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泉婕妤，端午安康～

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打赏啦！打赏啦！打赏此楼 359 楼作者步月金币 66 个！

你们不要进来啊。~ ~

我们去啾啾日记贴开车呀

### The Graph Brain Project & Big Mathematics

#### Overview

The Graph Brain Project is an innovative experiment aiming to transform mathematical research by leveraging automated discovery software, large-scale collaboration, and systematic investigation. The project focuses on using computational tools to generate conjectures, solve mathematical problems, and promote a collaborative approach similar to research labs in other sciences.

#### Key Concepts

1. **Automated Mathematical Discovery**:

- The project utilizes software to generate invariant-relation and property-relation conjectures in various areas of mathematics. This software produces conjectures not implied by existing published theorems, pushing forward mathematical research.

2. **Large-Scale Collaboration**:

- The project envisions forming research groups of various sizes to work on specific mathematical problems using automated tools, databases, and systematic approaches. This collaborative model aims to achieve more efficient and effective use of published research.

3. **Systematic Investigation**:

- By coding many graph-theoretic concepts and graphs, the project investigates small open problems in graph theory. The modular nature of the experiment allows other researchers to supplement and build upon the initial code.

4. **Big Mathematics**:

- This concept refers to large, systematic, collaborative research on significant mathematical problems. The idea is to combine skills, tools, and results systematically to make substantial progress in mathematical research.

#### Key Achievements and Goals

- **Conjecture Generation**:

- The project demonstrates that automated programs can exhaustively generate and evaluate expressions formed from standard mathematical ingredients. These expressions can lead to new conjectures and theorems.

- **Collaboration and Network Effects**:

- Researchers from different sub-fields can contribute and benefit from shared code and data, creating a multiplier effect. The aim is to build code-bases of mathematical knowledge that are easy to use and extend.

- **Historical Context and Motivation**:

- The project draws inspiration from early work in computer science and AI, such as Turing's "Intelligent Machinery" and the development of automated theorem proving in the 1950s. The goal is to build on this foundation to enhance modern mathematical research.

#### Case Study: Independence Number of Graphs

- **Independence Number**:

- The independence number (or stability number) of a graph is the largest set of vertices with no edges between them. It is a central concept in graph theory, linked to problems like the P vs. NP question and Hadwiger’s Conjecture.

- **Bounds and Conjectures**:

- The project has generated new conjectured bounds for the independence number of a graph, not implied by existing bounds. These conjectures are tested and either proved or disproved, leading to new mathematical knowledge.

#### Future Directions

- **Expansion of Research**:

- The project plans to expand its scope by incorporating more invariants, objects, and properties into the conjecturing program. The goal is to systematically use theoretical knowledge to generate new conjectures that push the boundaries of current mathematical research.

- **Building a Collaborative Infrastructure**:

- By creating a collaborative infrastructure, the project aims to facilitate large-scale mathematical research, involving coding, conjecture generation, testing, and proving.

For detailed insights and more information, you can access the full document: [The Graph Brain Project & Big Mathematics](http://www.people.vcu.edu/~clarson/AC7-GBP-171019.pdf).

If you need further analysis or have specific questions about the project, please let me know!

### The Graph Brain Project & Big Mathematics

#### Project Overview

The Graph Brain Project is an ambitious initiative focused on revolutionizing mathematical research through automation, large-scale collaboration, and systematic investigation. The project leverages automated discovery software to generate conjectures and solve mathematical problems, advocating for a collaborative approach akin to research labs in other scientific disciplines.

#### Key Components and Objectives

1. **Automated Discovery Software**:

- The software used in the project is designed to generate invariant-relation and property-relation conjectures across various mathematical fields. It aims to produce conjectures that are not implied by existing theorems, thereby advancing mathematical knowledge.

2. **Large-Scale Collaboration**:

- The project promotes forming research groups to tackle specific mathematical problems. These groups utilize automated tools, databases, and systematic approaches to enhance research efficiency and effectiveness.

3. **Systematic Research**:

- By coding numerous graph-theoretic concepts and graphs, the project investigates open problems in graph theory. The modular nature of the experiment allows other researchers to build upon and expand the initial work.

4. **Big Mathematics**:

- This concept involves large, systematic, collaborative research on significant mathematical problems. The aim is to combine collective skills, tools, and results to achieve substantial progress in mathematics.

#### Key Achievements and Goals

- **Conjecture Generation**:

- Automated programs in the project generate and evaluate mathematical expressions, leading to new conjectures and theorems. This approach leverages computational power to exhaustively explore possible conjectures.

- **Collaboration and Network Effects**:

- Researchers from different mathematical sub-fields contribute to and benefit from shared code and data, creating a multiplier effect. The project aims to build extensive code-bases of mathematical knowledge that are easy to use and extend.

- **Historical Context and Motivation**:

- The project draws inspiration from early computer science and AI work, including Turing's ideas on "Intelligent Machinery" and the development of automated theorem proving. It aims to build on this foundation to enhance modern mathematical research.

#### Case Study: Independence Number of Graphs

- **Independence Number**:

- The independence number (stability number) of a graph is the largest set of vertices with no edges between them. This concept is central to graph theory and relates to major problems like the P vs. NP question and Hadwiger’s Conjecture.

