DIKWP Semantic Mathematics: A Step-by-Step Handbook 

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

DIKWP Semantic Mathematics is a comprehensive mathematical framework designed to model and process the cognitive transformations between the five core components: Data (D), Information (I), Knowledge (K), Wisdom (W), and Purpose (P). This framework provides precise mathematical representations and operations that enable systematic handling of semantic content across different cognitive spaces, ensuring consistency and interoperability in artificial intelligence and cognitive systems.

1. Core Components of DIKWP Semantic Mathematics

Each component of the DIKWP model is defined within the context of three cognitive spaces:

• Concept Space (ConC)

• Cognitive Space (ConN)

• Semantic Space (SemA)

We will detail each component, their definitions, and mathematical representations step by step.

1.1 Data (D) Conceptualization

Definition:

• In Concept Space (ConC): Data concepts represent specific facts or observations confirmed by their semantic correspondence in the cognitive entity's semantic space. Data concepts are recognized by sharing the same semantic attributes.

Mathematical Representation:

• Semantic Attribute Set:

  S={f1,f2,...,fn}S = \{f_1, f_2, ..., f_n\}S={f1​,f2​,...,fn​}

  Where fif_ifi​ represents a semantic feature of the data.

• Data Concept Set:

  $$D=\{d\mid d\text{ shares }S\}$$

  Each data element $d\in D$ is an instance that shares the semantic attribute set $S$.

Processing in Cognitive Space (ConN):

• Cognitive processes extract shared semantics to label data concepts, unifying them based on corresponding shared semantics.

Example:

• Recognizing different sheep as instances of the concept "sheep" by identifying shared semantic attributes such as four legs, wool, and herbivorous diet.

1.2 Information (I) Conceptualization

Definition:

• In Concept Space (ConC): Information concepts correspond to one or more "differences" in semantics within cognition.

Mathematical Representation:

• Information Semantics Processing Function:

  FI:X→YF_I: X \rightarrow YFI​:X→Y

• $X$: Input DIKWP content semantics (Data, Information, Knowledge, Wisdom, Purpose).

• $Y$: Output new DIKWP content semantics.

• Purpose-Driven Processing:

  Cognitive entities use their purpose to process input semantics, identifying differences and generating new semantic associations.

Processing in Cognitive Space (ConN):

• Identifying differences between input content and existing cognitive objects.

• Generating new information semantics through cognitive purpose-driven processing.

Example:

• Observing a parking lot full of cars (Data) and noticing the differences in parking times, owners, or models to generate specific information about each car.

1.3 Knowledge (K) Conceptualization

Definition:

• In Concept Space (ConC): Knowledge concepts correspond to one or more "complete" semantics, representing structured understanding formed through abstraction and generalization.

Mathematical Representation:

• Knowledge Graph:

  $$K=(N,E)$$

• N={n1,n2,...,nk}N = \{n_1, n_2, ..., n_k\}N={n1​,n2​,...,nk​}: Set of concept nodes.

• E={e1,e2,...,em}E = \{e_1, e_2, ..., e_m\}E={e1​,e2​,...,em​}: Set of edges representing relationships between concepts.

• Edges Representation:

  es=(ni,nj,r)e_s = (n_i, n_j, r)es​=(ni​,nj​,r)

• ni,nj∈Nn_i, n_j \in Nni​,nj​∈N: Concepts.

• $r$: Semantic relationship between nin_ini​ and njn_jnj​.

Processing in Cognitive Space (ConN):

• Formation of knowledge rules through higher-order cognitive activities.

• Assigning "complete" semantics to observations, forming systematic understanding.

Example:

• Generalizing that "all swans are white" based on observed instances, assigning completeness to partial observations.

1.4 Wisdom (W) Conceptualization

Definition:

• In Concept Space (ConC): Wisdom corresponds to information regarding ethics, social morals, and human values, integrating DIKWP content to guide decision-making.

Mathematical Representation:

• Decision Function:

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

• D∗D^*D∗: Optimal decision.

• Wisdom function $W$ processes all DIKWP components to generate decisions.

