Traditional Invention and Innovation Theory 1946-TRIZ Does Not Adapt to the Digital Era

-Innovative problem-solving methods combining DIKWP model and classic TRIZ

Purpose driven Integration of data, information, knowledge, and wisdom Invention and creation methods: DIKWP-TRIZ

(Chinese people's own original invention and creation methods:DIKWP - TRIZ)

Collatz Conjecture's Semantic Mathematics Exploration

Prof. Yucong Duan

Benefactor: Shiming Gong

DIKWP-AC Artificial Consciousness Laboratory

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

(Email：duanyucong@hotmail.com)

Catalogue

1 Background and Introduction

2 Semantic Explanation of Collatz Conjecture

2.1 Semantic angle of type instance

2.2 Semantic perspective of the overall type

3 Construction and exploration of semantic space

3.1 Semantic association of examples

3.2 Semantic association of types

3.3 Semantic association of overall types

4 The importance of bounded semantics

5 Conclusion

5.1 Contribution of Semantic Space

5.2 The practicability of bounded semantics

5.3 The new direction of mathematical research

6 Outlook

Reference

1 Background and Introduction

Collatz Conjecture, or 3x+1 problem, is a challenging problem in the fields of mathematics and computer science. The expression of this problem is relatively simple: for any positive integer n, if n is an even number, divide it by 2; If n is odd, multiply it by 3 and add 1. After several iterations, eventually n will become 1. The core question of Collatz Conjecture is whether this process can always converge to 1 for any positive integer n.

Although the problem of Collatz Conjecture is simple, so far, a general method has not been found to prove whether it is true or not. This problem has puzzled mathematicians and computer scientists for decades. Although numerical calculation shows that it still holds true for a huge value of n, it has not yet found a universally applicable proof.

The purpose of this technical report is to provide a brand-new explanation and in-depth exploration of Collatz Conjecture by introducing the frameworks of Existence Computing and Reasoning (EXCR) and Essence Computing and Reasoning (ESCR) and building a semantic space (SCR). We will examine this problem from different angles, thus providing new ideas and tools for solving it.

2 Semantic Explanation of Collatz Conjecture

2.1 Semantic angle of type instance

First of all, we will explain this problem from the semantic point of view of an instance of the type of CollatzConjunction. Suppose we have a natural number n, which is an instance of Collatz Conjecture. We can use the instantiation relation INS(N) to represent an instance of n, where n represents a specific numerical value of n. According to Collatz Conjecture, for any instance of natural number n, it is either an odd O instance (INS(O)=o) or an even E instance (INS(E)=e). According to the rules of Collatz Conjecture, if n is an odd O, multiply it by 3 and add 1, that is, n: = 3o+1; If n is an even number e, divide it by 2, that is, n:=e/2. Repeat this operation, and finally you can get n=1.

This can be expressed in the following ways:

scssCopy codeINS(N) := ASS({INS(O), INS(E)}, {REL(+), REL(/)})

Here, INS(O) and INS(E) represent examples of odd o and even e respectively, and o and e are their specific values respectively. By the relationship between REL(+) and REL(/), we can express the operations of multiplying by 3, adding 1 and dividing by 2. Finally, through this series of operations, we get n=1, which is the termination condition of Collatz Conjecture.

2.2 Semantic perspective of the overall type

Next, we will consider the explanation of Collatz Conjecture from the semantic point of view of the overall type of the instance. For any instance of natural number n, we can equivalently deduce the corresponding semantic association at the type level through the semantic association at the instance level of odd O or even E across types, according to the basic assumption axiom of EXCR, that is, the Conservation of Existence Set, CEX).

In this case, odd type O or even type E can be related to each other through semantic association N(E):=N(O)+1 at the type level. This is because the relationship between odd and even numbers is established by N(E):=N(O)+1.

scssCopy codeASS(TYPE(O), TYPE(E)) := ASS((TYPE(O), TYPE(E)), {REL(+), REL(/)})

Type-level semantic association N(E):=N(O)+1 implies the semantic equivalence between odd type O and even type E.

scssCopy codeN(E) := N(O) + 1 => EXCR(TYPE(O)) := EXCR(TYPE(E)) => EXCR(O) := EXCR(E)

From this, we can rely on the conservation axiom CEX of EXCR's basic hypothesis axiom to determine the semantic equivalence of the existence between natural number type N and odd number type O and even number type E.

