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Mathematizing the DIKWP Whitebox Test Using DIKWP×DIKW（初学者版）

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Mathematizing the DIKWP Whitebox Test Using DIKWP×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

To mathematize the DIKWP whitebox tests for the cognitive development stages (Months 12–24), we'll represent the internal processes and transformations within the DIKWP framework using mathematical expressions. The DIKWP model comprises five modules:

D: Data
I: Information
K: Knowledge
W: Wisdom
P: Purpose

By using DIKWP×DIKWP consecutive expressions, we model the iterative and recursive nature of cognitive development, where outputs from one stage serve as inputs for the next, reflecting the continuous growth and refinement of cognitive abilities.

Mathematical Representation of DIKWP ModulesNotation and Definitions

Let $D$ represent raw data input.
$I = f_D(D)$ : Information derived from data through processing function $f_D$ .
$K = f_I(I)$ : Knowledge constructed from information using function $f_I$ .
$W = f_K(K)$ : Wisdom obtained by applying knowledge via function $f_K$ .
$P = f_W(W)$ : Purpose alignment achieved through wisdom using function $f_W$ .

Each function $f$ represents the transformation process between modules.

Consecutive Expressions

The consecutive application of DIKWP can be expressed as:

$P = f_W(f_K(f_I(f_D(D))))$

For iterative development, we model the process over time $t$ with stages $n$ :

$P_n = f_W(f_K(f_I(f_D(D_n))))$

Stage 1: Months 12–15Test Case 1.1: Responding to Simple WordsMathematical Modeling

Data Acquisition (D):

$"Dada"D_{\text{audio}} = \text{Audio signal of words "Mama" and "Dada"}$
$caregiversD_{\text{visual}} = \text{Images of respective caregivers}$

Information Processing (I):

$recognitionI_{\text{audio}} = f_D(D_{\text{audio}}) = \text{Phoneme extraction and pattern recognition}$
$recognitionI_{\text{visual}} = f_D(D_{\text{visual}}) = \text{Facial feature recognition}$

Knowledge Construction (K):

$facesK_{\text{association}} = f_I(I_{\text{audio}}, I_{\text{visual}}) = \text{Association between words and faces}$

Wisdom Application (W):

$caregiverW_{\text{response}} = f_K(K_{\text{association}}) = \text{Directed attention or gesture towards caregiver}$

Purpose Alignment (P):

$recognitionP_{\text{communication}} = f_W(W_{\text{response}}) = \text{Effective communication of recognition}$

Expression:

$Pcommunication=fW(fK(fI(fD(Daudio,Dvisual))))P_{\text{communication}} = f_W(f_K(f_I(f_D(D_{\text{audio}}, D_{\text{visual}}))))$

Test Case 3.1: Cause and Effect UnderstandingMathematical Modeling

Data Acquisition (D):

$interactionD_{\text{toy}} = \text{Sensory data from toy interaction}$

Information Processing (I):

$sound)I_{\text{action}} = f_D(D_{\text{toy}}) = \text{Mapping actions to sensory feedback (e.g., shaking produces sound)}$

Knowledge Construction (K):

$relationshipK_{\text{cause-effect}} = f_I(I_{\text{action}}) = \text{Understanding the cause-effect relationship}$

Wisdom Application (W):

$effectW_{\text{repetition}} = f_K(K_{\text{cause-effect}}) = \text{Intentional repetition of action to achieve desired effect}$

Purpose Alignment (P):

$behaviorP_{\text{exploration}} = f_W(W_{\text{repetition}}) = \text{Engagement in exploratory behavior}$

Expression:

$Pexploration=fW(fK(fI(fD(Dtoy))))P_{\text{exploration}} = f_W(f_K(f_I(f_D(D_{\text{toy}}))))$

Stage 2: Months 15–18Test Case 1.2: Two-Word CombinationsMathematical Modeling

Data Acquisition (D):

$toy)D_{\text{needs}} = \text{Internal states indicating desires (e.g., wanting a toy)}$

Information Processing (I):

$vocabularyI_{\text{lexicon}} = f_D(D_{\text{needs}}) = \text{Selection of relevant words from vocabulary}$

Knowledge Construction (K):

$car")K_{\text{syntax}} = f_I(I_{\text{lexicon}}) = \text{Formation of syntactic structures (e.g., "want car")}$

