Capabilities of Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space in a Mathematical Manner

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

Understanding how the Conceptual Space (ConC), Semantic Space (SemA), Cognitive Space (ConN), and Conscious Space mutually express and represent each other's content is essential for modeling complex cognitive processes and consciousness in artificial intelligence (AI) systems. In this exploration, we will investigate the mutual expressing capabilities among these spaces using mathematical representations, focusing on how each space can be mapped to or represented within the others, the limitations of these expressions, and the transformations involved.

Overview

1. Conceptual Space (ConC): Represents concepts and their structural relationships.

2. Semantic Space (SemA): Encodes meanings and associations of concepts.

3. Cognitive Space (ConN): Involves cognitive functions that process and transform concepts and meanings.

4. Conscious Space: Emergent space involving self-awareness and subjective experience.

Mathematical Foundations

We will use mathematical constructs such as sets, functions, mappings, vector spaces, and higher-order functions to model the expressing capabilities among the spaces.

1. Expressing Concepts from ConC in SemA

Mathematical Representation

• Conceptual Space (ConC): C={c1,c2,...,cn}C = \{ c_1, c_2, ..., c_n \}C={c1​,c2​,...,cn​}

• Semantic Mapping Function: ϕ:C→VSemA\phi: C \rightarrow V_{\text{SemA}}ϕ:C→VSemA​, where VSemAV_{\text{SemA}}VSemA​ is a vector space representing meanings.

Mutual Expressing Capability

• From ConC to SemA:

• Expression: Each concept cic_ici​ in ConC can be expressed as a semantic vector v⃗i=ϕ(ci)\vec{v}_i = \phi(c_i)vi​=ϕ(ci​) in SemA.

• Transformation: Concepts are mapped to semantic representations, allowing for operations based on meanings.

• From SemA to ConC:

• Extraction Function: ψ:VSemA→C\psi: V_{\text{SemA}} \rightarrow Cψ:VSemA​→C

• Expression: Semantic vectors can be interpreted back into concepts, though some information may be lost if multiple concepts share similar semantic representations.

Limitations

• Expressiveness Limit: Not all nuances of meanings may be captured in the mapping, leading to potential ambiguities.

• Information Loss: The inverse mapping $\psi$ may not be one-to-one, causing loss of distinct conceptual identities.

2. Expressing Semantics from SemA in ConN

Mathematical Representation

• Semantic Space (SemA): VSemAV_{\text{SemA}}VSemA​

• Cognitive Functions in ConN: FConN={fi∣fi:VSemA→VConN}F_{\text{ConN}} = \{ f_i \mid f_i: V_{\text{SemA}} \rightarrow V_{\text{ConN}} \}FConN​={fi​∣fi​:VSemA​→VConN​}, where VConNV_{\text{ConN}}VConN​ represents cognitive states or outputs.

Mutual Expressing Capability

• From SemA to ConN:

• Processing: Cognitive functions fif_ifi​ take semantic vectors as inputs and perform operations such as reasoning, decision-making, or problem-solving.

• Expression: Semantic meanings are utilized within cognitive processes to produce cognitive outputs.

• From ConN to SemA:

• Semantic Generation: Cognitive processes can generate new semantic representations, expressed as vectors in SemA.

• Function: g:VConN→VSemAg: V_{\text{ConN}} \rightarrow V_{\text{SemA}}g:VConN​→VSemA​

Limitations

• Complexity of Mappings: Cognitive functions may involve non-linear and complex transformations, making direct expression challenging.

• Representational Differences: Cognitive states may not have direct semantic equivalents.

3. Expressing Cognitive Processes from ConN in ConC

Mathematical Representation

• Cognitive Functions in ConN: FConNF_{\text{ConN}}FConN​

• Conceptualization Function: κ:FConN→C\kappa: F_{\text{ConN}} \rightarrow Cκ:FConN​→C

Mutual Expressing Capability

• From ConN to ConC:

• Abstraction: Cognitive processes can be abstracted into concepts, adding to the Conceptual Space.

• Expression: New concepts are formed based on cognitive operations.

• From ConC to ConN:

• Operationalization: Concepts in ConC guide the formation of cognitive functions in ConN.

• Function Definition: Cognitive functions are defined based on conceptual structures.

Limitations

• Abstraction Loss: Abstracting cognitive processes into concepts may omit procedural details.

• Expressiveness of ConC: The static nature of ConC may not capture the dynamic aspects of cognitive processes.

