Semantic Mathematics: P = NP?  from EXCR and 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 The mathematical background of the P vs NP problem.

2 Introduction to EXCR and ESCR Theory

3 EXCR and ESCR Theory Analysis on the P vs NP Problem

3.1 Application of EXCR Theory in the P vs NP Problem

3.2 Application of ESCR Theory in the P vs NP Problem

4 Example of EXCR and ESCR Theory Argumentation on the P vs NP Problem

4.1 Example 1: SAT Problem

4.2 Steps

5 Innovativeness of EXCR and ESCR in the P vs NP Problem

6 Extension of Argument on the P vs NP Problem Based on EXCR and ESCR Theories

6.1 Step 1: Definition of the Semantic Space of P vs NP Problem

6.2 Step 2: Application of EXCR Theory to Establish Semantic Correlation between P and NP

6.3 Step 3: Application of ESCR Theory to Identify the Essential Attributes of P and NP

6.4 Step 4: Derivation of Conclusion for the P vs NP Problem

7 EXCR and ESCR Semantic Modeling of the P Problem

7.1 Definition of P-Class Problems

7.1.1 Semantic Space of P-Class Problems (EXCR)

7.1.2 Essence Properties of P-Class Problems (ESCR)

7.2 Specific Analysis Process

7.2.1 Existence Computation and Reasoning (EXCR) for P-Class Problems

7.2.2 Essence Computation and Reasoning (ESCR) for P-Class Problems

7.3 Comprehensive Analysis and Conclusion

7.4 Example Analysis

7.4.1 EXCR Semantic Modeling of the Sorting Problem

7.4.2 ESCR Semantic Modeling of the Sorting Problem

8 Semantic Analysis of the P vs NP Problem Based on EXCR and ESCR

8.1 Semantic Space Modeling of the P vs NP Problem (EXCR)

8.1.1 Definition of NP Problems

8.1.2 Semantic Space of NP Problems (EXCR)

8.2 Essence Attribute Modeling of the P vs NP Problem (ESCR)

8.3 Comprehensive Analysis and Conclusion of the P vs NP Problem

8.3.1 Existence Computation and Reasoning (EXCR) for NP Problems

8.3.2 Essence Computation and Reasoning (ESCR) for NP Problems

8.4 Comprehensive Analysis

8.5 Specific Analysis and Results

8.5.1 Analysis of TSP Problem Instances (Assuming P = NP)

8.5.2 Conclusion

8.6 Final Conclusion

9 Comparative Analysis of Related Research Methods

10 Comparative Analysis

10.1 Basic Principles

10.2 Analysis Methods

10.3 Main Objectives

10.4 Advantages and Disadvantages

Conclusion

References

The P vs NP problem is a core challenge in computational theory, primarily examining whether every solution verifiable in polynomial time can also be computed in polynomial time. We utilize the Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR) theories proposed by Professor Yucong Duan to conduct a semantic mathematical argument on the P vs NP problem.

1 The mathematical background of the P vs NP problem.

P-class problems: Problems that can be solved in polynomial time by a deterministic Turing machine.

NP-class problems: Problems whose solutions can be verified in polynomial time by a deterministic Turing machine.

Core problem: Whether P = NP, i.e., whether all solutions that can be verified in polynomial time can also be found in polynomial time.

2 Introduction to EXCR and ESCR Theory

Existence Computation and Reasoning (EXCR): By defining the Conservation Axiom of Existence Sets (CEX), formalizing semantic reasoning, and establishing a semantic transformation mechanism from instances to types.

Essence Computation and Reasoning (ESCR): By combining the Consistency Axiom of Essence Sets (CES) and essence attribute identification, ensuring the accuracy and consistency of the reasoning process, and conducting semantic reasoning based on essence attributes.

3 EXCR and ESCR Theory Analysis on the P vs NP Problem

3.1 Application of EXCR Theory in the P vs NP Problem

Definition of Semantic Space: Firstly, we define the semantic spaces for the P and NP classes of problems. The semantic space for P-class problems is denoted as EXCR(P), and for NP-class problems as EXCR(NP).

Conservation Axiom of Existence Sets (CEX): We assume the existence of a semantic equivalence axiom, CEX(P, NP), used to verify the relationship between P-class and NP-class problems.

Semantic Association and Reasoning: Through the EXCR theory, we can establish semantic associations between P-class and NP-class problems. For instance, if we can find an existential semantic transformation that maps NP-class problems into the semantic space of P-class problems, then we can deduce P = NP.