Wisdom Application (W):

$needsW_{\text{communication}} = f_K(K_{\text{syntax}}) = \text{Verbal expression of needs}$

Purpose Alignment (P):

$communicationP_{\text{need fulfillment}} = f_W(W_{\text{communication}}) = \text{Achieving desired outcome through communication}$

Expression:

$fulfillment=fW(fK(fI(fD(Dneeds))))P_{\text{need fulfillment}} = f_W(f_K(f_I(f_D(D_{\text{needs}}))))$

Test Case 3.1: Shape SorterMathematical Modeling

Data Acquisition (D):

$sorterD_{\text{shapes}} = \text{Visual and tactile data from shapes and sorter}$

Information Processing (I):

$angles)I_{\text{properties}} = f_D(D_{\text{shapes}}) = \text{Extraction of shape properties (edges, angles)}$

Knowledge Construction (K):

$correspondenceK_{\text{matching}} = f_I(I_{\text{properties}}) = \text{Understanding of shape-hole correspondence}$

Wisdom Application (W):

$holesW_{\text{problem-solving}} = f_K(K_{\text{matching}}) = \text{Placement of shapes into correct holes}$

Purpose Alignment (P):

$taskP_{\text{achievement}} = f_W(W_{\text{problem-solving}}) = \text{Satisfaction from completing the task}$

Expression:

$Pachievement=fW(fK(fI(fD(Dshapes))))P_{\text{achievement}} = f_W(f_K(f_I(f_D(D_{\text{shapes}}))))$

Stage 3: Months 18–21Test Case 1.2: Simple SentencesMathematical Modeling

Data Acquisition (D):

$running)D_{\text{experience}} = \text{Perceived actions or events (e.g., dog running)}$

Information Processing (I):

$actionsI_{\text{semantic}} = f_D(D_{\text{experience}}) = \text{Identification of subjects and actions}$

Knowledge Construction (K):

$sentencesK_{\text{language}} = f_I(I_{\text{semantic}}) = \text{Grammatical structuring of sentences}$

Wisdom Application (W):

$runs")W_{\text{expression}} = f_K(K_{\text{language}}) = \text{Verbalizing observations ("Dog runs")}$

Purpose Alignment (P):

$othersP_{\text{communication}} = f_W(W_{\text{expression}}) = \text{Sharing information with others}$

Expression:

$Pcommunication=fW(fK(fI(fD(Dexperience))))P_{\text{communication}} = f_W(f_K(f_I(f_D(D_{\text{experience}}))))$

Test Case 3.1: Memory RecallMathematical Modeling

Data Acquisition (D):

$visitD_{\text{event}} = \text{Sensory data from the park visit}$

Information Processing (I):

$memoryI_{\text{encoding}} = f_D(D_{\text{event}}) = \text{Encoding experiences into memory}$

Knowledge Construction (K):

$detailsK_{\text{memory}} = f_I(I_{\text{encoding}}) = \text{Storage of event details}$

Wisdom Application (W):

$eventsW_{\text{retrieval}} = f_K(K_{\text{memory}}) = \text{Recalling and articulating past events}$

Purpose Alignment (P):

$storytellingP_{\text{sharing}} = f_W(W_{\text{retrieval}}) = \text{Engaging in social interaction through storytelling}$

Expression:

$Psharing=fW(fK(fI(fD(Devent))))P_{\text{sharing}} = f_W(f_K(f_I(f_D(D_{\text{event}}))))$

Stage 4: Months 21–24Test Case 2.1: Retelling StoriesMathematical Modeling

Data Acquisition (D):

$storyD_{\text{story}} = \text{Auditory data from listening to the story}$

Information Processing (I):

$elementsI_{\text{comprehension}} = f_D(D_{\text{story}}) = \text{Understanding narrative elements}$

Knowledge Construction (K):

$sequenceK_{\text{narrative}} = f_I(I_{\text{comprehension}}) = \text{Internal representation of story sequence}$

Wisdom Application (W):

$storyW_{\text{retelling}} = f_K(K_{\text{narrative}}) = \text{Reconstructing and verbalizing the story}$

Purpose Alignment (P):

$othersP_{\text{communication}} = f_W(W_{\text{retelling}}) = \text{Demonstrating understanding and engaging with others}$

Expression:

