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https://www.researchgate.net/publication/358722876_Existence_Computation_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_based_Revelation_of_Collatz_Conjecture

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

• February 2022

• DOI: 

• 10.13140/RG.2.2.28299.36647

• Projects: 

• From Existence Computation to Semantic Computation through Frequency Defined Fully Typed Resources

• 2021 Data, Information, Knowledge and Wisdom Conference (DIKW 2021)

• Yucong Duan

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

基于存在计算(EXCR)与语义计算(ESCR)的关于3x+1问题的语义空间(SCR)解释

By Yucong Duan,

DIKW research group, Hainan University

Email: duanyucong@hotmail.com

Abstract: From a cognitive perspective in the semantic space, we proposed the revelation of the semantics of Collatz Conjecture or the 3x+1 Problem based on our proposed Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) mechanism following our previous revelation of the semantics axioms of Conservation of Existence Set Axiom (CEX  ,  Consistency of Compounded Essential Set Axiom (CES) and Inheritance of Existence Semantics Axiom (IHES ).

Collatz Conjecture的语义解释：

(a)Collatz Conjecture语义可以从类型的实例的语义角度等价于:

从类型语义的实例层面任何一个自然数N的实例INS(N)=n，它或者是一个奇数O的实例INS(O)=o，或者是一个偶数E的实例INS(E)=e。当n是奇数o，则对它乘3再加1获得n:=3o+1，当n是偶数e，则对它除以2获得n:=e/2，如此循环，最终都能够得到n=1。

INS(N)

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

:=ASS({INS(O)*3+1, INS(E)/2})

:=ASS({INS(O)*3+1, INS(E)/2})

:=ASS({o*3+1, e/2}, 1)

:=ASS({n*3+1, n/2}, 1)

=>n->1

(b)Collatz Conjecture语义可以从实例的整体类型的语义角度等价于:

从实例的整体类型语义的层面任何一个自然数N的实例TYPE(INS(N)=n都可以在确认自身的存在语义的基础上，由跨类型的奇数O或偶数E实例层面的语义INS(N):=ASS({INS(O),INS(E)},{REL(+),REL(/)})关联，根据存在计算与推理EXCR的基础假设公理就是存在的守恒公理(CEX, Conservation of Existence Set)等价推导出类型层面的对应语义关联TYPE(N):=ASS({TYPE(O),TYPE(E})。

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

=>TYPE(N):=ASS({TYPE(O),TYPE(E)},{REL(+),REL(/)})

TYPE(N):=ASS({TYPE(O),TYPE(E})蕴含类型层面自然数类型N与奇数类型O与偶数类型E整体之间的存在语义上的等价性。

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

=>EXCR(N):=EXCR(TYPE(O),TYPE(E))

=>EXCR(N):=EXCR(O,E)

奇数类型O或偶数类型E由于可以通过类型层面语义关联N(E):=N(O)+1建立相互之间的联系。

ASS(TYPE(O),TYPE(E))

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

:=ASS(((TYPE(O),TYPE(E)),1),REL(+))

=>N(E):=N(O)+1

类型层面语义关联N(E):=N(O)+1，根据存在计算与推理EXCR的基础假设公理就是存在的守恒公理(CEX, Conservation of Existence Set)等价推导出类型层面奇数类型O与偶数类型E之间的存在语义上的等价性。

N(E):=N(O)+1

=>EXCR(TYPE(O)):=EXCR(TYPE(E))

=>EXCR(O):=EXCR(E)

由EXCR(N):=EXCR(O,E)以及EXCR(O):=EXCR(E)，我们可以可以依托存在计算与推理EXCR的基础假设公理存在的守恒公理CEX确定自然数类型N与奇数类型O与偶数类型E整体之间的存在语义上的等价性。

ASS(EXCR(N):=EXCR(O,E), EXCR(O):=EXCR(E))

=>EXCR(N):=EXCR(O):=EXCR(E)

结合存在计算与推理EXCR的基础假设公理就是存在的守恒公理(CEX, Conservation of Existence Set)与本质计算与推理ESCR的基础假设公理就是本质集合整体完整性的组合一致性公理(CES, Consistency of Compounded Essential Set),我们面向存在语义提出存在语义继承公理(IHES, Inheritance of Existence Semantics)：类型层面的存在语义在纯类型层面的语义处理过程中，对于具有存在语义依赖关系或者语义等价关系的目标A和目标B，例如类型A的存在语义依赖于或等价于类型B的存在语义语义集合EX(B) 或{ex(b)}，目标A继承或保有目标B的所有存在语义EX(A) 或{ex(a)}。

IHES：

ASS({EXCR(A):=EXCR(B), EXCR(A)=>EXCR(B)})

=>EX(B)=>EX(A)

结合使用存在计算与推理EXCR的基础假设公理就是存在的守恒公理(CEX, Conservation of Existence Set)，本质计算与推理ESCR的基础假设公理就是本质集合整体完整性的组合一致性公理(CES, Consistency of Compounded Essential Set)，与存在语义继承公理(IHES, Inheritance of Existence Semantics)， 依据自然数类型N的实例INS(N)之间的自然数操作加Z(+)、乘法Z(*)、除法Z(/)，也即ASS(INS(N),{Z(+), Z(*), Z(/)} )，作用在自然数类型N的连续实例INS(N)的整体{INS(N)}，上不改变其中任何一个类型的本质语义关系ASS(CES(EXCS(N)))的存在语义我们可以推出一系列的本质语义关系。

ASS(CES(EXCS(N)))

:=ASS(EXCR(CES(N)))

:=CES(ASS(EXCS(N)))

:=CES(EXCS(ASS(N)))

:=EXCS(CES(ASS(N)))

