Mathematical DIKWP Model through Cognitive Function Modeling

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)

Acknowledgment of Correction in Previous Papers.

The mapping for Wisdom should not be limited to D∗D^*D∗ but should encompass all elements of the DIKWP model. The correct mapping is:

W:{D,I,K,W,P}→{D,I,K,W,P}∗W: \{D, I, K, W, P\} \rightarrow \{D, I, K, W, P\}^*W:{D,I,K,W,P}→{D,I,K,W,P}∗

This means that the Wisdom function $W$ maps elements from the set of Data, Information, Knowledge, Wisdom, and Purpose to one or more elements within the same set, enabling interconversion among the semantics of the DIKWP model. This correction ensures that Wisdom can influence and transform any component within the DIKWP framework, aligning with the model's emphasis on interconnectedness and semantic evolution.

Introduction

In the DIKWP model, the Cognitive Space (ConN) is a multidimensional and dynamic environment where cognitive processing functions transform inputs into outputs. These cognitive functions, denoted as $R$, are crucial for simulating human cognitive processes within artificial intelligence systems. By examining the specific cognitive functions $R$ in the cognitive space, we can understand how the DIKWP model mathematically represents and simulates human cognition.

1. Cognitive Space (ConN) Overview

The Cognitive Space is defined as:

• Function Set: R={fConN1,fConN2,…,fConNn}R = \{f_{\text{ConN}_1}, f_{\text{ConN}_2}, \dots, f_{\text{ConN}_n}\}R={fConN1​​,fConN2​​,…,fConNn​​}

• Input Space: Inputi\text{Input}_iInputi​, consisting of various data or information sources.

• Output Space: Outputi\text{Output}_iOutputi​, containing higher cognitive products after processing.

Each cognitive function fConNif_{\text{ConN}_i}fConNi​​ transforms inputs into outputs, simulating specific cognitive processes such as perception, attention, memory, learning, reasoning, and decision-making.

2. Specific Cognitive Functions $R$ in the Cognitive Space

Let's examine the specific cognitive functions and how they simulate human cognitive processes.

Purpose: Simulates the human ability to perceive and interpret sensory information.

Mathematical Representation:

fPerception:Inputsensory→Outputperceptsf_{\text{Perception}}: \text{Input}_{\text{sensory}} \rightarrow \text{Output}_{\text{percepts}}fPerception​:Inputsensory​→Outputpercepts​

• Input: Raw sensory data ($D$)

• Output: Percepts, which are structured representations of sensory inputs.

Explanation:

• Process: Transforms raw data into meaningful patterns.

• Simulation: In AI, this could be image recognition or signal processing.

Purpose: Models the human ability to focus on specific information while ignoring others.

Mathematical Representation:

fAttention:(Percepts,Context)→Focused Perceptsf_{\text{Attention}}: (\text{Percepts}, \text{Context}) \rightarrow \text{Focused Percepts}fAttention​:(Percepts,Context)→Focused Percepts

• Input: Percepts and context information.

• Output: A subset of percepts that are the focus of processing.

Explanation:

• Process: Filters percepts based on relevance.

• Simulation: Attention mechanisms in neural networks, such as in Transformer models.

Purpose: Simulates storage and retrieval of information.

Mathematical Representation:

fMemory:{Store:(Input)→Memory StateRetrieve:(Query)→Retrieved Informationf_{\text{Memory}}: \left\{ \begin{array}{ll} \text{Store}: (\text{Input}) \rightarrow \text{Memory State} \\ \text{Retrieve}: (\text{Query}) \rightarrow \text{Retrieved Information} \end{array} \right.fMemory​:{Store:(Input)→Memory StateRetrieve:(Query)→Retrieved Information​

Explanation:

• Process: Stores inputs into memory and retrieves information when needed.

• Simulation: Database systems, knowledge bases, or memory networks in AI.

Purpose: Models the ability to learn from data and experiences.