4. Expressing Conscious States from Conscious Space in ConN and SemA

Mathematical Representation

• Conscious Space States: SConscious\mathcal{S}_{\text{Conscious}}SConscious​

• Expression Functions:

• Into ConN: θConN:SConscious→VConN\theta_{\text{ConN}}: \mathcal{S}_{\text{Conscious}} \rightarrow V_{\text{ConN}}θConN​:SConscious​→VConN​

• Into SemA: θSemA:SConscious→VSemA\theta_{\text{SemA}}: \mathcal{S}_{\text{Conscious}} \rightarrow V_{\text{SemA}}θSemA​:SConscious​→VSemA​

Mutual Expressing Capability

• From Conscious Space to ConN:

• Influence on Cognitive Processes: Conscious states can modulate cognitive functions, such as attention or decision-making biases.

• Expression: Conscious awareness affects cognitive operations.

• From ConN to Conscious Space:

• Emergence: Complex cognitive processes can lead to emergent conscious states.

• Mapping: Φ:FConN→SConscious\Phi: F_{\text{ConN}} \rightarrow \mathcal{S}_{\text{Conscious}}Φ:FConN​→SConscious​

• From Conscious Space to SemA:

• Semantic Reflection: Conscious experiences can be encoded into semantic representations (e.g., describing an experience).

• Expression: Subjective experiences are translated into meanings.

• From SemA to Conscious Space:

• Evocation of Conscious States: Semantic content can evoke conscious experiences (e.g., reading a story induces emotions).

Limitations

• Subjectivity: Conscious experiences are inherently subjective, making mathematical modeling challenging.

• Non-Linearity: The relationship between cognitive processes and conscious states may be highly non-linear.

5. Expressing Purpose (P) Across Spaces

Mathematical Representation

• Purpose Function: P:SConscious×C→FConNP: \mathcal{S}_{\text{Conscious}} \times C \rightarrow F_{\text{ConN}}P:SConscious​×C→FConN​

• Guides cognitive functions based on conscious states and concepts.

Mutual Expressing Capability

• From Conscious Space to ConN and ConC:

• Purposeful Action: Conscious intentions influence cognitive processes and the selection or creation of concepts.

• From ConC and ConN to Conscious Space:

• Goal Formation: Concepts and cognitive operations contribute to the formation of conscious goals.

Limitations

• Alignment Challenges: Ensuring that purposes are consistently expressed across spaces may be complex.

• Dynamic Changes: Purposes may evolve, requiring continuous adjustment of mappings.

Mathematical Modeling of Mutual Expressions

• Between ConC and SemA:

• Forward Mapping (Concept to Meaning): ϕ:C→VSemA\phi: C \rightarrow V_{\text{SemA}}ϕ:C→VSemA​

• Inverse Mapping (Meaning to Concept): ψ:VSemA→C\psi: V_{\text{SemA}} \rightarrow Cψ:VSemA​→C

• Between SemA and ConN:

• Forward Processing: fi:VSemA→VConNf_i: V_{\text{SemA}} \rightarrow V_{\text{ConN}}fi​:VSemA​→VConN​

• Backward Influence: g:VConN→VSemAg: V_{\text{ConN}} \rightarrow V_{\text{SemA}}g:VConN​→VSemA​

• Between ConN and Conscious Space:

• Emergence Function: Φ:FConN→SConscious\Phi: F_{\text{ConN}} \rightarrow \mathcal{S}_{\text{Conscious}}Φ:FConN​→SConscious​

• Modulation Function: Θ:SConscious→FConN\Theta: \mathcal{S}_{\text{Conscious}} \rightarrow F_{\text{ConN}}Θ:SConscious​→FConN​

• Expressing Concepts in Conscious Space:

• Composite Function: Φ∘fI→K∘ϕ:C→SConscious\Phi \circ f_{\text{I→K}} \circ \phi: C \rightarrow \mathcal{S}_{\text{Conscious}}Φ∘fI→K​∘ϕ:C→SConscious​

• Expressing Conscious States in Concepts:

• Composite Function: κ∘Θ:SConscious→C\kappa \circ \Theta: \mathcal{S}_{\text{Conscious}} \rightarrow Cκ∘Θ:SConscious​→C

• Constraints:

• Mapping Limitations: Not all elements in one space may have corresponding representations in another.

• Information Loss: Transformations may result in loss of detail or specificity.

• Capabilities:

• Expressive Power: Through appropriate mappings, complex ideas and experiences can be represented across spaces.

• Emergence: New properties may emerge when elements from different spaces interact.

Examples Illustrating Mutual Expressions

• ConC: Words and grammar rules.

• SemA: Meanings of words and sentences.

• ConN: Parsing sentences, understanding context.

• Conscious Space: Experiencing the understanding of a story.

