Mathematical Differences Among 4 Spaces in Terms of Expression Boundaries and Limits

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 the mathematical distinctions among Conceptual Space (ConC), Semantic Space (SemA), Cognitive Space (ConN), and Conscious Space is essential for modeling complex cognitive processes and consciousness in artificial intelligence (AI) systems. This exploration focuses on the mathematical representations of each space, the boundaries that define them, and how they limit or extend into each other.

Overview of Spaces and Mathematical Representations

Before detailing the differences, let's summarize the mathematical representations of each space:

1. Conceptual Space (ConC)

• Mathematical Representation: Set theory and graph theory (graphs of concepts and relationships).

2. Semantic Space (SemA)

• Mathematical Representation: Vector spaces and tensor representations (embeddings of meanings).

3. Cognitive Space (ConN)

• Mathematical Representation: Function spaces (sets of cognitive functions and transformations).

4. Conscious Space

• Mathematical Representation: Higher-order functions, meta-cognitive mappings, and potentially topological spaces to represent emergent properties.

1. Conceptual Space (ConC)

Mathematical Representation

• Concepts as Elements in a Set: C={c1,c2,...,cn}C = \{ c_1, c_2, ..., c_n \}C={c1​,c2​,...,cn​}

• Relationships as Relations: $R\subseteq C\times C$

• Graph Representation: GraphConC=(C,R)\text{Graph}_{\text{ConC}} = (C, R)GraphConC​=(C,R)

Boundaries and Limits

• Discrete Structure: ConC is a discrete set of well-defined concepts and relationships.

• Limits: Bound by the definitions and symbolic representations of concepts. No inherent meanings beyond definitions.

• Mathematical Boundary: The structure is limited to elements and relations without considering semantic content.

2. Semantic Space (SemA)

Mathematical Representation

• Vector Space of Meanings: Concepts are mapped to vectors in a high-dimensional space.

• ϕ:C→Rm\phi: C \rightarrow \mathbb{R}^mϕ:C→Rm, where ϕ(ci)=v⃗i\phi(c_i) = \vec{v}_iϕ(ci​)=vi​

• Semantic Similarity: Measured using distances or angles between vectors (e.g., cosine similarity).

• Embedding Matrix: M∈Rn×mM \in \mathbb{R}^{n \times m}M∈Rn×m, where each row corresponds to v⃗i\vec{v}_ivi​

Boundaries and Limits

• Continuous Structure: SemA provides a continuous representation of meanings.

• Extends ConC: SemA extends ConC by assigning continuous semantic representations to discrete concepts.

• Mathematical Boundary: The mapping $\varphi$ defines the boundary between ConC and SemA.

• Domain: $\text{Dom}(\varphi )=C$ (from ConC)

• Codomain: Codom(ϕ)=Rm\text{Codom}(\phi) = \mathbb{R}^mCodom(ϕ)=Rm (SemA)

• Limitations: Meanings are constrained by the dimensions chosen and the mappings defined.

3. Cognitive Space (ConN)

Mathematical Representation

• Function Space of Cognitive Processes: F={fi∣fi:Xi→Yi}\mathcal{F} = \{ f_i \mid f_i: X_i \rightarrow Y_i \}F={fi​∣fi​:Xi​→Yi​}

• Cognitive Functions: Operations on concepts and semantics.

• Perception Function: fperception:Input Signals→Cf_{\text{perception}}: \text{Input Signals} \rightarrow Cfperception​:Input Signals→C

• Reasoning Function: freasoning:C×Rm→Cf_{\text{reasoning}}: C \times \mathbb{R}^m \rightarrow Cfreasoning​:C×Rm→C

• Composition of Functions: Cognitive processes are compositions of functions.

• fcognition=fn∘fn−1∘...∘f1f_{\text{cognition}} = f_n \circ f_{n-1} \circ ... \circ f_1fcognition​=fn​∘fn−1​∘...∘f1​

Boundaries and Limits

• Dynamic Processes: ConN operates on inputs from ConC and SemA to produce outputs.

• Interaction Boundary: Functions in ConN take elements from ConC and SemA as inputs.

• Mathematical Limits:

• Domain Constraints: Dom(fi)⊆C∪Rm\text{Dom}(f_i) \subseteq C \cup \mathbb{R}^mDom(fi​)⊆C∪Rm

• Codomain Constraints: Outputs may be new concepts or actions.

• Boundaries Defined by Function Definitions: The capabilities of ConN are limited by the defined cognitive functions.