Mathematical Representation:

fLearning:(Data,Prior Knowledge)→Updated Knowledgef_{\text{Learning}}: (\text{Data}, \text{Prior Knowledge}) \rightarrow \text{Updated Knowledge}fLearning​:(Data,Prior Knowledge)→Updated Knowledge

• Input: Data and prior knowledge ($K$).

• Output: Updated knowledge structures.

Explanation:

• Process: Adjusts knowledge based on new information.

• Simulation: Machine learning algorithms, such as supervised or unsupervised learning.

Purpose: Simulates logical reasoning and inference.

Mathematical Representation:

fReasoning:(Knowledge,Context)→Inferencesf_{\text{Reasoning}}: (\text{Knowledge}, \text{Context}) \rightarrow \text{Inferences}fReasoning​:(Knowledge,Context)→Inferences

• Input: Knowledge and context.

• Output: New inferences or conclusions.

Explanation:

• Process: Applies logical rules to derive new information.

• Simulation: Rule-based systems, logic programming, or probabilistic reasoning models.

Purpose: Models the process of making choices based on information and preferences.

Mathematical Representation:

fDecision:(Inferences,Preferences)→Decisionf_{\text{Decision}}: (\text{Inferences}, \text{Preferences}) \rightarrow \text{Decision}fDecision​:(Inferences,Preferences)→Decision

• Input: Inferences and preferences (which may include Wisdom $W$ and Purpose $P$).

• Output: A decision or action.

Explanation:

• Process: Evaluates options and selects the optimal one.

• Simulation: Decision trees, utility functions, or reinforcement learning.

Purpose: Reflects on and regulates one's own cognitive processes.

Mathematical Representation:

fMetacognition:(Cognitive States)→Regulation Actionsf_{\text{Metacognition}}: (\text{Cognitive States}) \rightarrow \text{Regulation Actions}fMetacognition​:(Cognitive States)→Regulation Actions

• Input: Current cognitive states or processes.

• Output: Adjustments to cognitive functions.

Explanation:

• Process: Monitors performance and makes adjustments.

• Simulation: Adaptive learning systems, self-tuning algorithms.

3. Modeling Cognitive Functions in the DIKWP Framework

Each cognitive function operates within the DIKWP framework, transforming and interacting with the components.

• Data ($D$): Raw inputs for perception and initial processing.

• Information ($I$): Processed data resulting from perception and attention.

• Knowledge ($K$): Stored information and learned patterns from memory and learning functions.

• Wisdom ($W$): Ethical and value-based considerations influencing reasoning and decision-making.

• Purpose ($P$): Goals guiding attention, learning, reasoning, and decision-making.

The overall cognitive process can be seen as a composition of functions:

Output=fDecision∘fReasoning∘fLearning∘fAttention∘fPerception(Input,P,W)\text{Output} = f_{\text{Decision}} \circ f_{\text{Reasoning}} \circ f_{\text{Learning}} \circ f_{\text{Attention}} \circ f_{\text{Perception}} (\text{Input}, P, W)Output=fDecision​∘fReasoning​∘fLearning​∘fAttention​∘fPerception​(Input,P,W)

• Composition: Each function's output becomes the next function's input.

• Inclusion of $P$ and $W$: Purpose and Wisdom influence multiple stages.

1. Perception:

   Percepts=fPerception(Data)\text{Percepts} = f_{\text{Perception}}(\text{Data})Percepts=fPerception​(Data)

2. Attention:

   Focused Percepts=fAttention(Percepts,P)\text{Focused Percepts} = f_{\text{Attention}}(\text{Percepts}, P)Focused Percepts=fAttention​(Percepts,P)

3. Learning and Memory Update:

   Updated Knowledge=fLearning(Focused Percepts,K)\text{Updated Knowledge} = f_{\text{Learning}}(\text{Focused Percepts}, K)Updated Knowledge=fLearning​(Focused Percepts,K)

4. Reasoning:

   Inferences=fReasoning(Updated Knowledge,W)\text{Inferences} = f_{\text{Reasoning}}(\text{Updated Knowledge}, W)Inferences=fReasoning​(Updated Knowledge,W)