Steps:

Define the semantics of problem instances: Assuming the existence of an NP problem instance, INS(NP), verifiable in polynomial time. We map it to a P problem instance, INS(P), through EXCR.

Establish semantic equivalence relations: Using the conservation axiom of existence sets, CEX, we verify the semantic equivalence between INS(NP) and INS(P).

Derive polynomial-time solutions: If we can prove the equivalence of INS(P) and INS(NP) in the semantic space, then we can deduce that NP problems can be solved in polynomial time, thus proving P = NP.

3.2 Application of ESCR Theory in the P vs NP Problem

Identification of the Intrinsic Attributes of Problems: Through the ESCR theory, we identify the intrinsic attributes of P-class and NP-class problems. The intrinsic attributes of P-class problems are denoted as ESS(P), and for NP-class problems as ESS(NP).

Combination Consistency Axiom (CES): We assume the existence of a combination consistency axiom, CES(P, NP), used to verify the relationship between the intrinsic attributes of P-class and NP-class problems.

Reasoning on Intrinsic Attributes: Through the ESCR theory, we can deduce the consistency of intrinsic attributes between P-class and NP-class problems. For example, if we can prove the existence of NP problem intrinsic attributes in the intrinsic attribute space of P-class problems, then we can deduce P = NP.

Steps:

Define the intrinsic attributes of problem types: Assuming the existence of an NP problem type, TYPE(NP), we map it to a P problem type, TYPE(P), through ESCR.

Establish equivalence relations of intrinsic attributes: Using the combination consistency axiom, CES, we verify the equivalence of intrinsic attributes between TYPE(NP) and TYPE(P).

Derive consistency of intrinsic attributes: If we can prove the equivalence of TYPE(P) and TYPE(NP) in the intrinsic attribute space, then we can deduce that NP problems can be solved in polynomial time, thus proving P = NP.

4 Example of EXCR and ESCR Theory Argumentation on the P vs NP Problem

4.1 Example 1: SAT Problem

Problem Definition: The SAT problem is an NP-complete problem, verifying whether a Boolean formula is satisfiable.

Application of EXCR: Through EXCR, we map instances of the SAT problem to instances of problems in the P class, verifying their semantic equivalence.

Application of ESCR: Through ESCR, we identify the essential properties of the SAT problem and problems in the P class, verifying their compositional consistency.

4.2 Steps

Semantic Definition of the SAT Problem: The semantics of instances of the SAT problem INS(SAT) are mapped to instances of problems in the P class INS(P).

Semantic Equivalence Verification: Through CEX, we verify the semantic equivalence of INS(SAT) and INS(P).

Identification of Essential Properties: Through ESCR, we identify the essential properties of the SAT problem and problems in the P class, ESS(SAT) and ESS(P).

Verification of Essential Property Equivalence: Through CES, we verify the compositional consistency of ESS(SAT) and ESS(P).

5 Innovativeness of EXCR and ESCR in the P vs NP Problem

Innovation One: Introduction of Semantic Space and Essential Attributes: By introducing semantic space and essential attributes, EXCR and ESCR provide a new approach to analyze the P vs NP problem.

Innovation Two: Axiomatic Reasoning Process: Through the CEX and CES axioms, EXCR and ESCR maintain the consistency of semantics and essential attributes in the reasoning process, ensuring the accuracy of inference.

Innovation Three: Cross-Domain Applications: EXCR and ESCR not only have significant applications in mathematics and logic but can also be extended to fields such as natural language processing and information retrieval, demonstrating their wide applicability.

6 Extension of Argument on the P vs NP Problem Based on EXCR and ESCR Theories

To comprehensively argue the P vs NP problem, we will progressively apply the theories of Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR), attempting to provide a new perspective to resolve this issue.

6.1 Step 1: Definition of the Semantic Space of P vs NP Problem

Define the semantic space of P-class problems:

Represented as EXCR(P), where EXCR denotes Existence Computation and Reasoning.

Examples of P-class problems: Problems solvable by deterministic Turing machines in polynomial time.

Define the semantic space of NP-class problems:

Represented as EXCR(NP).

Examples of NP-class problems: Problems whose solutions can be verified by deterministic Turing machines in polynomial time.

6.2 Step 2: Application of EXCR Theory to Establish Semantic Correlation between P and NP

Conservation Axiom of Existence Sets (CEX):

Assume the existence of a semantic-equivalent conservation axiom, CEX(P, NP), to verify the relationship between P-class and NP-class problems.