- **Bounds and Conjectures**:

- The project has generated new bounds for the independence number of a graph. These conjectures are tested, leading to either proofs or counterexamples, thus advancing mathematical knowledge.

### Academies and Institutions Involved

#### Virginia Commonwealth University (VCU)

- **Primary Institution**:

- The project is spearheaded by researchers at Virginia Commonwealth University. The team is supported by various grants and funding programs, including the Simons Foundation Mathematics and Physical Sciences–Collaboration Grants for Mathematicians Award and VCU’s Presidential Research Inception Program (PRIP).

#### Key Researchers and Contributors

- **Core Team**:

- R. Barden, N. Bushaw, C. Callison, A. Fernandez, B. Harris, I. Holden, C. E. Larson, D. Muncy, C. O’Shea, J. Shive, J. Raines, P. Rana, N. Van Cleemput, B. Ward, and N. Wilcox-Cook.

### Future Directions and Expansion

- **Expansion of Research**:

- The project plans to incorporate more invariants, objects, and properties into the conjecturing program. The goal is to systematically use theoretical knowledge to generate new conjectures that push the boundaries of current mathematical research.

- **Building Collaborative Infrastructure**:

- The project aims to create a robust infrastructure to facilitate large-scale mathematical research. This includes coding, conjecture generation, testing, and proving, all conducted collaboratively.

For further details and insights, you can access the full document: [The Graph Brain Project & Big Mathematics](http://www.people.vcu.edu/~clarson/AC7-GBP-171019.pdf).

If you have any specific questions or need more information, feel free to ask!

泉婕妤你每天这日记合格的么……

以下是引用狐狐我呀在2024/6/13 16:43:39的发言：
泉婕妤你每天这日记合格的么……

谁告诉你我每天写日记了

Sure, here are some of the main theorems in computability theory and complexity theory:

**Computability Theory:**

1. **Church-Turing Thesis**: This is not a theorem but a hypothesis which states that a function on the natural numbers is computable by a human being following an algorithm, ignoring resource limitations, if and only if it is computable by a Turing machine1.

2. **Halting Problem**: This theorem states that there is no general algorithm that can correctly determine whether arbitrary programs halt1.

3. **Rice's Theorem**: This theorem states that all non-trivial, semantic properties of programs are undecidable1.

4. **Recursion Theorem**: This theorem states that a Turing machine can effectively "make a copy" of its own description, allowing it to pass that description on to other Turing machines2.

**Complexity Theory:**

1. **Time Hierarchy Theorem**: This theorem formally states that given more time, a Turing machine can solve more problems?.

2. **Space Hierarchy Theorem**: Similar to the Time Hierarchy Theorem, this theorem states that given more space, a Turing machine can solve more problems?.

3. **P vs NP Problem**: This is an open problem in complexity theory. It asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer?.

4. **Cook-Levin Theorem**: This theorem states that the Boolean satisfiability problem (SAT) is NP-complete, meaning it is in NP and all problems in NP can be reduced to it in polynomial time?.

5. **Savitch’s Theorem**: This theorem provides an upper bound for the space complexity of non-deterministic Turing machines?.

These theorems form the foundation of our understanding of what can and cannot be computed, and how efficiently these computations can be performed. Please let me know if you need more information on any of these topics!

以下是引用獾獾我呀在2024/6/13 16:48:08的发言：

谁告诉你我每天写日记了

看来不是只有本狐不写日记啊

图片点击可在新窗口打开查看

Absolutely, the study of theoretical computation theory has profound implications for the advancement of computer science. Here are a few ways how:

1. **Understanding Limits**: Theoretical computation helps us understand the limits of what can and cannot be computed. This understanding is crucial because it helps us avoid pursuing solutions to problems that are proven to be non-computable.

2. **Algorithmic Efficiency**: The study of complexity theory, a subset of computation theory, allows us to understand the efficiency of algorithms. This understanding can lead to the development of more efficient algorithms, which can solve problems faster or using less resources.

3. **Problem Classification**: Computation theory helps in classifying problems based on their complexity. This classification (P, NP, NP-complete, etc.) helps in understanding the inherent difficulty of problems and guides us in seeking appropriate solutions.

4. **Cryptography and Security**: Many modern cryptographic systems are based on computational hardness assumptions that come from complexity theory. Understanding these theories is fundamental to building secure cryptographic systems.

5. **Quantum Computing**: Theoretical computer science is also essential in the development of quantum computing. Concepts like quantum bits (qubits) and quantum gates are deeply rooted in theoretical computation.

6. **Software Verification**: Theoretical computation theory also plays a role in software verification, where we want to prove that a program behaves as expected. Concepts from computation theory can help in creating methods to formally verify software.

In summary, theoretical computation theory forms the bedrock of computer science. It provides the fundamental principles that guide the design of algorithms, data structures, programming languages, and even hardware architectures. It's an essential field of study for anyone looking to deeply understand and contribute to the field of computer science.

以下是引用狐狐我呀在2024/6/13 16:50:33的发言：

看来不是只有本狐不写日记啊

图片点击可在新窗口打开查看

正常人谁写日记啊

以下是引用獾獾我呀在2024/6/13 16:50:57的发言：

正常人谁写日记啊

Emmm……本狐明天开始恢复正常

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