Semantic Mathematics: Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) Theories

Yucong Duan

Benefactor: Shiming Gong

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

DIKWP-AC Artificial Consciousness Standardization Committee

World Conference on Artificial Consciousness

World Artificial Consciousness Association

(Email：duanyucong@hotmail.com)

Catalog

1 Existence Computation and Reasoning (EXCR)

1.1 Theoretical Overview

1.2 Core Concepts

1.3 Theoretical Applications

1.3.1 Semantic Interpretation of the Goldbach Conjecture

1.3.2 Semantic Interpretation of the Four Color Theorem

1.3.3 Semantic Interpretation of the Collatz Conjecture

2 Essence Computation and Reasoning (ESCR)

2.1 Theoretical Overview

2.2 Core Concepts

2.3 Theoretical Applications

2.3.1 Semantic Interpretation of Points, Lines, and Planes

2.3.2 Semantic Interpretation of the Goldbach Conjecture

2.3.3 Semantic Interpretation of the Collatz Conjecture

3 Theoretical Analysis and Comparison

4 Key Research of Professor Yucong Duan's Papers

5 The Innovation of Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) in the Fields of Mathematics and Logic

5.1 Technological Evolution in Mathematics and Logic

5.1.1 Set Theory and Axiomatic Systems

5.1.2 Mathematical Logic and Model Theory

5.1.3 Computational Theory and Complexity

5.2 Innovation of EXCR and ESCR Theories

5.2.1 Innovation 1: Existence and Essence Computation at the Semantic Level

5.2.2 Innovation 2: Application Extension of Semantic Reasoning

5.2.3 Innovation 3: Multidisciplinary Integration

5.3 Detailed Applications of EXCR and ESCR in Mathematics and Logic Fields

5.3.1 Goldbach's Conjecture

5.3.2 Four Color Theorem

5.3.3 Collatz Conjecture

6 Application Comparison of Technologies Based on EXCR and ESCR Theories in Doctor-Patient Interaction Cases

6.1 Overview of Technologies

6.1.1 Existence Computation and Reasoning (EXCR)

6.1.2 Essence Computation and Reasoning (ESCR)

6.1.3 Traditional Natural Language Processing (NLP)

6.1.4 Expert Systems

6.2 Comparison of Processing Technologies

6.2.1 Case 1: Misdiagnosis of Heart Disease as Gastroesophageal Reflux Disease

6.2.2 Case 2: Misdiagnosis of Chronic Cough as Asthma

6.3 Comparison of Advantages and Disadvantages

6.3.1 Existence Computation and Reasoning (EXCR)

6.3.2 Essence Computation and Reasoning (ESCR)

6.3.3 Traditional Natural Language Processing (NLP)

6.3.4 Expert Systems

References

1 Existence Computation and Reasoning (EXCR)

1.1 Theoretical Overview

Existence Computation and Reasoning (EXCR) is a method of semantic association and reasoning by computing Existential Semantics. The EXCR theory aims to handle complex semantics and polysemy problems through semantic similarity calculations and Conservation of Existence Set (CEX) for probabilistic confirmation.

1.2 Core Concepts

Existential Semantics: The existence state of the analyzed object and its position in the semantic space.

Semantic Similarity Calculation: Evaluating the similarity between different semantic entities for semantic association.

Conservation of Existence Set (CEX): Probabilistic confirmation based on the existence characteristics of semantic entities to ensure the reasonableness and consistency of semantic associations.

1.3 Theoretical Applications

1.3.1 Semantic Interpretation of the Goldbach Conjecture

Explaining the Goldbach conjecture from the perspectives of type semantics and instance level, achieving semantic deduction through the association of types and instances.

Using Existence Computation and Reasoning to prove the semantic equivalence of representing even numbers as the sum of two primes.

1.3.2 Semantic Interpretation of the Four Color Theorem

Explaining the distinction of regions on a plane through the analysis of existential semantics.

Using Existence Computation and Reasoning to prove the semantic reasonableness of the Four Color Theorem.

1.3.3 Semantic Interpretation of the Collatz Conjecture

Analyzing the iterative process of natural numbers using Existence Computation and Reasoning to prove its semantic finiteness.