3 Construction and exploration of semantic space

When discussing the semantic Explanation of Collatz Conjecture, we not only consider the semantics of a single instance, but also pay attention to the construction and exploration of Semantic Space, SCR). Semantic space is a conceptual space, which contains all semantic information related to Collatz Conjecture, thus helping us to understand the essence of the problem more deeply.

In the semantic space, we can regard each natural number n as a node, and the nodes are connected with each other through different semantic associations. These semantic associations include semantic associations of instances, types and overall types. These associations describe the relationship between natural numbers and their transformation rules in Collatz Conjecture.

3.1 Semantic association of examples

In the semantic space, each natural number n has its corresponding instantiation relation INS(N), which describes whether it is odd or even, and performs corresponding operations according to the rules of Collatz Association. The semantic association of these examples forms an important part in the semantic space, which helps us to understand the behavior of a single natural number n. Through continuous iteration, the relationship between examples constitutes a complex semantic network, which reflects the transformation logic between values.

3.2 Semantic association of types

In addition to the semantic association of instances, the semantic association of types also plays a key role in semantic space. By associating the instance type of natural number n with the instance types of odd type O and even type E, we establish a higher level semantic relationship. These relations show the conversion rules between odd and even numbers, and explain why the operation of Collatz Conjecture can finally converge to 1. This high-level relationship embodies the mathematical logic of Collatz Conjecture operation and provides us with a deeper understanding.

3.3 Semantic association of overall types

Finally, the semantic association of the whole type provides a global perspective. By connecting the whole of different numerical types, we get a more comprehensive understanding of collatz conjugate. This global perspective reveals the semantic unity of different numerical types in the Collatz process. This unity emphasizes the internal consistency of the whole problem, which helps us to understand the Collatz Conjecture more clearly.

4 The importance of bounded semantics

There is a key concept in the semantic Explanation of Collatz Conjecture, namely bounded semantics. Bounded semantics means that the processing of a problem will not lead to an infinite increase or decrease in numerical value, but will eventually tend to a limited range of values. This concept is very important in the research of Collatz Conjecture, because it is directly related to the solution of the problem.

The definition of bounded semantics enables us to better understand why Collatz Convergence converges to 1 at a certain point without infinite loop or infinite increase. It helps us to determine the key feature of the problem, which is widely used in mathematical research and calculation theory. The concept of bounded semantics makes the solution of Collatz Conjecture more clear and feasible.

5 Conclusion

By introducing the framework of EXCR and ESCR, this technical report provides a new explanation and in-depth exploration for Collatz Conjecture. This method not only helps to solve specific mathematical problems, but also provides new ideas and tools for mathematical research.

5.1 Contribution of Semantic Space

The construction of semantic space provides a comprehensive perspective for understanding and solving Collatz Conjecture. It presents the problem from different angles, which enables us to explore the essence of Collatz Conjecture more deeply. Through the semantic space, we can more clearly understand the examples of natural number n, the relationship between types and the consistency of the overall types. This comprehensive perspective is helpful for us to understand the Collatz Conjecture more comprehensively.

5.2 The practicability of bounded semantics

The concept of bounded semantics has important practicability in the research of Collatz Conjecture. It not only helps us to understand why the collatz convergence converges to 1 at a certain point, but also provides us with clues to solve this problem. Bounded semantics makes the problem clearer and helps us to promote the final solution of the problem. This concept can also be applied in other mathematical problems and calculation theories, which provides a new way of thinking for mathematical research.