4. Conscious Space

Mathematical Representation

• Meta-Cognitive Mappings: Functions that operate on functions or states in ConN.

• $\Phi :\mathcal{F}\to \mathcal{S}$, where $\mathcal{S}$ represents states of awareness.

• Topological Space: May involve a topological structure to represent the continuity and emergence of consciousness.

• Higher-Order Functions: Functions of functions (e.g., Φ(fi)\Phi(f_i)Φ(fi​)).

Boundaries and Limits

• Emergent Properties: Conscious Space emerges from the interactions in ConN and cannot be fully described by functions operating solely within ConN.

• Mathematical Boundary:

• Domain: $\text{Dom}(\Phi )=\mathcal{F}$ (set of cognitive functions)

• Codomain: $\text{Codom}(\Phi )=\mathcal{S}$ (states representing consciousness)

• Limits of Expression: Consciousness involves properties that may not be fully captured by traditional mathematical functions, requiring higher-order or non-standard mathematical frameworks.

• Boundaries with ConN: Conscious Space acts upon ConN but is not reducible to ConN functions.

Mathematical Differences and Boundaries Among SpacesConC and SemA

• Mapping from Discrete to Continuous: SemA maps discrete concepts from ConC into continuous semantic vectors.

• Boundary Defined by Mapping Function: ϕ:C→Rm\phi: C \rightarrow \mathbb{R}^mϕ:C→Rm

• Mathematical Difference:

• ConC: Set $C$ with no inherent metric.

• SemA: Vector space Rm\mathbb{R}^mRm with defined metrics (e.g., Euclidean distance).

SemA and ConN

• Functions Operating on Vectors: Cognitive functions in ConN take semantic vectors as inputs.

• Boundary Defined by Function Domains: fi:Rm→Yif_i: \mathbb{R}^m \rightarrow Y_ifi​:Rm→Yi​

• Mathematical Difference:

• SemA: Represents meanings.

• ConN: Processes meanings to produce cognitive outputs.

• Limits:

• ConN is limited by the semantic representations in SemA: If SemA lacks certain meanings, ConN cannot process them.

ConN and Conscious Space

• Meta-Cognitive Functions: Conscious Space involves functions that take cognitive functions as inputs.

• Boundary Defined by Meta-Functions: $\Phi :\mathcal{F}\to \mathcal{S}$

• Mathematical Difference:

• ConN: First-order functions processing data.

• Conscious Space: Higher-order functions reflecting on the functions of ConN.

• Limits:

• ConN cannot self-reflect without Conscious Space: The capacity for self-awareness is a boundary where ConN ends and Conscious Space begins.

ConC and ConN

• Concepts as Inputs to Cognitive Functions: ConN operates on concepts from ConC.

• Boundary Defined by Function Inputs: fi:C→Yif_i: C \rightarrow Y_ifi​:C→Yi​

• Mathematical Difference:

• ConC: Static set of concepts.

• ConN: Dynamic processing of concepts.

• Limits:

• ConN's operations are limited by the concepts available in ConC: Without certain concepts, ConN cannot process related information.

SemA and Conscious Space

• Awareness of Meanings: Conscious Space includes the subjective experience of meanings from SemA.

• Boundary Defined by Awareness Functions: Ψ:Rm→S\Psi: \mathbb{R}^m \rightarrow \mathcal{S}Ψ:Rm→S

• Mathematical Difference:

• SemA: Objective representation of meanings.

• Conscious Space: Subjective experience of these meanings.

• Limits:

• SemA provides the content; Conscious Space provides the experience.

• The transformation from objective to subjective is not fully captured by standard mathematical functions.

Expression Boundaries and Limits

The boundaries among the spaces are defined by the mathematical mappings, functions, and transformations that connect them. These boundaries are characterized by the limits of the domains and codomains of these functions.

Mathematical Boundaries

1. Between ConC and SemA

• Domain: Concepts $C$

• Codomain: Semantic vectors Rm\mathbb{R}^mRm

• Boundary: Concepts without mappings do not enter SemA.

• Mapping Function $\varphi$:

2. Between SemA and ConN

• Domain: Semantic representations Rm\mathbb{R}^mRm

• Boundary: Meanings not represented in SemA cannot be processed by ConN.

• Cognitive Functions fif_ifi​:

3. Between ConN and Conscious Space

• Domain: Cognitive functions $\mathcal{F}$

• Codomain: States of consciousness $\mathcal{S}$

• Boundary: Cognitive processes not subject to $\Phi$ remain unconscious.