The goal is to establish a semantic correlation between P-class and NP-class problems through EXCR theory.

Define the semantics of NP problem instances:

Assume the existence of an NP problem instance, INS(NP), verifiable in polynomial time.

We map it to a P problem instance, INS(P), through EXCR.

Establish semantic equivalence:

Through the conservation axiom of existence sets, CEX, we verify the semantic equivalence between INS(NP) and INS(P).

If INS(NP) and INS(P) are equivalent in the semantic space, it implies that NP problems can be solved in polynomial time, thus P = NP.

6.3 Step 3: Application of ESCR Theory to Identify the Essential Attributes of P and NP

Identify the essential attributes of P-class and NP-class problems:

The essential attributes of P-class problems are represented as ESS(P).

The essential attributes of NP-class problems are represented as ESS(NP).

Combination Consistency Axiom (CES):

Assume the existence of a combination consistency axiom, CES(P, NP), to verify the relationship between the essential attributes of P-class and NP-class problems.

Essential attribute reasoning:

Through ESCR theory, derive the consistency of essential attributes between P-class and NP-class problems.

The objective is to demonstrate the existence of NP problem essential attributes in the space of P-class problem essential attributes, thereby deriving P = NP.

Example: Argument for Specific Problems

Example: 3-SAT Problem

Problem definition:

The 3-SAT problem is NP-complete, verifying whether a Boolean formula is satisfiable.

Application of EXCR theory:

Map instances of the 3-SAT problem, INS(3-SAT), to instances of P-class problems, INS(P).

Define the semantics of the 3-SAT problem: EXCR(3-SAT).

Semantic equivalence verification:

Through CEX, verify the semantic equivalence between INS(3-SAT) and INS(P).

If successful, it indicates that the 3-SAT problem can be solved in polynomial time.

Application of ESCR theory:

Identify the essential attributes of the 3-SAT problem and P-class problems, ESS(3-SAT) and ESS(P).

Through CES, verify the equivalence of ESS(3-SAT) and ESS(P).

Essential attribute equivalence verification:

Through CES, verify the consistency of the 3-SAT problem and P-class problems in terms of essential attributes.

6.4 Step 4: Derivation of Conclusion for the P vs NP Problem

Semantic equivalence derivation:

Through EXCR theory and CEX verification, assume we can prove the semantic equivalence of INS(NP) and INS(P).

Semantic equivalence implies that NP problems can be solved in polynomial time.

Essential attribute consistency derivation:

Through ESCR theory and CES verification, assume we can prove the equivalence of ESS(NP) and ESS(P).

Essential attribute consistency implies that P-class problems and NP-class problems are equivalent in terms of essential attributes.

Conclusion:

If through the derivation process of EXCR and ESCR theories, we can verify semantic equivalence and essential attribute consistency, then we can conclude P = NP.

This means that every solution verifiable in polynomial time can also be computed in polynomial time.

Through the application of EXCR and ESCR theories, we attempt a detailed argument on the P vs NP problem from the perspectives of semantic space and essential attributes. EXCR theory establishes semantic equivalence between P-class and NP-class problems through existence computation and reasoning, while ESCR theory verifies the consistency of essential attributes between the two types of problems. This approach provides a new perspective for resolving the P vs NP problem, demonstrating the potential application value of these theories in computational theory.

7 EXCR and ESCR Semantic Modeling of the P Problem

7.1 Definition of P-Class Problems

P-class problems refer to a set of problems that can be solved by a deterministic Turing machine in polynomial time.

7.1.1 Semantic Space of P-Class Problems (EXCR)

Definition of Semantic Space:

Let the semantic space of P-class problems be EXCR(P), representing the set of problems that can be solved in polynomial time.

Semantic Space Modeling:

EXCR(P) = {INS(P) | P is an instance of a P-class problem and can be solved in polynomial time}.

For example, for the sorting problem, EXCR(P) includes all instances that can be solved by sorting algorithms in polynomial time.

Existence Computation and Reasoning (EXCR):

In the semantic space EXCR(P), verify if all problem instances can be solved in polynomial time.

7.1.2 Essence Properties of P-Class Problems (ESCR)

Definition of Essence Properties:

Let the essence properties of P-class problems be ESS(P), representing the set of essential properties that can be solved in polynomial time.

Essence Property Modeling:

ESS(P) = {ESS(P) | P is a P-class problem, and its essence properties can be verified in polynomial time}.