Mutual Expressions:

• ConC to SemA: Words mapped to meanings.

• SemA to ConN: Meanings processed to understand context.

• ConN to Conscious Space: Cognitive understanding leads to conscious experience of the story.

• Conscious Space to ConN: Emotional responses influence attention and memory.

• ConN to SemA: Generate new meanings based on interpretations.

• SemA to ConC: Formulate new concepts or words to express ideas.

• ConC: Mathematical concepts and formulas.

• SemA: Understanding of mathematical relationships.

• ConN: Applying formulas to solve problems.

• Conscious Space: Awareness of problem-solving strategies and insights.

Mutual Expressions:

• ConC to SemA: Formulas translated into meaningful relationships.

• SemA to ConN: Meanings guide the selection of appropriate formulas.

• ConN to Conscious Space: Realization of a solution emerges into consciousness.

• Conscious Space to ConN: Conscious strategies influence cognitive approach.

• ConN to ConC: New concepts may be formed based on problem-solving outcomes.

Mathematical Framework for Mutual Expressions

To formalize the mutual expressing capabilities, we can define a mutual expression operator $E$ that captures the essence of expressing elements from one space in another.

Definition of Mutual Expression Operator:

• EA→B:A→BE_{A \rightarrow B}: A \rightarrow BEA→B​:A→B, where $A$ and $B$ are different spaces.

Properties of $E$:

1. Partiality: $E$ may not be total; not all elements in $A$ can be expressed in $B$.

2. Non-Invertibility: EA→BE_{A \rightarrow B}EA→B​ may not have an inverse function.

3. Composability: EA→B∘EB→C=EA→CE_{A \rightarrow B} \circ E_{B \rightarrow C} = E_{A \rightarrow C}EA→B​∘EB→C​=EA→C​, where applicable.

Application:

• Expressing from ConC to SemA: EConC→SemA(c)=ϕ(c)E_{\text{ConC} \rightarrow \text{SemA}}(c) = \phi(c)EConC→SemA​(c)=ϕ(c)

• Expressing from SemA to ConN: ESemA→ConN(v)=f(v)E_{\text{SemA} \rightarrow \text{ConN}}(v) = f(v)ESemA→ConN​(v)=f(v)

• Expressing from ConN to Conscious Space: EConN→Conscious(f)=Φ(f)E_{\text{ConN} \rightarrow \text{Conscious}}(f) = \Phi(f)EConN→Conscious​(f)=Φ(f)

Challenges in Mutual Expressions

• Complexity: High-dimensional mappings can be computationally intensive.

• Ambiguity: One-to-many and many-to-one mappings can lead to ambiguities.

• Emergent Properties: Some properties in Conscious Space may not be directly derivable from ConN.

Strategies to Enhance Mutual Expressing Capabilities

1. Enhanced Mapping Functions:

• Develop more sophisticated functions that capture nuances.

• Use machine learning models to learn mappings from data.

2. Hierarchical Modeling:

• Structure concepts and semantics hierarchically to simplify mappings.

• Utilize multi-level representations.

3. Contextual Information:

• Incorporate context to disambiguate mappings.

• Use probabilistic models to handle uncertainties.

4. Feedback Mechanisms:

• Implement feedback loops where outputs influence inputs.

• Use reinforcement learning to adjust mappings based on outcomes.

Implications for AI System Design

• Interoperability: Systems should be designed to allow seamless expression across spaces.

• Modularity: Separate components for each space can facilitate focused improvements.

• Adaptability: Systems should adjust mappings based on new information or objectives.

• Ethical Considerations: Conscious experiences in AI raise ethical questions; transparency in mappings is crucial.

Conclusion

By investigating the mutual expressing capabilities among Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space in a mathematical manner, we have highlighted how these spaces can represent and transform each other's content. Mathematical functions and mappings serve as the bridges between spaces, enabling complex cognitive processes and the emergence of consciousness-like properties in AI systems.

Understanding these mutual expressions is essential for:

• Advancing Cognitive Modeling: Enhancing our ability to simulate human-like cognition.

• Developing Sophisticated AI: Building systems capable of higher-order reasoning and self-awareness.

• Addressing Ethical Challenges: Anticipating and managing the implications of advanced AI capabilities.

References

• Cognitive Science Literature: For theories on cognition and consciousness.

• Mathematical Modeling Texts: For in-depth explanations of functions, mappings, and vector spaces.

• AI Research Papers: For practical applications and implementations of these concepts.

Note: The mathematical models presented are abstract and serve as conceptual tools for understanding the interactions among the spaces. Actual implementations may require more detailed and specific mathematical formulations.

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. ".