5. Decision-Making:

   Decision=fDecision(Inferences,P,W)\text{Decision} = f_{\text{Decision}}(\text{Inferences}, P, W)Decision=fDecision​(Inferences,P,W)

6. Metacognition (Optional):

   Regulation Actions=fMetacognition(Cognitive States)\text{Regulation Actions} = f_{\text{Metacognition}}(\text{Cognitive States})Regulation Actions=fMetacognition​(Cognitive States)

4. Simulating Human Cognitive Processes

The cognitive functions $R$ are designed to emulate human cognitive abilities. Let's explore how they simulate specific human processes.

• Human Process: Sensory input is filtered and focused based on relevance and interest.

• Simulation: AI systems use sensors and attention mechanisms to prioritize important data.

• Human Process: Experiences are stored and recalled; learning adjusts understanding.

• Simulation: Machine learning algorithms update models based on new data, storing patterns.

• Human Process: Logical thinking and problem-solving based on knowledge.

• Simulation: AI applies algorithms to deduce new information or predict outcomes.

• Human Process: Choices are made considering preferences, ethics, and goals.

• Simulation: AI uses optimization algorithms and utility functions to select actions.

• Human Process: Reflecting on one's own thinking to improve performance.

• Simulation: AI systems monitor their own processes to adjust parameters dynamically.

5. Mathematical Modeling of Cognitive Functions

Each cognitive function can be formally defined using mathematical constructs.

• Perception Function Space:

  fPerception∈FPerception:D→Pf_{\text{Perception}} \in \mathcal{F}_{\text{Perception}}: \mathcal{D} \rightarrow \mathcal{P}fPerception​∈FPerception​:D→P

  Where $\mathcal{D}$ is the data space, and $\mathcal{P}$ is the percept space.

• Attention Function Space:

  fAttention∈FAttention:P×C→P′f_{\text{Attention}} \in \mathcal{F}_{\text{Attention}}: \mathcal{P} \times \mathcal{C} \rightarrow \mathcal{P}'fAttention​∈FAttention​:P×C→P′

  Where $\mathcal{C}$ is the context space, and P′⊆P\mathcal{P}' \subseteq \mathcal{P}P′⊆P.

• Learning Function Space:

  fLearning∈FLearning:P′×K→K′f_{\text{Learning}} \in \mathcal{F}_{\text{Learning}}: \mathcal{P}' \times \mathcal{K} \rightarrow \mathcal{K}'fLearning​∈FLearning​:P′×K→K′

  Where $\mathcal{K}$ is the knowledge space.

• Reasoning Function Space:

  fReasoning∈FReasoning:K′×C→If_{\text{Reasoning}} \in \mathcal{F}_{\text{Reasoning}}: \mathcal{K}' \times \mathcal{C} \rightarrow \mathcal{I}fReasoning​∈FReasoning​:K′×C→I

  Where $\mathcal{I}$ is the inference space.

• Decision Function Space:

  fDecision∈FDecision:I×P×W→Af_{\text{Decision}} \in \mathcal{F}_{\text{Decision}}: \mathcal{I} \times \mathcal{P} \times \mathcal{W} \rightarrow \mathcal{A}fDecision​∈FDecision​:I×P×W→A

  Where $\mathcal{A}$ is the action or decision space, and $\mathcal{W}$ is the wisdom space.

• Perception: Convolutional neural networks for image recognition.

• Attention: Attention layers in Transformer models for sequence processing.

• Learning: Gradient descent algorithms in machine learning.

• Reasoning: Logical inference engines or probabilistic models.

• Decision-Making: Reinforcement learning algorithms optimizing a reward function.

• Metacognition: Meta-learning algorithms that adjust learning strategies.

6. Applications in Artificial Intelligence

• Perception: Tokenization and embedding of text data.

• Attention: Attention mechanisms to focus on relevant words.

• Learning: Language models learning grammar and semantics.