$Pcommunication=fW(fK(fI(fD(Dstory))))P_{\text{communication}} = f_W(f_K(f_I(f_D(D_{\text{story}}))))$

Test Case 4.1: Role-PlayingMathematical Modeling

Data Acquisition (D):

$teacher)D_{\text{social}} = \text{Observations of social roles (e.g., doctor, teacher)}$

Information Processing (I):

$rolesI_{\text{role}} = f_D(D_{\text{social}}) = \text{Understanding functions and behaviors associated with roles}$

Knowledge Construction (K):

$characteristicsK_{\text{role-play}} = f_I(I_{\text{role}}) = \text{Internalization of role characteristics}$

Wisdom Application (W):

$playW_{\text{simulation}} = f_K(K_{\text{role-play}}) = \text{Enacting roles through pretend play}$

Purpose Alignment (P):

$empathyP_{\text{social development}} = f_W(W_{\text{simulation}}) = \text{Enhancing social understanding and empathy}$

Expression:

$development=fW(fK(fI(fD(Dsocial))))P_{\text{social development}} = f_W(f_K(f_I(f_D(D_{\text{social}}))))$

Consecutive DIKWP×DIKWP Expressions

To represent the iterative nature of cognitive development, we can model consecutive applications of the DIKWP cycle, where the output of one cycle feeds into the next:

First Iteration (n):
$P_n = f_W(f_K(f_I(f_D(D_n))))$
Second Iteration (n+1), using $P_n$ as input:
$P_{n+1} = f_W(f_K(f_I(f_D(D_{n+1} + P_n))))$

This expression indicates that the system's previous purpose-driven actions ( $P_n$ ) influence new data acquisition and subsequent processing, reflecting learning and development over time.

Mathematical Representation of Developmental Progression

By expressing cognitive development as a function of time $t$ , we model the cumulative effect of consecutive DIKWP cycles:

$f_W(f_K(f_I(f_D(D(t) + \int_{0}^{t} P(\tau) d\tau))))$

Here, $∫0tP(τ)dτ\int_{0}^{t} P(\tau) d\tau$ represents the accumulated purposeful actions up to time $t$ , influencing current data processing.

Handling the 3-No Problem Mathematically

The system addresses incomplete, imprecise, and inconsistent data through functions that generate hypotheses and adjust processing:

Hypothesis Generation Function (H):

For incomplete data:
$H_{\text{incomplete}}(D)$

Data Abstraction Function (A):

For imprecise data:
$D^{''} = A (D^{'})$

Consistency Adjustment Function (C):

For inconsistent data:
$D^{'''} = C (D^{''})$

Incorporating these into the DIKWP process:

$P = f_W(f_K(f_I(f_D(C(A(D + H(D)))))))$

Example: Test Case Incorporating the 3-No ProblemTest Case: Vocabulary Expansion with Incomplete DataMathematical Modeling

Data Acquisition with Incomplete Data (D):

$\text{Partial auditory signals of new words}$

Hypothesis Generation (H):

$H_{\text{incomplete}}(D) = \text{Predicted missing phonemes}$

Data Abstraction (A):

$\text{Focus on key phonetic features}$

Information Processing (I):

$f_D(D'') = \text{Extraction of word meanings}$

Knowledge Construction (K):

$f_I(I) = \text{Integration into existing vocabulary}$

Wisdom Application (W):

$f_K(K) = \text{Usage of new word in context}$

Purpose Alignment (P):

$f_W(W) = \text{Effective communication and learning}$

Expression:

$f_W(f_K(f_I(f_D(A(D + H_{\text{incomplete}}(D))))))$

Conclusion

By mathematizing the DIKWP whitebox tests using DIKWP×DIKWP consecutive expressions, we model the cognitive development of the artificial consciousness system in a precise and structured manner. This mathematical representation captures:

The sequential and iterative nature of cognitive processes.
The integration of previous experiences into current processing.
The handling of incomplete, imprecise, and inconsistent data through specific functions.
The continuous alignment of actions with purpose, reflecting the growth of consciousness.

This approach provides a formal framework to analyze and understand the system's internal workings, facilitating further development and refinement of artificial cognitive architectures.

Note: The functions $f_D, f_I, f_K, f_W, H, A, C$ are abstract representations of complex processes within the system. In a practical implementation, these would correspond to specific algorithms or neural network operations designed to perform the transformations between modules.

References for Further Reading

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
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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