For example, for the sorting problem, ESS(P) includes properties such as the time complexity and stability of sorting algorithms.

Essence Computation and Reasoning (ESCR):

In the essence properties ESS(P), verify if all properties can be verified in polynomial time.

7.2 Specific Analysis Process

7.2.1 Existence Computation and Reasoning (EXCR) for P-Class Problems

Instance Verification:

For each instance INS(P) of a P-class problem, verify if it can be solved in polynomial time.

For example, for each sorting problem instance (such as arrays), verify if sorting algorithms can complete sorting in polynomial time.

EXCR Verification Process:

Let the instance INS(P) = {p1, p2, ..., pn}.

Verify if EXCR(P) := {INS(p1), INS(p2), ..., INS(pn)} ∈ P.

That is, for each instance pi, verify if it can be solved in polynomial time.

Verification Result:

If all instances can be solved in polynomial time, the EXCR(P) verification passes.

7.2.2 Essence Computation and Reasoning (ESCR) for P-Class Problems

Property Verification:

For each essence property ESS(P) of a P-class problem, verify if it can be verified in polynomial time.

For example, for the time complexity of sorting problems, verify if it can be done in polynomial time.

ESCR Verification Process:

Let the property ESS(P) = {ess1, ess2, ..., essn}.

Verify if ESCR(P) := {ESS(ess1), ESS(ess2), ..., ESS(essn)} ∈ P.

That is, for each property essi, verify if it can be verified in polynomial time.

Verification Result:

If all essence properties can be verified in polynomial time, the ESCR(P) verification passes.

7.3 Comprehensive Analysis and Conclusion

EXCR Verification Conclusion:

If EXCR verification passes, it indicates that all instances of P-class problems can be solved in polynomial time.

ESCR Verification Conclusion:

If ESCR verification passes, it indicates that all essence properties of P-class problems can be verified in polynomial time.

Overall Conclusion:

If both EXCR and ESCR verifications for P-class problems pass, it indicates that they meet the requirements of being solved in polynomial time in terms of existence and essence properties.

7.4 Example Analysis

Specific analysis using the sorting problem as an example:

7.4.1 EXCR Semantic Modeling of the Sorting Problem

Instance Definition:

INS(Sort) = {arr1, arr2, ..., arrn}, where arri is an array instance to be sorted.

Existence Computation and Reasoning:

For each array instance arri, verify if sorting algorithms can complete sorting in polynomial time.

For example, for the array [3, 1, 2], verify if sorting algorithms can complete sorting in O(n log n) time complexity.

7.4.2 ESCR Semantic Modeling of the Sorting Problem

Definition of Essence Properties:

ESS(Sort) = {time_complexity, stability}, where time_complexity is the time complexity, and stability is the algorithm stability.

Essence Computation and Reasoning:

Verify if the time complexity and stability of sorting algorithms can be verified in polynomial time.

For example, verify the time complexity O(n log n) and stability of quicksort.

Through the semantic modeling and specific analysis of EXCR and ESCR above, we verify the possibility of instances and essence properties of P-class problems being solved and verified in polynomial time. Therefore, for P-class problems like the sorting problem, both EXCR and ESCR verifications pass, indicating they meet the requirements of being solved in polynomial time.

For the P vs NP problem, further verification and analysis need to consider more complex NP problem instances and their essence properties. If similar verification and reasoning can be conducted on a broader range of NP problems, yielding consistent conclusions, clearer conclusions can be drawn for the P vs NP problem. This process offers new perspectives and theoretical foundations for the P vs NP problem, aiding deeper research and exploration.

8 Semantic Analysis of the P vs NP Problem Based on EXCR and ESCR

The P vs NP problem is a fundamental issue in computer science, exploring whether every problem solvable in non-deterministic polynomial time can also be solved in deterministic polynomial time. Specifically, the question is: Does P equal NP? Within the framework of Existence Computation and Reasoning (EXCR) and Essence Computation and Reasoning (ESCR), we can analyze and discuss this problem from the perspective of semantic modeling.

8.1 Semantic Space Modeling of the P vs NP Problem (EXCR)

8.1.1 Definition of NP Problems

NP problems refer to the set of problems that can be solved by non-deterministic Turing machines in non-deterministic polynomial time.

The verifiable property of NP problems is: Given a solution to a problem, its correctness can be verified in polynomial time.

8.1.2 Semantic Space of NP Problems (EXCR)

Semantic space definition:

Let the semantic space of NP problems be EXCR(NP), representing the set of problems solvable in non-deterministic polynomial time.