Deriving the semantic interpretation of the Collatz Conjecture through the semantic association at the type level.

2 Essence Computation and Reasoning (ESCR)

2.1 Theoretical Overview

Essence Computation and Reasoning (ESCR) diagnoses and predicts by identifying and reasoning about the essential attributes of objects. ESCR focuses on verifying the Consistency of Compounded Essential Set (CES) of the essential attributes of objects to ensure the accuracy and consistency of the reasoning process.

2.2 Core Concepts

Essential Attributes: The core features and attributes of an object used to describe its essence.

Consistency of Compounded Essential Set (CES): Verifying whether the combination of essential attributes conforms to a certain pattern or rule for reasoning.

Essential Semantic Reasoning: Combining essential attributes with a knowledge base for deep reasoning and diagnosis.

2.3 Theoretical Applications

2.3.1 Semantic Interpretation of Points, Lines, and Planes

Explaining the existence relationships of geometric elements in the semantic space through Essence Computation and Reasoning.

Using Consistency of Compounded Essential Set to prove the semantic associations and existence significance among geometric elements.

2.3.2 Semantic Interpretation of the Goldbach Conjecture

Analyzing the semantic relationship between even numbers and prime numbers from the perspective of essential attributes.

Proving the semantic equivalence of the Goldbach Conjecture at the type level through Essence Computation and Reasoning.

2.3.3 Semantic Interpretation of the Collatz Conjecture

Explaining the semantic changes of natural numbers in the iterative process through the analysis of essential attributes.

Using Essence Computation and Reasoning to verify the semantic reasonableness and finiteness of the Collatz Conjecture.

3 Theoretical Analysis and Comparison

4 Key Research of Professor Yucong Duan's Papers

Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) Based Revelation of Goldbach's Conjecture

Proposes to explain Goldbach's Conjecture from the perspectives of type semantics and instance semantics.

Uses EXCR and ESCR for semantic deduction to prove the reasonableness of Goldbach's Conjecture in semantic space.

Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) Based Revelation of the Four Color Theorem

Explains the Four Color Theorem through existence computation and reasoning.

Uses essence computation and reasoning to prove the reasonableness of the Four Color Theorem in semantic space.

Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) Based Revelation of Collatz Conjecture

Analyzes the Collatz Conjecture using EXCR and ESCR theories to prove its semantic finiteness.

Explains the semantic changes of natural numbers in the iterative process through the reasoning of essential attributes.

Professor Yucong Duan's EXCR and ESCR theories provide new methods and frameworks for solving complex semantic problems. Through existence computation and reasoning and essence computation and reasoning, more accurate semantic associations and reasoning can be achieved, especially excelling in handling polysemy, semantic ambiguity, and context dependence issues. These theories have significant applications not only in proving mathematical conjectures and geometric explanations but also in fields such as medical diagnosis and information retrieval. In the future, by integrating multiple technologies and building high-quality semantic knowledge bases, EXCR and ESCR are expected to play an important role in more fields, promoting the development of information processing and cognitive science.

5 The Innovation of Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) in the Fields of Mathematics and Logic

This chapter studies the innovation and academic value of the Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) theories proposed by Professor Yucong Duan in the fields of mathematics and logic from various aspects such as historical evolution, technological breakthroughs, theoretical frameworks, and practical applications.

5.1 Technological Evolution in Mathematics and Logic

5.1.1 Set Theory and Axiomatic Systems

Historical Background: Since Cantor proposed set theory, mathematics has gradually entered the axiomatic era. Set theory provided a unified foundation for mathematics but also led to issues such as Russell's paradox, prompting mathematicians to further explore the perfection of axiomatic systems.

Technological Breakthrough: Hilbert proposed the vision of formalized mathematics, and von Neumann and Gödel, through in-depth studies of axiomatic systems, revealed the intrinsic limitations of mathematical logic, such as Gödel's incompleteness theorems.

5.1.2 Mathematical Logic and Model Theory

Historical Background: The development of mathematical logic began with the work of Frege and Russell, laying the foundation of modern logic through Boolean algebra and first-order logic.