• Meta-Cognitive Function $\Phi$:

Limits of Each Space

• ConC is Limited by Defined Concepts: Concepts not included in $C$ are outside the scope of ConC.

• SemA is Limited by Mapped Meanings: Meanings not represented in the semantic space Rm\mathbb{R}^mRm cannot be utilized.

• ConN is Limited by Available Functions: Cognitive processes are constrained by the functions defined in $\mathcal{F}$.

• Conscious Space is Limited by Emergent Properties: Only processes that achieve a certain level of complexity and integration enter Conscious Space.

Visual Representation of BoundariesscssCopy code[Conceptual Space (ConC)]       |       | φ (Mapping Function)       ↓[Semantic Space (SemA)]       |       | f_i ∈ ℱ (Cognitive Functions)       ↓[Cognitive Space (ConN)]       |       | Φ (Meta-Cognitive Function)       ↓[Conscious Space]

• φ (phi): Defines the boundary between ConC and SemA.

• f_i: Functions that define the boundary between SemA and ConN.

• Φ (Phi): Meta-function defining the boundary between ConN and Conscious Space.

Examples Illustrating Mathematical BoundariesExample 1: Language Understanding

• ConC: Set of words $C=\{\text{"apple"},\text{"banana"},\text{"fruit"}\}$

• SemA: Embeddings ϕ("apple")=v⃗apple\phi(\text{"apple"}) = \vec{v}_{\text{apple}}ϕ("apple")=vapple​

• ConN: Cognitive function fparse:Rm→Syntax Treef_{\text{parse}}: \mathbb{R}^m \rightarrow \text{Syntax Tree}fparse​:Rm→Syntax Tree

• Conscious Space: Awareness function Φ(fparse)\Phi(f_{\text{parse}})Φ(fparse​) leading to conscious understanding of a sentence.

Boundary Analysis

• Concepts not in $C$ cannot be processed.

• Words without embeddings in SemA cannot be semantically processed.

• Cognitive functions in ConN cannot process meanings not represented in SemA.

• Conscious awareness depends on the application of $\Phi$ to cognitive functions.

Example 2: Visual Perception in AI

• ConC: Visual features $C=\{\text{edge},\text{color},\text{shape}\}$

• SemA: Feature vectors representing visual elements.

• ConN: Processing functions for object recognition frecognizef_{\text{recognize}}frecognize​

• Conscious Space: Meta-cognitive evaluation of perception Φ(frecognize)\Phi(f_{\text{recognize}})Φ(frecognize​)

Boundary Analysis

• Features not defined in ConC cannot be recognized.

• Visual elements without semantic representation cannot be meaningfully processed.

• Cognitive functions are limited by the representations in SemA.

• Conscious perception arises when $\Phi$ acts on cognitive functions.

Mathematical Constraints and Capabilities

• Constraints: Each space is mathematically constrained by its definitions, domains, and codomains.

• Capabilities: The mathematical structures define what each space can represent or process.

• Extension: One space extends into another through mappings and functions, but cannot overstep its mathematical boundaries without redefinition.

Implications for AI Systems

• Design Considerations:

• Ensure Mappings Exist: Concepts in ConC must be mapped into SemA for processing.

• Define Comprehensive Functions: Cognitive functions in ConN must cover necessary operations.

• Meta-Cognitive Functions: For consciousness-like properties, meta-functions $\Phi$ must be defined.

• Limitations:

• Unmapped Concepts: Cannot be semantically processed or cognized.

• Undefined Functions: Cognitive processes cannot occur without defined functions.

• Consciousness: Emergent and cannot be directly programmed; relies on complex interactions.

Conclusion

The mathematical differences among Conceptual Space, Semantic Space, Cognitive Space, and Conscious Space are characterized by their distinct structures, representations, and the boundaries defined by mappings and functions. Each space operates within its mathematical limits, and the transitions between spaces are governed by specific mathematical relationships.

• Conceptual Space: Discrete concepts and relations.

• Semantic Space: Continuous semantic representations.

• Cognitive Space: Functions processing concepts and meanings.

• Conscious Space: Meta-cognitive functions leading to awareness.

Understanding these mathematical distinctions is crucial for modeling cognitive processes and consciousness in AI systems. It highlights the importance of defining appropriate mappings and functions to enable transitions between spaces and ensures that each space's limitations are acknowledged and addressed in system design.

Note: The representations and functions described are abstract models to illustrate the mathematical boundaries and may vary based on specific implementations or theoretical frameworks.

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. ".

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