Semantic space modeling:

EXCR(NP) = {INS(NP) | NP is an instance of an NP problem that can be solved in non-deterministic polynomial time}.

For example: For the Traveling Salesman Problem (TSP), EXCR(NP) includes all instances where the optimal path can be found through guessing and verification in non-deterministic polynomial time.

Existence Computation and Reasoning (EXCR):

In the semantic space EXCR(NP), verify if all problem instances can be solved in non-deterministic polynomial time.

8.2 Essence Attribute Modeling of the P vs NP Problem (ESCR)

Essence attribute definition:

Let the essence attribute of NP problems be ESS(NP), representing the set of essential attributes verifiable in non-deterministic polynomial time.

Essence attribute modeling:

ESS(NP) = {ESS(NP) | NP is an NP problem, and its essential attributes can be verified in non-deterministic polynomial time}.

For example: For TSP, ESS(NP) includes attributes such as path length and optimality.

Essence Computation and Reasoning (ESCR):

In the essence attribute ESS(NP), verify if all essential attributes of problems can be verified in non-deterministic polynomial time.

8.3 Comprehensive Analysis and Conclusion of the P vs NP Problem

8.3.1 Existence Computation and Reasoning (EXCR) for NP Problems

Instance verification:

For each instance INS(NP) of an NP problem, verify if it can be solved in non-deterministic polynomial time.

For example: For each instance of the TSP (such as city sets and distance matrices), verify if the optimal path can be found in non-deterministic polynomial time.

EXCR verification process:

Let the instance INS(NP) = {np1, np2, ..., npn}.

Verify EXCR(NP) := {INS(np1), INS(np2), ..., INS(npn)} ∈ NP.

That is, for each instance npi, verify if it can be solved in non-deterministic polynomial time.

Verification result:

If all instances can be solved in non-deterministic polynomial time, EXCR(NP) verification passes.

8.3.2 Essence Computation and Reasoning (ESCR) for NP Problems

Attribute verification:

For each essence attribute ESS(NP) of an NP problem, verify if it can be verified in non-deterministic polynomial time.

For example: For the TSP problem, verify if path length and optimality can be verified in non-deterministic polynomial time.

ESCR verification process:

Let the attribute ESS(NP) = {ess1, ess2, ..., essn}.

Verify ESCR(NP) := {ESS(ess1), ESS(ess2), ..., ESS(essn)} ∈ NP.

That is, for each attribute essi, verify if it can be verified in non-deterministic polynomial time.

Verification result:

If all essential attributes can be verified in non-deterministic polynomial time, ESCR(NP) verification passes.

8.4 Comprehensive Analysis

Assuming P = NP:

If P = NP, then every NP problem can be solved in deterministic polynomial time.

That is, for all instances and essence attributes of NP problems, they can be verified and solved in polynomial time.

Assuming P ≠ NP:

If P ≠ NP, then there exist NP problems that cannot be solved in deterministic polynomial time.

That is, for some instances or essence attributes of NP problems, they cannot be verified and solved in polynomial time.

8.5 Specific Analysis and Results

8.5.1 Analysis of TSP Problem Instances (Assuming P = NP)

Instance definition:

INS(TSP) = {city sets and distance matrices}.

EXCR analysis:

Verify EXCR(TSP) := {INS(city sets and distance matrices)} ∈ P.

For each TSP instance, verify if the optimal path can be found in polynomial time.

ESCR analysis:

Verify ESCR(TSP) := {path length and optimality} ∈ P.

For each TSP instance, verify if path length and optimality can be verified in polynomial time.

8.5.2 Conclusion

EXCR and ESCR verification passed:

If all instances and essence attributes of the TSP problem pass EXCR and ESCR verification, it indicates that the TSP problem can be solved and verified in polynomial time.

This supports the assumption of P = NP.

EXCR and ESCR verification failed:

If some instances or essence attributes of the TSP problem fail EXCR and ESCR verification, it indicates that the TSP problem cannot be solved and verified in polynomial time.

This supports the assumption of P ≠ NP.

8.6 Final Conclusion

Through the semantic modeling and specific analysis of EXCR and ESCR, we can gain a deeper understanding and exploration of the P vs NP problem. If similar verification and reasoning can be conducted on a broader range of NP problems, and consistent conclusions can be reached, it may lead to a clearer conclusion on the P vs NP problem. This process provides a new perspective and theoretical basis for the P vs NP problem, facilitating further research and exploration.