Technological Breakthrough: Gödel and Tarski, in their research on model theory, explored the satisfiability and truth of logical propositions through the correspondence between logical systems and models.

5.1.3 Computational Theory and Complexity

Historical Background: The introduction of Turing machines marked the birth of computational theory, providing a theoretical foundation for studying the computability and complexity of algorithms.

Technological Breakthrough: From Turing and von Neumann to Cook and Karp, computational complexity theory gradually improved, with the P vs NP problem becoming a core challenge in computational theory.

5.2 Innovation of EXCR and ESCR Theories

5.2.1 Innovation 1: Existence and Essence Computation at the Semantic Level

Existence Computation and Reasoning (EXCR)

Theoretical Framework: EXCR theory, by defining the Conservation of Existence Set (CEX) axiom, formalizes semantic reasoning and constructs a semantic conversion mechanism from instances to types. EXCR not only achieves type semantic reasoning in mathematical logic but also solves semantic ambiguity and polysemy issues through semantic similarity calculation.

Technological Breakthrough: In the proof of Goldbach's Conjecture, EXCR establishes the semantic relationship of numerical equivalence through the semantic association of even and prime number types, achieving innovation from numerical computation to semantic proof.

Essence Computation and Reasoning (ESCR)

Theoretical Framework: ESCR, through the Combinatorial Consistency Axiom (CES) and essential attribute recognition, ensures the accuracy and consistency of the reasoning process. ESCR, starting from essential attributes, conducts semantic reasoning, solving complex semantic association problems that traditional logical systems find difficult to handle.

Technological Breakthrough: In the semantic explanation of the Four Color Theorem, ESCR verifies the reasonableness of the Four Color Theorem in semantic space through the analysis of essential attributes of regional division, demonstrating its application value in geometric explanation.

5.2.2 Innovation 2: Application Extension of Semantic Reasoning

Natural Language Processing

Theoretical Framework: EXCR and ESCR apply semantic reasoning to natural language processing through semantic association and essential attribute recognition. By solving polysemy and semantic ambiguity problems, the accuracy of semantic understanding is improved.

Technological Breakthrough: In medical diagnosis scenarios, semantic parsing of patient symptom descriptions achieves personalized semantic association and misunderstanding elimination, demonstrating its effectiveness in practical applications.

Mathematical Logic Proof

Theoretical Framework: EXCR and ESCR provide new methods for mathematical logic proof through semantic reasoning and essential attribute analysis, solving complex mathematical conjectures and theorem proofs.

Technological Breakthrough: In the proof of the Collatz Conjecture, semantic analysis of the iterative process of natural numbers proves its semantic finiteness, demonstrating the powerful application potential of EXCR and ESCR in mathematical reasoning.

5.2.3 Innovation 3: Multidisciplinary Integration

Cross-disciplinary Application

Theoretical Framework: EXCR and ESCR theories integrate multidisciplinary knowledge such as mathematics, logic, computer science, and natural language processing, providing a new cross-disciplinary research approach.

Technological Breakthrough: Through application verification in different disciplines, EXCR and ESCR demonstrate broad applicability and theoretical innovation, providing new ideas and methods for cross-disciplinary research.

5.3 Detailed Applications of EXCR and ESCR in Mathematics and Logic Fields

5.3.1 Goldbach's Conjecture

EXCR Application: Explains Goldbach's Conjecture from the perspectives of type semantics and instance semantics, proving the semantic equivalence of representing even numbers as the sum of two prime numbers through existence computation and reasoning.

ESCR Application: Analyzes the semantic relationship between even numbers and prime numbers from the perspective of essential attributes, verifying the semantic equivalence of Goldbach's Conjecture at the type level through essence computation and reasoning.

5.3.2 Four Color Theorem

EXCR Application: Explains the color differentiation of regions on a plane through existence semantic analysis, proving the reasonableness of the Four Color Theorem in semantic space using existence computation and reasoning.

ESCR Application: Verifies the semantic association and existence significance of the Four Color Theorem at the type level through essence computation and reasoning.