Processing in Cognitive Space (ConN):

• Considering ethical, moral, and feasibility factors.

• Constructing a human-centered value system to guide decisions.

Example:

• Deciding on a medical treatment plan by integrating patient data (D), medical information (I), clinical knowledge (K), ethical considerations (W), and treatment goals (P).

1.5 Purpose (P) Conceptualization

Definition:

• In Concept Space (ConC): Purpose represents stakeholders' understanding of a phenomenon (Input) and the objectives they aim to achieve (Output).

Mathematical Representation:

• Purpose Tuple:

  $$P=(\text{Input},\text{Output})$$

• Both Input and Output consist of DIKWP content semantics.

• Transformation Function:

  $$T:\text{Input}\to \text{Output}$$

Processing in Cognitive Space (ConN):

• Goal-oriented transformation of DIKWP content semantics.

• Learning and adapting to achieve predefined goal semantics.

Example:

• In an AI planning system, the purpose is to transform the current state (Input) into a desired goal state (Output) through a sequence of actions.

2. Conceptual Spaces in DIKWP Semantic Mathematics

The DIKWP components operate within three interconnected spaces, each with specific roles and mathematical representations.

2.1 Concept Space (ConC)

Definition:

• The cognitive representation of concepts, including definitions, features, and relationships, expressed through language and symbols.

Mathematical Representation:

• Graph Structure:

  GraphConC=(VConC,EConC)\text{Graph}_{\text{ConC}} = (V_{\text{ConC}}, E_{\text{ConC}})GraphConC​=(VConC​,EConC​)

• VConCV_{\text{ConC}}VConC​: Set of concept nodes.

• EConCE_{\text{ConC}}EConC​: Set of edges representing relationships.

• Operations:

• Query:

  Q(VConC,EConC,q)→{v1,v2,...,vm}Q(V_{\text{ConC}}, E_{\text{ConC}}, q) \rightarrow \{v_1, v_2, ..., v_m\}Q(VConC​,EConC​,q)→{v1​,v2​,...,vm​}

• Add:

  Add(VConC,v)\text{Add}(V_{\text{ConC}}, v)Add(VConC​,v)

• Update:

  Update(VConC,v,A(v))\text{Update}(V_{\text{ConC}}, v, A(v))Update(VConC​,v,A(v))

Role:

• Organizes and categorizes DIKWP components.

• Facilitates mapping between components through conceptual relationships.

2.2 Cognitive Space (ConN)

Definition:

• A dynamic processing environment where DIKWP components are transformed into understanding and actions through cognitive processing functions.

Mathematical Representation:

• Function Set:

  R={fConN1,fConN2,...,fConNn}R = \{ f_{\text{ConN}_1}, f_{\text{ConN}_2}, ..., f_{\text{ConN}_n} \}R={fConN1​​,fConN2​​,...,fConNn​​}

• Each function fConNi:Inputi→Outputif_{\text{ConN}_i}: \text{Input}_i \rightarrow \text{Output}_ifConNi​​:Inputi​→Outputi​

• Sub-steps of Cognitive Processing:

  fConNi=fConNi(n)∘fConNi(n−1)∘...∘fConNi(1)f_{\text{ConN}_i} = f_{\text{ConN}_i}^{(n)} \circ f_{\text{ConN}_i}^{(n-1)} \circ ... \circ f_{\text{ConN}_i}^{(1)}fConNi​​=fConNi​(n)​∘fConNi​(n−1)​∘...∘fConNi​(1)​

• Cognitive processing is broken down into sub-functions representing different processing stages.

Role:

• Processes DIKWP components through functions like Data preprocessing, pattern recognition, reasoning, and decision-making.

• Transforms inputs from the external environment into cognitive outputs.

2.3 Semantic Space (SemA)

Definition:

• The network of semantic associations between concepts within the cognitive subject's mind, including relationships and dependencies.