5.3 The new direction of mathematical research

The methods and tools proposed in this technical report provide new ideas for exploring a wider range of mathematical problems. By transforming the problem into the analysis of existential computation and semantic computation, we can explore the essence of mathematical problems more deeply. Collatz Conjecture, as a concerned problem of number theory, has always attracted the interest of mathematicians. We hope that this new Explanation method can provide more inspiration for solving this problem, and also provide a new tool and perspective for mathematical research.

Throughout the discussion, we emphasized the basic assumptions of EXCR and ESCR, such as the conservation axiom of existence (CEX), the combinatorial consistency axiom of global integrity of essential sets (CES) and the semantic inheritance axiom of existence (IHES). These basic assumptions build a solid semantic foundation for us, which enables us to deeply explore the semantic essence of Collatz Conjecture.

6 Outlook

By transforming the problem into the analysis of existential computation and semantic computation, we can not only solve the concrete problem of Collatz Conjecture, but also provide new research directions and methods for other problems in the field of mathematical research and computational theory. The following are some possible prospects and future research directions:

Wider research on mathematical problems: We can apply the framework of existential computing and semantic computing to study other mathematical problems. These problems may include various unsolved problems in the fields of number theory, graph theory and set theory. This new method may help us to better understand and solve complex mathematical problems.

Application of computer science: the framework of existential computing and semantic computing is not only applicable to the field of mathematics, but also to problems in computer science. For example, in the theory of algorithm design and computational complexity, this framework may be helpful to analyze the performance and efficiency of the algorithm and find a better algorithm solution.

Machine learning and artificial intelligence: Introducing the ideas of existential computing and semantic computing into the fields of machine learning and artificial intelligence may help to develop more intelligent algorithms and systems. This can include applications in automatic reasoning, knowledge representation and autonomous decision-making.

Mathematics education: Introducing this new explanation method into mathematics education can help students better understand the essence and logic of mathematics problems. This is helpful to train mathematicians and computational scientists with more creative and critical thinking ability.

Interdisciplinary research: the framework of existential computing and semantic computing has interdisciplinary characteristics, which can promote cooperation and exchanges in different fields. Mathematicians, computer scientists, philosophers and researchers in other fields can discuss the application and potential of this framework in their respective fields.

In a word, the semantic Explanation of Collatz Conjecture and the exploration of semantic space provide us with a brand-new way to think about mathematical problems. By transforming the problem into the context of existential computing and semantic computing, we can understand the essence of the problem more deeply and look for clues to the solution. This method is not only valuable for solving specific problems, but also provides a new direction and tool for mathematical research and exploration in other fields. In the future, we can expect to see more research work on existential computing and semantic computing, and the wide application of this framework in different fields.

Reference

[1] Duan Y. Which characteristic does GPT-4 belong to? An analysis through DIKWP model. DOI: 10.13140/RG.2.2.25042.53447. https://www.researchgate.net/publication/375597900_Which_characteristic_does_GPT-4_belong_to_An_analysis_through_DIKWP_model_GPT-4_shishenmexinggeDIKWP_moxingfenxibaogao. 2023.

[2] Duan Y. DIKWP Processing Report on Five Personality Traits. DOI: 10.13140/RG.2.2.35738.00965. https://www.researchgate.net/publication/375597092_wudaxinggetezhide_DIKWP_chulibaogao_duanyucongYucong_Duan. 2023.

[3] Duan Y. Research on the Application of DIKWP Model in Automatic Classification of Five Personality Traits. DOI: 10.13140/RG.2.2.15605.35047. https://www.researchgate.net/publication/375597087_DIKWP_moxingzaiwudaxinggetezhizidongfenleizhongdeyingyongyanjiu_duanyucongYucong_Duan. 2023.

[4] Duan Y, Gong S. DIKWP-TRIZ method: an innovative problem-solving method that combines the DIKWP model and classic TRIZ. DOI: 10.13140/RG.2.2.12020.53120. https://www.researchgate.net/publication/375380084_DIKWP-TRIZfangfazongheDIKWPmoxinghejingdianTRIZdechuangxinwentijiejuefangfa. 2023.