5.3.3 Collatz Conjecture

EXCR Application: Analyzes the iterative process of natural numbers using existence computation and reasoning, proving its semantic finiteness, and derives the semantic explanation of the Collatz Conjecture through type-level semantic association.

ESCR Application: Explains the semantic changes of natural numbers in the iterative process through the analysis of essential attributes, and verifies the reasonableness and finiteness of the Collatz Conjecture in semantics using essence computation and reasoning.

Professor Yucong Duan's EXCR and ESCR theories, through innovative semantic reasoning methods, provide new research tools and ideas for the fields of mathematics and logic: semantic mathematics. These theories demonstrate their effectiveness not only in solving complex mathematical problems but also in natural language processing, information retrieval, and cross-disciplinary research. In the future, with the continuous improvement of semantic knowledge bases and computational models, EXCR and ESCR, as the foundational theories of semantic mathematics, are expected to play an important role in more fields, promoting the development and innovation of information processing and cognitive science.

6 Application Comparison of Technologies Based on EXCR and ESCR Theories in Doctor-Patient Interaction Cases

In this section, we will compare the application of several related processing technologies in cases of enhancing understanding in doctor-patient cognitive interactions. Special attention is given to the Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) proposed by Professor Yucong Duan, as well as traditional Natural Language Processing (NLP) and expert systems. By comparing the effectiveness of these technologies in handling patient symptom descriptions, eliminating misunderstandings, and providing diagnostic suggestions, we will analyze the strengths and weaknesses of each technology.

6.1 Overview of Technologies

6.1.1 Existence Computation and Reasoning (EXCR)

Emphasizes semantic association and reasoning through Existential Semantics.

Uses semantic similarity calculation and Conservation of Existence Set (CEX) for probability confirmation.

Suitable for handling complex semantics and polysemy issues.

6.1.2 Essence Computation and Reasoning (ESCR)

Focuses on Essential Attributes and their Consistency of Compounded Essential Set (CES).

Extracts key attributes from descriptions and performs diagnostics through essential semantic reasoning.

Used for identifying essential characteristics of symptoms and potential causes.

6.1.3 Traditional Natural Language Processing (NLP)

Processes patient descriptions based on rule-based or machine learning models.

Mainly uses keyword matching, syntax analysis, and statistical models for semantic understanding.

Limited ability to handle polysemy and complex contexts.

6.1.4 Expert Systems

Diagnoses based on predefined knowledge bases and reasoning rules.

Uses if-then rules and expert experience for symptom analysis and reasoning.

Requires a large amount of domain knowledge and predefined rules.

6.2 Comparison of Processing Technologies

6.2.1 Case 1: Misdiagnosis of Heart Disease as Gastroesophageal Reflux Disease

Scenario Description: The patient describes, "I feel a bit tight in my chest, especially when lying down at night." The doctor initially diagnoses it as gastroesophageal reflux disease and suggests the patient take antacids.

EXCR Technology:

Semantic Association: Determines the specific positions of "tight chest" and "lying down at night" in the semantic space through semantic similarity calculation.

Probability Confirmation: Uses CEX to evaluate the association probability of symptoms with different diseases (e.g., heart disease, gastroesophageal reflux disease).

Misunderstanding Elimination: Further questions the patient, combines detailed descriptions and physical examination results for comprehensive analysis, and confirms the possibility of heart issues.

ESCR Technology:

Essential Attribute Recognition: Extracts key attributes such as "tight chest" and "lying down at night."

Combination of Essential Attributes: Verifies whether the combination of these attributes conforms to a certain disease pattern through CES.

Knowledge Reasoning: Combines with the medical knowledge base to infer the potential risk of heart disease and suggests further examinations.

NLP Technology:

Keyword Matching: Identifies keywords in the description such as "tight chest" and "lying down at night."

Syntax Analysis: Analyzes the sentence structure to understand the patient's symptom description.

Statistical Model: Determines the association of these keywords with certain diseases based on training data.

Limitations: May not fully consider polysemy and complex contexts, leading to a higher risk of misdiagnosis.