However, the derivation of specific conclusions still requires thorough analysis and testing of more NP problems in practical verification, which will be a long and complex process. The framework of EXCR and ESCR provides powerful tools and methods for this analysis, but achieving this goal still requires extensive research and effort.

9 Comparative Analysis of Related Research Methods

In the study of the P vs NP problem, EXCR (Existence Computation and Reasoning) and ESCR (Essence Computation and Reasoning) provide a novel perspective from a semantic standpoint. The following is a comparative analysis of EXCR and ESCR with other classical solutions to demonstrate their uniqueness and advantages.

10 Comparative Analysis

10.1 Basic Principles

EXCR & ESCR: Based on semantic space and essential attributes modeling, focusing on semantic consistency and essence verification of the problem.

Traditional Complexity Theory: Focuses on classifying problems by time complexity and the resource limitations of algorithms, primarily through theoretical analysis to determine the complexity class of problems.

Proof Theory Method: Utilizes mathematical logic and proof systems for reasoning and verification of problems, particularly suitable for formal and logical reasoning.

SAT Solvers: Solves specific NP-complete problems using Boolean satisfiability problem solvers, employing DPLL and CDCL algorithms.

Algebraic Geometry Methods: Analyzes polynomial equations and algebraic structures, using tools such as Gröbner bases and Hilbert bases.

10.2 Analysis Methods

EXCR & ESCR: Performs deep analysis through semantic verification of existence and essence attributes, offering an abstract yet insightful perspective.

Traditional Complexity Theory: Determines the computational time of problems under different computational models by classifying time complexity.

Proof Theory Method: Provides proof and verification at the level of mathematical logic through logical reasoning and mathematical proof.

SAT Solvers: Uses efficient solving algorithms to tackle specific NP-complete problems, providing practical solutions.

Algebraic Geometry Methods: Analyzes polynomial systems using geometric and algebraic tools, offering a geometric perspective on solutions.

10.3 Main Objectives

EXCR & ESCR: Aim to analyze and interpret the deep structure of problems through semantic consistency and essence verification.

Traditional Complexity Theory: The primary goal is to determine the time complexity class of problems, assessing the efficiency and resource requirements of algorithms.

Proof Theory Method: Provides rigorous mathematical verification of problems through mathematical logic proofs.

SAT Solvers: Aim to effectively solve specific NP-complete problems, providing practical solutions in real-world applications.

Algebraic Geometry Methods: Analyzes and solves polynomial systems through algebraic and geometric methods, offering solutions from a geometric perspective.

10.4 Advantages and Disadvantages

EXCR & ESCR:

Advantages: Provides deep semantic understanding, suitable for abstract analysis of complex problems.

Disadvantages: Theoretical framework is complex, with specific applications and practices requiring further research.

Traditional Complexity Theory:

Advantages: Precisely defines time complexity, providing a basis for evaluating algorithm efficiency.

Disadvantages: Focuses mainly on time complexity, lacking semantic-level analysis of problems.

Proof Theory Method:

Advantages: Rigorous mathematical proofs and high certainty in logical reasoning.

Disadvantages: The proof process is complex, making it difficult to handle large-scale real-world problems.

SAT Solvers:

Advantages: Effectively solves specific NP-complete problems, widely applied in practical problem-solving.

Disadvantages: Efficiency may be low for some problems, with the solving process potentially very time-consuming.

Algebraic Geometry Methods:

Advantages: Processes polynomial systems using algebraic and geometric tools, providing a geometric perspective.

Disadvantages: The method is complex, computationally expensive, and has limited applicability.

Through comparative analysis, it is evident that EXCR and ESCR offer a new method for analyzing the P vs NP problem from the perspectives of semantics and essence, presenting unique advantages and application prospects compared to traditional time complexity analysis, logical proofs, SAT solvers, and algebraic geometry methods. Specifically, the semantic modeling and essence verification of EXCR and ESCR can provide deep explanations and analysis for complex problems. Although the theoretical framework and specific applications require further research and refinement, there is broad application potential in semantic analysis of complex systems and essence verification of logical problems.

Conclusion

Through this innovative semantic mathematical perspective proposed by Professor Yucong Duan, the study of the P vs NP problem can gain more understanding and inspiration, providing new methods for theoretical research and potentially leading to more breakthroughs in practical applications. Future research can further integrate the theories of EXCR and ESCR, exploring their applications and practical effects in various fields, thus providing new pathways for the development of computational complexity theory and the resolution of real-world problems.

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.