[5] Duan Y. The Technological Prospects of Natural Language Programming in Large-scale AI Models: Implementation Based on DIKWP. DOI: 10.13140/RG.2.2.19207.57762. https://www.researchgate.net/publication/374585374_The_Technological_Prospects_of_Natural_Language_Programming_in_Large-scale_AI_Models_Implementation_Based_on_DIKWP_duanyucongYucong_Duan. 2023.

[6] Duan Y. The Technological Prospects of Natural Language Programming in Large-scale AI Models: Implementation Based on DIKWP. DOI: 10.13140/RG.2.2.19207.57762. https://www.researchgate.net/publication/374585374_The_Technological_Prospects_of_Natural_Language_Programming_in_Large-scale_AI_Models_Implementation_Based_on_DIKWP_duanyucongYucong_Duan. 2023.

[7] Duan Y. Exploring GPT-4, Bias, and its Association with the DIKWP Model. DOI: 10.13140/RG.2.2.11687.32161. https://www.researchgate.net/publication/374420003_tantaoGPT-4pianjianjiqiyuDIKWPmoxingdeguanlian_Exploring_GPT-4_Bias_and_its_Association_with_the_DIKWP_Model. 2023.

[8] Duan Y. DIKWP language: a semantic bridge connecting humans and AI. DOI: 10.13140/RG.2.2.16464.89602. https://www.researchgate.net/publication/374385889_DIKWP_yuyanlianjierenleiyu_AI_deyuyiqiaoliang. 2023.

[9] Duan Y. The DIKWP artificial consciousness of the DIKWP automaton method displays the corresponding processing process at the level of word and word granularity. DOI: 10.13140/RG.2.2.13773.00483. https://www.researchgate.net/publication/374267176_DIKWP_rengongyishide_DIKWP_zidongjifangshiyiziciliducengjizhanxianduiyingdechuliguocheng. 2023.

[10] Duan Y. Implementation and Application of Artificial wisdom in DIKWP Model: Exploring a Deep Framework from Data to Decision Making. DOI: 10.13140/RG.2.2.33276.51847. https://www.researchgate.net/publication/374266065_rengongzhinengzai_DIKWP_moxingzhongdeshixianyuyingyongtansuocongshujudaojuecedeshendukuangjia_duanyucongYucong_Duan. 2023.

Data can be regarded as a concrete manifestation of the same semantics in our cognition. Often, Data represents the semantic confirmation of the existence of a specific fact or observation, and is recognised as the same object or concept by corresponding to some of the same semantic correspondences contained in the existential nature of the cognitive subject's pre-existing cognitive objects. When dealing with data, we often seek and extract the particular identical semantics that labels that data, and then unify them as an identical concept based on the corresponding identical semantics. For example, when we see a flock of sheep, although each sheep may be slightly different in terms of size, colour, gender, etc., we will classify them into the concept of "sheep" because they share our semantic understanding of the concept of "sheep". The same semantics can be specific, for example, when identifying an arm, we can confirm that a silicone arm is an arm based on the same semantics as a human arm, such as the same number of fingers, the same colour, the same arm shape, etc., or we can determine that the silicone arm is not an arm because it doesn't have the same semantics as a real arm, which is defined by the definition of "can be rotated". It is also possible to determine that the silicone arm is not an arm because it does not have the same semantics as a real arm, such as "rotatable".

Information, on the other hand, corresponds to the expression of different semantics in cognition. Typically, Information refers to the creation of new semantic associations by linking cognitive DIKWP objects with data, information, knowledge, wisdom, or purposes already cognised by the cognising subject through a specific purpose. When processing information, we identify the differences in the DIKWP objects they are cognised with, corresponding to different semantics, and classify the information according to the input data, information, knowledge, wisdom or purpose. For example, in a car park, although all cars can be classified under the notion of 'car', each car's parking location, time of parking, wear and tear, owner, functionality, payment history and experience all represent different semantics in the information. The different semantics of the information are often present in the cognition of the cognitive subject and are often not explicitly expressed. For example, a depressed person may use the term "depressed" to express the decline of his current mood relative to his previous mood, but this "depressed" is not the same as the corresponding information because its contrasting state is not the same as the corresponding information. However, the corresponding information cannot be objectively perceived by the listener because the contrasting state is not known to the listener, and thus becomes the patient's own subjective cognitive information.