由于从存在语义层面自然数类型的实例INS(N)=n之间，例如n与n+1具有有界的连续性或有限性，也就是ASS(n, n+1)在一个有限的自然数操作加Z(+)、乘法Z(*)、除法Z(/)之后获得的边界bound(ASS(n, n+1))会与ASS(n-1, n)以及ASS(n+1, n+2)对应的边界bound(ASS(n-1, n))以及bound(ASS(n+1, n+2))相邻。这个相邻语义的递归实现蕴含着每个bound(ASS(n, n+1))的有限性。这个有限性按CEX公理，在纯类型变换ASS({n*3+1, n/2}, 1)下也将保持不变，ASS({n*3+1, n/2}, 1)=>bound(ASS({n*3+1, n/2}, 1))。由存在语义继承公理(IHES, Inheritance of Existence Semantics)，基于自然数类型N与奇数类型O与偶数类型E整体之间的存在语义上的等价性可得EX(N)=>EX(O)和EX(N)=>EX(E)。将EX(N)=>EX(O)和EX(N)=>EX(E)代入ASS({n*3+1, n/2}, 1)=>bound(ASS({n*3+1, n/2}, 1))

，即可得到Collatz Conjecture的有界语义bound(ASS({O*3+1, E/2}, 1)):

ASS({n*3+1, n/2}, 1)=>bound(ASS({n*3+1, n/2}, 1))

=>

ASS({O*3+1, E/2}, 1)=>bound(ASS({O*3+1, E/2}, 1))

综上，bound(ASS({O*3+1, E/2}, 1))的有限性也就蕴含着相关操作的处理过程的有限性，也即得证Collatz Conjecture。

References:

(1) Yucong Duan: Towards a Periodic Table of conceptualization and formalization on State, Style, Structure, Pattern, Framework, Architecture, Service and so on. SNPD 2019: 133-138

(2) Yucong Duan: Existence Computation: Revelation on Entity vs. Relationship for Relationship Defined Everything of Semantics. SNPD 2019: 139-144

(3) Yucong Duan: Applications of Relationship Defined Everything of Semantics on Existence Computation. SNPD 2019: 184-189

(4) Yucong Duan, Xiaobing Sun, Haoyang Che, Chunjie Cao, Zhao Li, Xiaoxian Yang: Modeling Data, Information and Knowledge for Security Protection of Hybrid IoT and Edge Resources. IEEE Access 7: 99161-99176 (2019)

(5) 段玉聪等, 跨界、跨 DIKW 模态、介尺度内容主客观语义融合建模与处理研究. 中国科技成果，2021年8月498期，45-48.

(6) Y. Duan, "Semantic Oriented Algorithm Design: A Case of Median Selection," 2018 19th IEEE/ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD), 2018, pp. 307-311, doi: 10.1109/SNPD.2018.8441053.

(7) Y. Duan, "A Constructive Semantics Revelation for Applying the Four Color Problem on Modeling," 2010 Second International Conference on Computer Modeling and Simulation, 2010, pp. 146-150, doi: 10.1109/ICCMS.2010.113.

(8) Yucong Duan, A dualism based semantics formalization mechanism for model driven engineering, IJSSCI, vol. 1, no. 4, pp. 90-110, 2009.

(9) Yucong Duan, "Efficiency from Formalization: An Initial Case Study on Archi3D" in Studies of Computing Intelligence, Springer, 2009.

(10) Yucong Duan, "Creation Ontology with Completeness for Identification of 3D Architectural Objects" in ICCTD, IEEE CS press, pp. 447-455, 2009.

(11) Y. Huang and Y. Duan, "Towards Purpose Driven Content Interaction Modeling and Processing based on DIKW," 2021 IEEE World Congress on Services (SERVICES), 2021, pp. 27-32, doi: 10.1109/SERVICES51467.2021.00032.

(12) T. Hu and Y. Duan, "Modeling and Measuring for Emotion Communication based on DIKW," 2021 IEEE World Congress on Services (SERVICES), 2021, pp. 21-26, doi: 10.1109/SERVICES51467.2021.00031.

(13) Duan Yucong, Christophe Cruz. Formalizing Semantic of Natural Language through Conceptualization from Existence. International Journal of Innovation, anagement and Technology, 2011, 2 (1), p. 37-42, ISSN: 2010-0248. ffhal-00625002

(14) Y. Duan, "A stochastic revelation on the deterministic morphological change of 3x+1," 2017 IEEE 15th International Conference on Software Engineering Research, Management and Applications (SERA), 2017, pp. 333-338, doi: 10.1109/SERA.2017.7965748.

(15) Yucong Duan, The end of "Objective" mathematics as a return to "Subjective". February 2022.DOI: 10.13140/RG.2.2.36171.87841.https://www.researchgate.net/publication/358 607773_The_end_of_Objective _mathematics_as_a_return_to_Subjective/stats

(16) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based revelation of the semantics of point, line and plane. February 2022. https://www.researchgate.net/publication/358608122_Existence_Computatio n_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_base d_revelation_of_the_semantics_of_point_line_and_plane

(17) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of the Four Color Theorem. February 2022. https://www.researchgate.net/publication/358608147_Existence_Computatio n_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_base d_Revelation_of_the_Four_Color_Theorem

(18) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of the Goldbach's conjecture. February 2022. https://www.researchgate.net/publication/358637942_Existence_Computation_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_based_Revelation_of_the_Goldbach's_conjecture

(19) Yucong Duan, Identifying Objective True/False from Subjective Yes/No Semantic based on OWA and CWA. July 2013. Journal of Computers 8(7)DOI: 10.4304/jcp.8.7.1847-1852.https://www.researchgate.net/publication/276240420_Identifying_Objective_ TrueFalse_from_Subjective_YesNo_Semantic_based_on_OWA_and_CWA/citations

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