Mathematical Representation:

• Graph Structure:

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

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

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

• Operations:

• Query:

  Query(VSemA,ESemA,q)→{v1,v2,...,vm}\text{Query}(V_{\text{SemA}}, E_{\text{SemA}}, q) \rightarrow \{v_1, v_2, ..., v_m\}Query(VSemA​,ESemA​,q)→{v1​,v2​,...,vm​}

• Add:

  Add(VSemA,v)\text{Add}(V_{\text{SemA}}, v)Add(VSemA​,v)

• Update:

  Update(ESemA,v,v′,e)\text{Update}(E_{\text{SemA}}, v, v', e)Update(ESemA​,v,v′,e)

Role:

• Represents semantic relationships and meanings.

• Supports semantic consistency in DIKWP transformations.

3. DIKWP Graphs and Their Interactions

Each DIKWP component can be represented as a graph, capturing the relationships and transformations between components.

3.1 Data Graph (DG)

Definition:

• A graph representing Data concepts and their relationships.

Mathematical Representation:

• Data Graph:

  DG=(VD,ED)\text{DG} = (V_D, E_D)DG=(VD​,ED​)

• VDV_DVD​: Set of Data nodes.

• EDE_DED​: Set of edges representing relationships.

Interactions:

• Receives inputs and updates from other graphs via transformation functions:

• TID,TKD,TWD,TPDT_{\text{ID}}, T_{\text{KD}}, T_{\text{WD}}, T_{\text{PD}}TID​,TKD​,TWD​,TPD​

Example:

• Data nodes representing sensor readings, updated by information processing (IG) and knowledge integration (KG).

3.2 Information Graph (IG)

Definition:

• A graph representing Information concepts and their semantic relationships.

Mathematical Representation:

• Information Graph:

  IG=(VI,EI)\text{IG} = (V_I, E_I)IG=(VI​,EI​)

• VIV_IVI​: Set of Information nodes.

• EIE_IEI​: Set of edges based on semantic relationships.

Interactions:

• Generated from DG via TDIT_{\text{DI}}TDI​.

• Adjusted by KG, WG, and PG via:

• TKI,TWI,TPIT_{\text{KI}}, T_{\text{WI}}, T_{\text{PI}}TKI​,TWI​,TPI​

Example:

• Information nodes representing extracted features or patterns from Data, connected based on semantic differences.

3.3 Knowledge Graph (KG)

Definition:

• A graph representing Knowledge concepts and their relationships.

Mathematical Representation:

• Knowledge Graph:

  KG=(VK,EK)\text{KG} = (V_K, E_K)KG=(VK​,EK​)

• VKV_KVK​: Set of Knowledge nodes.

• EKE_KEK​: Set of edges representing conceptual relationships.

Interactions:

• Formed from IG via TIKT_{\text{IK}}TIK​.

• Influences DG, IG, and WG via:

• TKD,TKI,TKWT_{\text{KD}}, T_{\text{KI}}, T_{\text{KW}}TKD​,TKI​,TKW​

Example:

• Knowledge nodes representing concepts like "gravity" or "evolution," connected through logical or causal relationships.

3.4 Wisdom Graph (WG)

Definition:

• A graph representing Wisdom concepts, integrating ethical and value-based considerations.

Mathematical Representation:

• Wisdom Graph:

  WG=(VW,EW)\text{WG} = (V_W, E_W)WG=(VW​,EW​)

• VWV_WVW​: Set of Wisdom nodes.

• EWE_WEW​: Set of edges representing ethical relationships.

Interactions:

• Formed from KG via TKWT_{\text{KW}}TKW​.

• Feeds back to KG and IG via:

• TWK,TWIT_{\text{WK}}, T_{\text{WI}}TWK​,TWI​

Example:

• Wisdom nodes representing ethical guidelines or moral principles influencing decision-making processes.

3.5 Purpose Graph (PG)

Definition:

• A graph representing goals and the strategies to achieve them.

Mathematical Representation:

• Purpose Graph:

  PG=(VP,EP)\text{PG} = (V_P, E_P)PG=(VP​,EP​)

• VPV_PVP​: Set of Purpose nodes (goals, objectives).

• EPE_PEP​: Set of edges representing strategies or steps.