Knowledge corresponds to the complete semantics in cognition. Knowledge is the understanding and explanation of the world acquired through observation and learning. In processing knowledge, we abstract at least one concept or schema that corresponds to a complete semantics through observation and learning. For example, we learn that all swans are white through observation, which is a complete knowledge of the concept "all swans are white" that we have gathered through a large amount of information.

Wisdom corresponds to information in the perspective of ethics, social morality, human nature, etc., a kind of extreme values from the culture, human social groups relative to the current era fixed or individual cognitive values. When dealing with Wisdom, we integrate this data, information, knowledge, and wisdom and use them to guide decision-making. For example, when faced with a decision-making problem, we integrate various perspectives such as ethics, morality, and feasibility, not just technology or efficiency.

Purpose can be viewed as a dichotomy (input, output), where both input and output are elements of data, information, knowledge, wisdom, or purpose. Purpose represents our understanding of a phenomenon or problem (input) and the goal we wish to achieve by processing and solving that phenomenon or problem (output). When processing purposes, the AI system processes the inputs according to its predefined goals (outputs), and gradually brings the outputs closer to the predefined goals by learning and adapting.

Yucong Duan, male, currently serves as a member of the Academic Committee of the School  of Computer Science and Technology at Hainan University. He is a professor and doctoral supervisor and is one of the first batch of talents selected into the South China Sea Masters Program of Hainan Province and the leading talents in Hainan Province. He graduated from the Software Research Institute of the Chinese Academy of Sciences in 2006, and has successively worked and visited Tsinghua University, Capital Medical University, POSCO University of Technology in South Korea, National Academy of Sciences of France, Charles University in Prague, Czech Republic, Milan Bicka University in Italy, Missouri State University in the United States, etc. He is currently a member of the Academic Committee of the School of Computer Science and Technology at Hainan University and he is the leader of the DIKWP (Data, Information, Knowledge, Wisdom, Purpose) Innovation Team at Hainan University, Distinguished Researcher at Chongqing Police College, Leader of Hainan Provincial Committee's "Double Hundred Talent" Team, Vice President of Hainan Invention Association, Vice President of Hainan Intellectual Property Association, Vice President of Hainan Low Carbon Economy Development Promotion Association, Vice President of Hainan Agricultural Products Processing Enterprises Association, Visiting Fellow, Central Michigan University, Member of the Doctoral Steering Committee of the University of Modena. Since being introduced to Hainan University as a D-class talent in 2012, He has published over 260 papers, included more than 120 SCI citations, and 11 ESI citations, with a citation count of over 4300. He has designed 241 serialized Chinese national and international invention patents (including 15 PCT invention patents) for multiple industries and fields and has been granted 85 Chinese national and international invention patents as the first inventor. Received the third prize for Wu Wenjun's artificial intelligence technology invention in 2020; In 2021, as the Chairman of the Program Committee, independently initiated the first International Conference on Data, Information, Knowledge and Wisdom - IEEE DIKW 2021; Served as the Chairman of the IEEE DIKW 2022 Conference Steering Committee in 2022; Served as the Chairman of the IEEE DIKW 2023 Conference in 2023. He was named the most beautiful technology worker in Hainan Province in 2022 (and was promoted nationwide); In 2022 and 2023, he was consecutively selected for the "Lifetime Scientific Influence Ranking" of the top 2% of global scientists released by Stanford University in the United States. Participated in the development of 2 international standards for IEEE financial knowledge graph and 4 industry knowledge graph standards. Initiated and co hosted the first International Congress on Artificial Consciousness (AC2023) in 2023.

Prof. Yucong Duan

DIKWP-AC Artificial Consciousness Laboratory

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

DIKWP research group, Hainan University

duanyucong@hotmail.com