Expert System:

Knowledge Base Matching: Judges that "tight chest" may be related to heart disease or gastroesophageal reflux disease based on predefined rules.

Rule Reasoning: Uses if-then rules to determine the severity of symptoms and potential causes.

Limitations: Requires a large number of predefined rules, has poor flexibility, and is limited in handling complex cases.

6.2.2 Case 2: Misdiagnosis of Chronic Cough as Asthma

Scenario Description: The patient describes, "I've been coughing a lot recently, especially at night." The doctor initially diagnoses it as asthma and prescribes the corresponding medication.

EXCR Technology:

Semantic Association: Determines the specific positions of "coughing" and "at night" in the semantic space through semantic similarity calculation.

Probability Confirmation: Uses CEX to evaluate the association probability of symptoms with different diseases (e.g., asthma, gastroesophageal reflux).

Misunderstanding Elimination: Further questions the patient, combines detailed descriptions and physical examination results for comprehensive analysis, and confirms other possible causes.

ESCR Technology:

Essential Attribute Recognition: Extracts key attributes such as "coughing" and "at night."

Combination of Essential Attributes: Verifies whether the combination of these attributes conforms to a certain disease pattern through CES.

Knowledge Reasoning: Combines with the medical knowledge base to infer the potential risk of gastroesophageal reflux or chronic bronchitis and suggests further examinations.

NLP Technology:

Keyword Matching: Identifies keywords in the description such as "coughing" and "at night."

Syntax Analysis: Analyzes the sentence structure to understand the patient's symptom description.

Statistical Model: Determines the association of these keywords with certain diseases based on training data.

Limitations: May not fully consider polysemy and complex contexts, leading to a higher risk of misdiagnosis.

Expert System:

Knowledge Base Matching: Judges that "coughing" may be related to asthma or gastroesophageal reflux based on predefined rules.

Rule Reasoning: Uses if-then rules to determine the severity of symptoms and potential causes.

Limitations: Requires a large number of predefined rules, has poor flexibility, and is limited in handling complex cases.

6.3 Comparison of Advantages and Disadvantages

6.3.1 Existence Computation and Reasoning (EXCR)

Advantages:

Capable of handling complex semantics and polysemy issues.

Based on semantic similarity calculation, can more accurately confirm probabilities.

Highly flexible and applicable to various scenarios.

Disadvantages:

Requires high-quality semantic knowledge bases and semantic computation models.

High implementation and computational complexity.

6.3.2 Essence Computation and Reasoning (ESCR)

Advantages:

Can identify essential characteristics of symptoms and potential causes.

Provides more accurate diagnostics through the consistency of compounded essential attributes.

Closely integrates with medical knowledge bases, suitable for deep reasoning.

Disadvantages:

Requires high-quality essential attribute recognition and combination models.

High implementation and computational complexity.

6.3.3 Traditional Natural Language Processing (NLP)

Advantages:

Fast processing speed, suitable for large-scale data processing.

Can continuously optimize and improve accuracy through machine learning models.

Disadvantages:

Limited ability to handle polysemy and complex contexts.

Relies on training data, performs poorly when dealing with new issues.

6.3.4 Expert Systems

Advantages:

Highly explainable based on predefined rules.

Provides stable and predictable diagnostic results.

Disadvantages:

Requires a large amount of domain knowledge and predefined rules.

Poor flexibility, difficult to handle complex and unforeseen cases.

Through comparative analysis, it can be seen that EXCR and ESCR have significant advantages in handling complex semantics and essential attribute recognition, enabling more accurate semantic association and reasoning. However, these technologies have high implementation and computational complexity and require high-quality semantic knowledge bases and computation models. Traditional NLP technologies and expert systems have advantages in processing speed and explainability but perform poorly in handling complex semantics and polysemy issues. In the future, combining multiple technologies to leverage their respective strengths may further improve the accuracy and efficiency of diagnosis and processing.

References

[1] Duan, Y. (2022). Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of the Goldbach's conjecture. ResearchGate.

[2] Duan, Y. (2022). Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of the Four Color Theorem. ResearchGate.

[3] Duan, Y. (2022). Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of Collatz Conjecture. ResearchGate.