Interactions:

• Formed from DG, IG, KG, and WG via:

• TDP,TIP,TKP,TWPT_{\text{DP}}, T_{\text{IP}}, T_{\text{KP}}, T_{\text{WP}}TDP​,TIP​,TKP​,TWP​

• Influences DG, IG, and KG via:

• TPD,TPI,TPKT_{\text{PD}}, T_{\text{PI}}, T_{\text{PK}}TPD​,TPI​,TPK​

Example:

• Purpose nodes representing desired outcomes, connected through planned actions or policies.

3.6 Interactions between Graphs

• Transformation Functions:

  TXY:YG→XG,X,Y∈{D,I,K,W,P},X≠YT_{XY}: Y_G \rightarrow X_G, \quad X, Y \in \{D, I, K, W, P\}, \quad X \ne YTXY​:YG​→XG​,X,Y∈{D,I,K,W,P},X=Y

• Content Models and Cognitive Models:

• Function $f$ transforms mappings between graphs:

  $$f:G\times G\to G$$

• Triplet Mapping:

• $S$: Semantic level.

• $C$: Conceptual level.

• $I$: Instance level.

• Each graph $g\in G$ is a triplet mapping:

  $$g:S\times C\times I$$

4. Mathematical Formulations of DIKWP Transformations

We will now detail the mathematical models and transformation functions between each DIKWP component.

4.1 Data to Information Transformation (D → I)

Objective:

• Convert Data concepts into Information by identifying differences and forming new semantic associations.

Transformation Function:

• Information Semantics Processing Function:

  FI:D→IF_I: D \rightarrow IFI​:D→I

Process:

• Cognitive entities use their purpose to process Data semantics, identifying differences, and generating Information semantics.

Example:

• From sensor Data (D), identify anomalies or trends (I) that represent meaningful information.

4.2 Information to Knowledge Transformation (I → K)

Objective:

• Organize Information into structured Knowledge, capturing "complete" semantics.

Transformation Function:

• Knowledge Formation Function:

  FK:I→KF_K: I \rightarrow KFK​:I→K

Process:

• Abstract and generalize Information to form Knowledge concepts.

• Construct Knowledge Graphs representing relationships and rules.

Example:

• From observed information about planetary motions (I), formulate the laws of gravitation (K).

4.3 Knowledge to Wisdom Transformation (K → W)

Objective:

• Integrate Knowledge with values and ethics to guide decision-making.

Transformation Function:

• Wisdom Decision Function:

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

Process:

• Apply ethical considerations and human values to Knowledge.

• Generate optimal decisions that align with moral principles.

Example:

• Using medical Knowledge (K) and ethical guidelines (W) to decide on patient treatment plans (D*).

4.4 Wisdom to Purpose Alignment (W → P)

Objective:

• Define objectives based on Wisdom to guide cognitive processes.

Transformation Function:

• Purpose Transformation Function:

  $$T:\text{Input}\to \text{Output}$$

Process:

• Align actions and decisions with overarching goals derived from Wisdom.

• Set goals that reflect ethical considerations and desired outcomes.

Example:

• Developing organizational strategies (P) based on ethical business practices (W).

4.5 Purpose to Data Influence (P → D)

Objective:

• Influence Data collection and interpretation based on Purpose.

Transformation Function:

• Purpose Feedback Function:

  TPD:P→DT_{\text{PD}}: P \rightarrow DTPD​:P→D

Process:

• Adjust Data gathering methods to align with goals.

• Prioritize Data relevant to achieving objectives.

Example:

• Collecting specific market research Data (D) to support a new product launch (P).

5. Detailed Step-by-Step Explanation

We will now walk through each DIKWP component in detail, explaining their mathematical representations and providing examples.

