of the network.

• Applications: Understanding how meanings diffuse through SemA.

• Word2Vec, GloVe: Techniques that map semantic units to vector spaces based on co-occurrence.

• Mathematical Representation:

  Embed:VSemA→Rn\text{Embed}: V_{\text{SemA}} \rightarrow \mathbb{R}^nEmbed:VSemA​→Rn

• Definition: Neural networks that operate on graph structures.

• Function:

  GNN(v)=σ(∑u∈N(v)wuvhu+b)\text{GNN}(v) = \sigma \left( \sum_{u \in \mathcal{N}(v)} w_{uv} h_u + b \right)GNN(v)=σ​u∈N(v)∑​wuv​hu​+b​

  Where huh_uhu​ is the feature vector of node $u$, $\mathcal{N}(v)$ is the neighborhood of $v$, $\sigma$ is an activation function, wuvw_{uv}wuv​ are trainable weights, and $b$ is a bias term.

6. Applications in Artificial Intelligence

• Semantic Parsing: Mapping sentences to semantic representations in SemA.

• Disambiguation: Using context in SemA to resolve word senses.

• Meaning Preservation: Ensuring that translations maintain the same semantic content.

• Alignment of SemA: Mapping between semantic spaces of different languages.

• Ontologies: Formal representations of knowledge domains within SemA.

• Inference Engines: Drawing conclusions based on relationships in SemA.

• Semantic Search: Retrieving information based on meaning rather than keywords.

• Personalization: Adapting recommendations based on the user's SemA.

• Simulating Human Thought: AI systems that mimic human semantic processing.

• Adaptive Learning: Updating SemA based on interactions and feedback.

7. Implications for Semantic Evolution and Cognitive Consistency

• Adaptability: SemA's ability to incorporate new meanings ensures AI systems remain relevant.

• Cultural and Social Influences: SemA reflects changes in language and societal norms.

• Alignment with Human Cognition: Modeling SemA after human semantic networks enhances AI-human interaction.

• Reducing Ambiguity: Consistent semantic processing minimizes misunderstandings.

• Bias and Fairness: Monitoring and adjusting SemA to prevent the propagation of biases.

• Transparency: Understanding the structure of SemA aids in explaining AI decisions.

8. Conclusion

The Semantic Space (SemA) within the DIKWP model is a dynamic and integral component that facilitates meaning-making through the formation and evolution of semantic associations. By representing semantic units and their relationships mathematically, SemA enables the modeling of complex cognitive processes and supports advanced artificial intelligence applications.

Key Takeaways:

• Formation: SemA is formed through mechanisms like association, categorization, and conceptual integration.

• Evolution: It evolves over time due to new experiences, learning, and contextual changes.

• Role in Meaning-Making: SemA is central to constructing and interpreting meanings, supporting inference, and managing ambiguity.

• Mathematical Modeling: Graph theory, network analysis, and machine learning techniques provide tools for modeling SemA dynamics.

• Applications: SemA is crucial in NLP, knowledge representation, information retrieval, and cognitive computing.

• Semantic Evolution and Cognitive Consistency: Understanding SemA's dynamics ensures AI systems evolve semantically in ways consistent with human cognition, promoting effective communication and ethical AI development.

Further Exploration

To deepen the understanding of SemA within the DIKWP framework:

• Semantic Network Analysis: Study large-scale semantic networks to identify patterns and structures.

• Cross-Linguistic Semantics: Explore how SemA can handle multiple languages and cultural contexts.

• Temporal Semantics: Investigate how meanings change over time and how SemA models this evolution.

• Integration with Cognitive Functions: Examine how SemA interacts with cognitive functions in the cognitive space.

• Ethical AI Development: Develop strategies to monitor and mitigate biases within SemA.

References for Further Reading

1. Semantic Networks in AI: "Semantic Networks and the Representation of Knowledge" by John F. Sowa.

2. Graph Theory Applications: "Networks, Crowds, and Markets: Reasoning About a Highly Connected World" by David Easley and Jon Kleinberg.

3. Natural Language Semantics: "Semantics" by Kate Kearns.

4. Graph Neural Networks: "Graph Neural Networks: A Review of Methods and Applications" by Zhou et al.

5. Dynamic Semantics: "Meaning in Flux: The Dynamics of Language" by Peter Gärdenfors.

6. 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

7. 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. ".