• Reasoning: Understanding context and making inferences.

• Decision-Making: Generating responses in dialogue systems.

• Perception: Sensor data processing for environment understanding.

• Attention: Focusing on critical obstacles or targets.

• Learning: Adapting to new environments through experience.

• Reasoning: Planning paths and actions.

• Decision-Making: Executing movements based on goals.

• Perception: Gathering data from various sources.

• Attention: Highlighting significant trends or anomalies.

• Learning: Updating models based on new data.

• Reasoning: Analyzing options and forecasting outcomes.

• Decision-Making: Recommending actions to users.

7. Cognitive Function Modeling in the DIKWP Context

The cognitive functions $R$ are integral to the interconversion among the semantics of the DIKWP model.

• Data to Information: Perception and attention functions process data into information.

• Information to Knowledge: Learning functions abstract information into knowledge.

• Knowledge to Wisdom: Reasoning functions integrate ethical considerations, guided by Wisdom.

• Wisdom Influencing Purpose: Metacognition and decision-making align actions with Purpose.

• Purpose Guiding Cognitive Functions: Purpose influences attention, learning priorities, and decision criteria.

• Transformation Functions:

  TDI:D→IT_{DI}: D \rightarrow ITDI​:D→ITIK:I→KT_{IK}: I \rightarrow KTIK​:I→KTKW:K→WT_{KW}: K \rightarrow WTKW​:K→WTWP:W→PT_{WP}: W \rightarrow PTWP​:W→PTPD:P→DT_{PD}: P \rightarrow DTPD​:P→D

  (And so on for all combinations.)

• Cognitive Functions as Transformation Operators:

  Each fConNif_{\text{ConN}_i}fConNi​​ can be seen as implementing a TXYT_{XY}TXY​ function.

8. Conclusion

The cognitive functions $R$ in the cognitive space of the DIKWP model are designed to simulate human cognitive processes mathematically. By defining specific functions that model perception, attention, memory, learning, reasoning, decision-making, and metacognition, the DIKWP framework provides a comprehensive and interconnected system for semantic processing and cognitive simulation.

These functions enable:

• Interconversion among DIKWP Components: Seamless transformation between Data, Information, Knowledge, Wisdom, and Purpose.

• Simulation of Human Cognition: Emulation of human cognitive abilities in AI systems.

• Cognitive Consistency: Alignment of AI processing with human cognitive patterns.

• Ethical and Purposeful AI: Incorporation of Wisdom and Purpose ensures AI actions align with human values and goals.

Implications for AI Development:

• Advanced Cognitive Modeling: Provides a foundation for creating AI systems with human-like understanding and reasoning.

• Enhanced Interaction: Improves AI's ability to interpret and respond to human inputs effectively.

• Ethical Considerations: Embeds ethical reasoning within AI decision-making processes.

• Goal Alignment: Ensures AI systems act in accordance with specified purposes and societal values.

Further Exploration

To deepen the understanding of cognitive function modeling within the DIKWP framework:

• Neuroscientific Correlations: Explore how these functions relate to neural processes in the human brain.

• Algorithmic Implementations: Develop specific algorithms that embody these cognitive functions.

• Experimental Validation: Test AI systems built upon this model in real-world scenarios.

• Ethical Frameworks: Integrate ethical theories into the Wisdom component for robust ethical AI.

• Adaptive Learning: Enhance metacognitive functions for AI systems to self-improve over time.

References for Further Reading

1. Cognitive Science Foundations: "Cognitive Psychology: A Student's Handbook" by Eysenck and Keane.

2. Artificial Intelligence Principles: "Artificial Intelligence: A Modern Approach" by Russell and Norvig.

3. Machine Learning Algorithms: "Pattern Recognition and Machine Learning" by Christopher Bishop.

4. Attention Mechanisms in AI: "Attention Is All You Need" by Vaswani et al.

5. Ethics in AI: "Ethics of Artificial Intelligence" edited by S. Matthew Liao.

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. ".