5.1 Step 1: Data Conceptualization

Mathematical Representation:

• Semantic Attribute Set:

  S={f1,f2,...,fn}S = \{f_1, f_2, ..., f_n\}S={f1​,f2​,...,fn​}

• Data Concept Set:

  $$D=\{d\mid d\text{ shares }S\}$$

Processing:

1. Observation: Collect raw data elements $d$.

2. Semantic Matching: Identify shared semantic attributes $S$ among $d$.

3. Concept Formation: Group $d$ into Data concepts based on $S$.

Example:

• Collecting various fruit samples (apples, oranges, bananas) and grouping them into the concept "fruit" by identifying shared attributes like being edible, seed-bearing, and sweet.

5.2 Step 2: Information Conceptualization

Mathematical Representation:

• Information Semantics Processing Function:

  FI:D→IF_I: D \rightarrow IFI​:D→I

Processing:

1. Identify Differences: Analyze Data concepts to find differences.

2. Purpose-Driven Processing: Use cognitive purpose to interpret differences.

3. Generate Information: Form new Information semantics representing these differences.

Example:

• Noticing that some fruits are citrus (lemons, oranges) and others are not, creating information about fruit categories.

5.3 Step 3: Knowledge Conceptualization

Mathematical Representation:

• Knowledge Graph:

  $$K=(N,E)$$

• Edges Representation:

  es=(ni,nj,r)e_s = (n_i, n_j, r)es​=(ni​,nj​,r)

Processing:

1. Abstract Concepts: Generalize Information to form higher-level concepts.

2. Establish Relationships: Define relationships between concepts.

3. Build Knowledge Graph: Represent concepts and relationships in a structured graph.

Example:

• Creating a taxonomy of fruits, understanding botanical relationships, and representing this knowledge in a graph.

5.4 Step 4: Wisdom Conceptualization

Mathematical Representation:

• Decision Function:

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

Processing:

1. Integrate Values: Incorporate ethical and moral considerations.

2. Analyze Knowledge: Evaluate Knowledge in the context of values.

3. Make Decisions: Generate optimal decisions aligning with Wisdom.

Example:

• Deciding to promote organic farming practices after considering environmental impact (Wisdom) and agricultural knowledge.

5.5 Step 5: Purpose Conceptualization

Mathematical Representation:

• Purpose Tuple:

  $$P=(\text{Input},\text{Output})$$

• Transformation Function:

  $$T:\text{Input}\to \text{Output}$$

Processing:

1. Define Goals: Specify desired outcomes (Output).

2. Assess Current State: Understand the current situation (Input).

3. Plan Actions: Develop strategies to transform Input into Output.

Example:

• Setting a goal to reduce carbon emissions (Output) and planning policies to shift from fossil fuels to renewable energy (Transformation).

6. Integration of Spaces and Graphs

The DIKWP components operate within and across the Concept Space, Cognitive Space, and Semantic Space, interconnected through their respective graphs.

6.1 Interactions within Spaces

• Concept Space: Organizes DIKWP components as concepts and relationships.

• Cognitive Space: Processes DIKWP components through cognitive functions.

• Semantic Space: Represents meanings and associations of DIKWP components.

6.2 Graphical Representations

• Graphs: Each DIKWP component is represented as a graph within the spaces.

• Mappings: Functions transform and map components across spaces and graphs.

• Triplet Mapping:

  $$g:S\times C\times I$$

6.3 Example of Integration

• Data Graph (DG) in Semantic Space maps Data semantics.

• Information Graph (IG) in Concept Space organizes Information concepts.

• Cognitive Functions in Cognitive Space process transformations between DG and IG.

7. Conclusion

DIKWP Semantic Mathematics provides a structured, step-by-step framework for modeling cognitive processes and transformations between Data, Information, Knowledge, Wisdom, and Purpose. By defining precise mathematical representations and processing functions, it enables consistent and interoperable implementations in AI and cognitive systems. Understanding and applying this framework allows for the development of intelligent systems that can process complex semantic content, make informed decisions, and align actions with ethical values and goals.

Note: This detailed explanation aligns precisely with the provided understanding of DIKWP components, their definitions, mathematical representations, and interactions within the Concept Space, Cognitive Space, and Semantic Space. It is designed to serve as a comprehensive reference for practitioners and researchers working with the DIKWP Semantic Mathematics framework.